Mathematical Physics

Virtual Presentations for the Icm 2 Conference in Prague: Poster 1

Authors: Andrew Beckwith
Comments: 4 Pages. 1st of six posters for the ICM 2 conference in Prague.

The following questions are raised in this document. First, can there be a stable (massive) graviton? If so, does this massive graviton, as modeled by KK DM, with a modification of slight 4 dimensional space mass, contribute to DE, at least in terms of re acceleration ? The answer, if one assumes that the square of a frequency for graviton mass is real valued and greater than zero appears to be affirmative. The author, when considering a joint DM – DE model finds evidence that re acceleration of the universe one billion years ago in a higher dimensional setting can be justified in terms of a slight modification of standard KK DM models, if one considers how an information exchange between present to prior universes occurs, which the author thinks mandates more than four dimensional space time geometry
Category: Mathematical Physics

Poster 2, for Icm2 Conference, Cz

Authors: Andrew Beckwith
Comments: 4 Pages. Poster or virtual paper for the ICM 2 conference

This paper uses the “Fjortoft theorem” for defining necessary conditions for instability. The point is that it does not apply in the vicinity of the big bang. We apply this theorem to what is called by T. Padmanabhan a thermodynamic potential which becomes would be unstable if conditions for the applications of “Fjortoft’s theorem” hold. In our case, there is no instability, so a different mechanism has to be appealed to. In the case of vacuum nucleation, we argue that conditions exist for the nucleation of particles as of the electroweak regime. Due to injecting material from a node point, in spacetime. This regime of early universe creation, coexits with the failure of applications of “Fjortoft” theorem in such a way as to give necessary and sufficient conditions for matter creation, in a way similar to the Higgs Boson
Category: Mathematical Physics

Poster 3 for the Icm 2 Conference, Cz

Authors: Andrew Beckwith
Comments: 4 Pages.

When initial radius R approaches zero if we use Stoica actually derived Einstein equations in a formalism which remove the big bang singularity pathology, then the reason for Planck length no longer holds. The implications of the initial radius approaching zero are the first part of this manuscript. Then the resolution is alluded to by work from Muller and Lousto, as to entanglement entropy implications of entanglement entropy. We present entanglement entropy in the early universe with a steadily shrinking scale factor, due to work from Muller and Lousto , and show that there are consequences due to initial entanged entropy for a time dependent horizon radius in cosmology, with for flat space conditions r(H)=conformal time. In the case of a curved, but not flat space version of entropy, we look at vacuum energy as proportional to the inverse of scale factor squared times the inverse of initial entropy, effectively when there is no initial time except with in line with the conformal time being almost zero. . The consequences for this initial entropy being entangled are elaborated in this manuscript. No matter how small the initial radial length gets, then for initial cosmological entropy if it is entanglement entropy, initial cosmological entropy will not go to zero.
Category: Mathematical Physics

Poster 4 for the Icm 2 Conference, Cz

Authors: Andrew Beckwith
Comments: 4 Pages.

First, we show through a numerical simulation that the massive Schwinger model used to formulate solutions to CDW transport is insufficient for transport of solitons (anti-solitons) through a pinning gap model of CDW transport. We show that a model Hamiltonian with Peierls condensation energy used to couple adjacent chains (or transverse wave vectors) permits formation of solitons (anti- solitons) which could be used to transport CDW through a potential barrier. We argue that there are analogies between this construction and the false vacuum hypothesis used for showing a necessary and sufficient condition for formation of CDW soliton – anti - soliton (S-S’) pairs in wave functionals presented in a prior publication
Category: Mathematical Physics

Poster 5, for the Icm 2 Conference, Cz

Authors: Andrew Beckwith
Comments: 4 Pages.

The author asks if octonion quantum gravity is relevant near the Planck scale. Furthermore, the question is raised if gravitational waves would be generated during the initial phase, , of the universe when an increase in degrees of freedom have in setting , so that the result can be observed by a gravitational detector.. The well appreciated .quantum gravity problem that the notion of a quantum state, representing the structure of spacetime at some instant, and the notion of the evolution of the state, does not get traction, since there are no real “instants”, is avoided by having the initial octonion geometry embedded in a larger, non linear “pilot model” (semi classical) embedding structure. The Penrose suggestion of re cycled space time avoiding a ‘big crunch’ is picked as the embedding structure, so as to avoid the ‘instants’ of time issue. In addition the favored idea is to avoid the well known string theory trap known as the dimensionality problem of an equation of motion (consistency condition) which is the reason why string theory dimensionality is either (10 or 26) depending upon if super symmetry is imposed. Getting octonion gravity as embedded in a larger, Pilot theory embedding structure may restore Quantum Gravity to its rightful place in early cosmology without the lunacy of then afterwards ‘Schrodinger equation ‘ states of the universe, forevermore afterwards. Setting , in a GW detector due to appropriate measurement procedures may allow the opportunity to find experimental clues as to this embedding structure in which octonion gravity may emerge in the Planckian regime.of evolutionary cosmology
Category: Mathematical Physics

Poster 6 for the Icm 2 Conference, CZ

Authors: Andrew Beckwith
Comments: 4 Pages.

The following questions is asked, If one takes the covariant derivative of a Stress-energy representation of early universe massive gravitons, is the derivative of the Graviton stress tensor equal to zero ? If so, then in what range of astrophysics does this occur, and when does this formalism break down? Lavenda and Davies argued that the derivative of a generalized GR stress energy tensor being zero in itself is insufficient to show that the 1st law of thermodynamics alone holds. If the full Stress-Energy tensor expression for GR is written out, there is a stress energy tensor component involving GW alone which we highlight. The problem as to this test is if the derivative of the Stress energy tensor, for gravitons as written by Visser is brought up. This Visser stress energy tensor for massive gravitons will not even satisfy the 1st thermodynamic law. We bring this up as a counter point to an article written by Lavenda and Davies purporting to claim that the Tolman test for a first law of thermodynamics which they generalize to first and second law of thermodynamics for inflationary cosmology. We show a breakdown of a zero value for the derivative of the Stress energy tensor for early universe massive gravitons and this derivative of the massive Graviton Stress energy tensor (Visser) will not even satisfy the first law of thermodynamics according to the Tolman criteria. Note that if the Visser Massive Graviton Stress energy tensor scenario does not hold then the Lavenda and Davies objection to inflation is upheld.
Category: Mathematical Physics

Orbital Averages and the Secular Variation of the Orbits

Authors: Maurizio M. D'Eliseo
Comments: 13 Pages.

Orbital averages are employed to compute the secular variation of the elliptical planetary elements in the orbital plane in presence of perturbing forces of various kinds. They are also useful as an aid in the computation of certain complex integrals. An extensive list of computed integrals is given.
Category: Mathematical Physics

Amending Maxwell’s Equations for Real and Complex Gauge Groups in Non-Abelian Form

Authors: Richard L. Amoroso, Elizabeth A. Rauscher
Comments: 6 Pages.

We have analyzed, calculated and extended the modification of Maxwell’s equations in a complex Minkowski metric, M4 in a C2 space using the SU2 gauge, SL(2,c) and other gauge groups, such as SUn for n >2 expanding the U1 gauge theories of Weyl. This work yields additional predictions beyond the electroweak unification scheme. Some of these are: 1) modified gauge invariant conditions, 2) short range non-Abelian force terms and Abelian long range force terms in Maxwell’s equations, 3) finite but small rest of the photon, and 4) a magnetic monopole like term and 5) longitudinal as well as transverse magnetic and electromagnetic field components in a complex Minkowski metric M4 in a C4 space.
Category: Mathematical Physics

The Structure of Information

Authors: S.E. Grimm
Comments: 9 Pages.

The contents of this essay describe the underlying reality of physic information. This reality turns out to be mathematical. All the information about the existence of phenomena and their mutually influences can be described and elucidated by mathematical concepts. The benefit of this mathematical structure is an impressive reduction of substantive concepts in physics (mainly special/general relativity and quantum theory). Moreover, the hypothesis belongs to the category “bottom up” approaches. Therefore, some well-known foundational questions in physics, mathematics, and cosmology will be reviewed.
Category: Mathematical Physics

A Complex and Triplex Framework for Encoding the Riemannian Dual Space-Time Topology Equipped with Order Parameter Fields

Authors: Nathan O. Schmidt
Comments: 16 pages, 2 figures, published in the Hadronic Journal

In this work, we forge a powerful, easy-to-visualize, flexible, consistent, and disciplined abstract vector framework for particle and astro physics that is compliant with the holographic principle. We demonstrate that the structural properties of the complex number and the sphere enable us to introduce and define the triplex number---an influential information structure that is similar to the 3D hyper-complex number by D. White and P. Nylander---which identifies a 3D analogue of (2D) complex space. Consequently, we engage the complex and triplex numbers as abstract vectors to systematically encode the state space of the Riemannian dual 3D and 4D space-time topologies, where space and time are dual and interconnected; we use the triplex numbers (with triplex multiplication) to extend 1D and 2D algebraic systems to 3D and 4D configurations. In doing so, we equip space-time with order parameter fields for topological deformations. Finally, to exemplify our motivation, we provide three example applications for this framework.
Category: Mathematical Physics

Time for New Cosmology

Authors: Dan Visser
Comments: 2 Pages.

The Higgs-particle could be a dark matter particle! It is not enough to confirm spin 0 and + parity for the Higgs-like particle to let it be the Higgs-particle. The Double Torus hypothesis should be involved. A dark matter particle in this new framework could also have spin 0 and + parity under condition it contributes to gravity, but its properties can also contribute as negative mass to anti-gravity.
Category: Mathematical Physics

The Schrödinger Equation in Complex Minkowski Space, Nonlocality and Anticipatory Systems

Authors: Richard L. Amoroso, Elizabeth A. Rauscher
Comments: 19 Pages.

We develop a formalism for the Schrödinger equation in an eight dimensional complex Minkowski space and discuss its relation to the Dirac equation, properties of nonlocality, remote connectedness, Young’s double slit experiment, Bell’s Theorem, the EPR paradox and anticipatory parameters of spacetime; and also identify an imaginary temporal component as a small nonlinear term and find soliton or solitary wave solutions. These coherent solutions can carry information over long distances, are consistent with Lorentz invariance and appear to provide a fundamental methodology for describing the issue of quantum measurement and a new context for the basis of quantum theory. In the Copenhagen view models of reality are not desirable. However our new approach may enable the redefinition of concepts of reality from a new nonlocal anticipatory quantum theory. Certainly the most desirable consequence of scientific discovery is the ability to redefine our concepts of reality.
Category: Mathematical Physics

Relativistic Physics in Complex Minkowski Space, Nonlocality, Aether Model and Quantum Physics

Authors: Richard L. Amoroso, Elizabeth A. Rauscher
Comments: 25 Pages.

Many naturally occurring phenomena require theoretical treatment utilizing complex analysis by methods such as the Cauchy-Riemann relations using hyper-geometrical spaces which treat inherently nonlinear, non-dispersive, collective nonlocal resonant states of a quantum system, so as to be consistent with the nonlinearity inherent in General Relativity. Typical quantum approaches form linear approximations limiting the ability to formulate a quantum consistent Relativity Theory.
Category: Mathematical Physics

The Genesis of Circle

Authors: Andrej Rehak
Comments: 4 Pages.

Geometric demonstration of free fall and rotation reveals space-time nature of the genesis of circle. We show that the operations with radius are equivalent to operations with velocity. Consequently, conventional spatial unit for radius (m), becomes equivalent to dynamic space-time unit for speed (m/s). We prove that the velocity is a physical equivalence of geometric idea of radius.
Category: Mathematical Physics

The Period of a Pendulum

Authors: Andrej Rehak
Comments: 8 Pages.

Geometric demonstration located in v, t diagram explains the nature of the period of a pendulum, i.e. the universal connection of time with velocity (radius) and acceleration. Conceptual nature of the principles proof points to its universal validity. In other words, if it is valid for a circle, it is valid. Through the geometry of free fall we describe physical nature of irrational numbers π and √2. We demonstrate physical matrix of scale √2nπ. Through the relationship between the variables of space, time and velocity, using the principle of the pendulum, we illustrate the foundation of the law of conservation of energy. Analyzing its motion we point to nature of distortion of Euclidean geometry.
Category: Mathematical Physics

Critical Analysis of the Mathematical Formalism of Theoretical Physics. I. Foundations of Differential and Integral Calculus

Authors: Temur Z. Kalanov
Comments: 8 Pages.

Critical analysis of the generally accepted (standard) foundations of differential and integral calculus is proposed. Methodological basis of the analysis is the unity of formal logic and of rational dialectics. It is shown that the generally accepted foundations are based on the logically and practically erroneous concepts “infinitesimal quantity (uninterruptedly diminishing quantity)”, “derivative”, “derivative as function of variable quantity” and, consequently, represent incorrect basis of mathematics and of theoretical physics.
Category: Mathematical Physics

Critical Analysis of the Mathematical Formalism of Theoretical Physics. II. Pythagorean Theorem

Authors: Temur Z. Kalanov
Comments: 10 Pages.

The critical analysis of the Pythagorean theorem and of the problem of irrational numbers is proposed. Methodological basis for the analysis is the unity of formal logic and of rational dialectics. It is shown that: 1) the Pythagorean theorem represents a conventional (conditional) theoretical proposition because, in some cases, the theorem contradicts the formal-logical laws and leads to the appearance of irrational numbers; 2) the standard theoretical proposition on the existence of incommensurable segments is a mathematical fiction, a consequence of violation of the two formal-logical laws: the law of identity of geometrical forms and the law of lack of contradiction of geometrical forms; 3) the concept of irrational numbers is the result of violation of the dialectical unity of the qualitative aspect (i.e. form) and quantitative aspect (i.e. content: length, area) of geometric objects. Irrational numbers represent a calculation process and, therefore, do not exist on the number scale. There are only rational numbers.
Category: Mathematical Physics

On the Natural-Science Foundations of Geometry

Authors: Temur Z. Kalanov
Comments: 11 Pages.

The work is devoted to solution of an actual problem – the problem of relation between geometry and natural sciences. Methodological basis of the method of attack is the unity of formal logic and of rational dialectics. It is shown within the framework of this basis that geometry represents field of natural sciences. Definitions of the basic concepts "point", "line", "straight line", "surface", "plane surface", and “triangle” of the elementary (Euclidean) geometry are formulated. The natural-scientific proof of the parallel axiom (Euclid’s fifth postulate), classification of triangles on the basis of a qualitative (essential) sign, and also material interpretation of Euclid’s, Lobachevski’s, and Riemann’s geometries are proposed.
Category: Mathematical Physics

"Hidden" Parameters Describing Internal Motion Within Extended Particle Elements

Authors: R.L. Amoroso, L.H. Kauffman, E.A. Rauscher, P. Rowlands J-P Vigier
Comments: 27 Pages.

Recent attempts to consider isolated particles and real constitutive wave elements as localized, extended spacetime structures (i.e., moving within time-like hypertubes or branes are developed within a causal extension of the Feynman-Gell-Mann electron model. These extended structures contain real internal motions, (i.e., internal hidden parameters) locally correlated with the "hidden parameters" describing the local collective motions of the corresponding pilot-waves. Recent experimental evidence is briefly discussed.
Category: Mathematical Physics

Empirical Protocol for Measuring Virtual Tachyon / Tardon Interactions in a Dirac Vacuum

Authors: Richard L. Amoroso, Elizabeth A. Rauscher
Comments: 18 Pages.

Here we present discussion for the utility of resonant interference in Calabi-Yau mirror symmetry as a putative empirical test of the existence of virtual tachyon / tardon interactions in a covariant Dirac polarized vacuum
Category: Mathematical Physics

P vs. NP and Riemann Hypothesis Solution (Revised)

Authors: Andrew Nassif
Comments: 2 Pages.

This problem has been solved by me over a year ago, and published. Today I am posting a more complex and complicated version, yet put in Layman terms. I hope by doing this it will garner the attention of Clay Mathematics as well as show logical input on its mathematical endeavors.
Category: Mathematical Physics

Some Thoughts on Special Relativity

Authors: Jeremy Dunning-Davies.
Comments: 6 Pages.

Here an introduction to Wesley’s neomechanics is presented. It is shown to produce some of the same results as Special Relativity but without both the mathematical and philosophical basis of that subject. As with other work in which results associated with General Relativity are obtained without recourse to the fundamental bases of that subject, so here too the pre-eminent place afforded Special Relativity in modern science is called into question. The opportunity is taken to extend Wesley’s ideas to the case where the mass of the body when at rest is not constant.
Category: Mathematical Physics

The Generalizations of the First Noether Theorem.

Authors: Vyacheslav Telnin
Comments: 6 Pages.

This paper deals with the generalizations of the First Noether theorem. It takes into account not only the first derivatives of the fields by the coordinates in Lagrangian, but also the second. And this theorem is generalized on the curved spaces. And also it's generalized on asymmetric metric tensors.
Category: Mathematical Physics

Calculate Universe 2

Authors: Branko Zivlak
Comments: 4 Pages. 1 Table

This article is about relations between fundamental physical constants.
Category: Mathematical Physics

Thoughts on Order, Etc. Occasioned by Writings of J. P. Wesley.

Authors: J. Dunning-Davies, D. Sands
Comments: 6 Pages.

Although rarely, if ever, acknowledged, there is real confusion existing over several of the fundamental ideas of classical macroscopic thermodynamics. Many of these surround the concept of entropy and one of the big questions never publicly asked is whether or not the functions called entropy in various branches of mathematics and physics are all the same? Here some thoughts are presented which have been provoked by reading some of the writings of J. P. Wesley in the hope that they, in turn, will provoke further examination of these notions, many of which are so basic to so many areas of physics and are causing so much trouble for students at the present time.
Category: Mathematical Physics

New Cosmological Hypothesis Match Observations by New Dark Energy-Time Applied to Dark Matter for the Existence of a Double Torus Universe.

Authors: Dan Visser
Comments: 13 Pages.

The Double Torus hypothesis stands for a new architecture for the Universe, which means another dark energy as dynamics and no Big Bang. New dark energy-time is applied to dark matter. It is ‘extra time’ originated from a time-scale smaller than the Planck-time. The implication is a Double Torus of the dark energy-time enclosing and intertwining an inner dark matter torus. It might sound controversial, but a new ‘dark energy-force formula’ in the hypothesis enables to make calculations, which match the observed dark matter-accelerations in galaxies and the Pioneer satellites 1 and 2. This new formula relates to Newton quantum-gravity force and dark matter-force both implemented in one product. The dark matter force could be gravitational (+) and anti-gravitational (-). The formula also sets the laboratory acceleration-limit for Newton-gravity, 5 x 10^-14 m/s^2, to a theoretically lower value of 2.8659 x10^-14 m/s^2. There is no other formula, or theory, to do that. The hypothesis-dynamic is developed by DAN Visser, Almere, the Netherlands, an independent cosmologist. Additional evidence comes from astronomical observations: Such as the asymmetry in the Cosmic Microwave Background (CMB) and a ‘Dark Flow’, which pleads for a cyclic-curved torus-shape of the universe. This also could explain the observed ‘cold spot’ in the CMB by to imagine dark matter is disappearing in the far end of the curved torus-shape. Hence, also a ‘hot spot’ in the CMB should exist. This should correspond to the approaching dark flow coming from the other curved torus far-end.
Category: Mathematical Physics

Calculate Universe 1

Authors: Branko Zivlak
Comments: 8 Pages. 11 relations, 1 figure

This paper is about relations between the mass of universe and mass of some elementary particles. Important is only the diagram in Figure 1, which you can understand in any language if you know the language of mathematics and physics.
Category: Mathematical Physics

Quantum Perturbation Theory in Stock Trading

Authors: Yingqiong Gu
Comments: 3 Pages.

It is a hot topic about how to trade a stock/group of stocks in a non-news day. Author tries to design safe, profitable automated stock trading agents using evolutionary algorithms[2]. In quantum mechanics, perturbation theory[4] is a set of approximation schemes directly related to mathematical perturbation for describing a complicated quantum system in terms of a simpler one. The idea is to start with a simple system for which a mathematical solution is known, and add an additional "perturbing" Hamiltonian representing a weak disturbance to the system. If the disturbance is not too large, the various physical quantities associated with the perturbed system (e.g. its energy levels and Eigen states) can, from considerations of continuity, be expressed as 'corrections' to those of the simple system. These corrections, being 'small' compared to the size of the quantities themselves, can be calculated using approximate methods such as asymptotic series. The complicated system can therefore be studied based on knowledge of the simpler one. In Stock Market, for a non-news trading day, stock prices will mostly depend on the initial price at given time, and bid-ask spread.
Category: Mathematical Physics

Quantization of Points and Energy on Dipole Vectors and on Spin

Authors: Markos Georgallides
Comments: 22 Pages.

Point , which is nothing and has not any Position , may be anywhere in Space , therefore the Primary point A , being nothing also in no Space , is the only Point and nowhere i.e. Primary Point is the only Space and from this all the others , so then this primary point A to exist at a second point B somewhere else , point A must move at point B , where then A ≡ B . Presupposition for Unit AB = ds means an Impulse P removes point A to B and since in each Restrained System (S) the Work done ( W ) by Impulse P on a Virtual displacement ( ds > 0) is zero , and then W = ∫ A-B [ P.ds ] = 0 → [ ds .( PA+P B ) = 0 ] and Point A is in Space [S] and point B is in Anti-Space [aS] which are self created and are Property and Essence from the same Unit AB, therefore these are also a Restrained System(S). Since the only two Elements of Universe are ds and P and are connected by Virtual Work Principle and since Unit AB = ds is a quantized dimension,so Impulse is also quantized.Quantization of Points becomes through Vector Unit dš = ÃB which is the first dimensional Unit [ Zenon Paradox 13]and this because Vector has Position and Direction .Quantization of Energy is done in bound States ( loops ) because bound states [11-16] withhold diffusion ( flow ) .The two fundamental dimensions ( quanta of Points (ds) and quanta of Energy (dP) , are connected on Primary dipole and on any dipole AiBi . [PNS] is self created and consists a space (x,y,z) Scalar field Sp which constructs a vector field Sa = Ñ.Sp . The position of dipole in the equilibrium Space Anti-Space keeps charge = momentum ( Energy→ the difference of primary Work which is bounded on edge points ) and Angular momentum ( Spin → angular momentum is the intrinsic twist of Space , Anti-Space ) which inextricably unify geometry of Space and motion . On any Dipole AB of [PNS] with dš = ( [J,E,B] and content © ) ,where motion occurs only as (+) → (-) Source (+) → Drain (-) ,Conservation State of Work exists on magnitude |dš| = √ J² + E² + B² of vector ř ( J,E, B ) ( which is the Sub-Space of AB ) and on the two perpendicular Fields Ê = Ñx J , B = Ñx Ê ( which are the two diffused equilibrium Spaces and Anti -Spaces of AB , Ñx Ê = 0 , Ñ x B = 0 ) . Since,Source (+) →Drain (-) , in [PNS] is the equilibrium of Space [S] and Anti-Space [aS] , therefore the only magnitude for motion is J , while E and B are produced , as this happens in the Magnetic fields where lines ( loops ) close in themselves and this because no sources exist ↑→ ® for magnetic fields B and so the two models [ the B-field Pole and Amperian loop ] are explained. Solving equation ds.( PA+ PB ) = 0 then ds.PA = - ds.PB and since , dš , is constant for Primary dipole , so the Work conserved on every point in Spaces is constant and equal to [ dš .Impulse P = constant ] . i.e. The applied Work [ Wi → 1 → ∞ ] = ( charge ± P ) on every point of the three , Spaces , Anti–Spaces and Sub–Spaces , which coexist in [PNS] is conserved on Points , meaning that every point of [PNS] consist the fundamental ± Scalar Field with variables the Constraint Forces and Angular momentum [ Spin = Wij →1→ ∞] , and is conserved always in [PS] , [ PaS] or, The two elements [Vector dš = AB (matter) , ± dP ,Work p , Spin = momentum ] of the Infinite Dipole Vectors [AB] = [ ĀB – PA , PB ] build Particles with Identity Card , the Constant Product → dš . p = constant …(k) ← so , the Possible Energy Range (m) Scales , of the two fundamental and quantized dimensions [ ds , - P , + P ] are the three Layers , (- i ) ↔ (0 ) = - P → [ ds , - P , ] → Black Holes Scale = k 1 → ds 1. dP 1 = k 1 (-i ) ↔ (+i) = 7 P → [ ds , - P , + P ] → Planck Scale Matter = k 2 → ds 2. dP 2 = k 2 (+i ) ↔ (0 ) = + P → [ ds , + P , ] → Dark Matter Scale = k 3 → ds 3. dP 3 = k 3 For any single particle of wavelength λ = ds and dP = Charge → momentum ( p ) exists : dš.dP = λ . p = constant = h → is the reduced Planck constant in Planck Scale Matter . For ds near zero and P = ∞ , is needed a new Type of Light to see what is happening below Planck length Level , Layer 10¯35 m .Primary Vector monad and Vector monads are , either Static ( equilibrium ) or Dynamic ( are moving ) , and keep their Conservation of State ( The Total work is equal to zero ) in the three equilibrium and perpendicular monads [ J , E , B ] which are in Subspaces , Spaces , Anti-Spaces ( and thus , not turn to non-existence ) and are interchanged by Pythagoras conservation law of Volume. The geometric concepts of [PNS] as two dimensional field is that , the third dimension represents the amplitude of the potential S . Considering x-y Plane as the base and the vertical (z) is the potential S , then contours of constant potential altitudes of [PNS] are Cylindrical or Solenoidal and for Position 0 ( which is the equilibrium Horizon of Space and anti-Space) is Sphere , therefore [PNS] with the included Space [S] and Anti-Space [aS] may be Sphere and Solenoidal simultaneously.
Category: Mathematical Physics

Nonlinear Prigozhin Theorem

Authors: Sergey Kamenshchikov
Comments: 6 Pages. Author is looking for postdoc position. kamphys@gmail.com

The basic purpose of this work was to develop correct disorder measure for phase transition of stochastic system. It was shown that control parameter of system evolution can be defined as relation between power, injected into system and power, dissipated in nonlinear mechanisms. It was suggested to define benchmark of system disorder, zero state entropy, through implicit condition of zero dynamic entropy. In terms of suggested control parameter Prigozhin nonlinear theorem was deduced on base of Klimontovich self – organization theorem. Phase space area, described by universal Prigozhin theorem, was defined as typical chaotic model - strange attractor.
Category: Mathematical Physics

New Gravity-Physics and Mathematics Calculate Dark Matter-Accelerations and Prove a Recalculated Double Torus Universe.

Authors: Dan Visser
Comments: 8 Pages.

New gravity-physics and -mathematics produce calculations of dark matter-accelerations and prove we live in a Double Torus Universe, which recalculates Quantum Gravity in vacuum. The set of equations to do that, prove the (new) dark energy force formula in the Double Torus Hypothesis is correct. The formula has been developed step by step and described in my Vixra-papers. It is an alternative for still unknown dark energy used in Big Bang cosmology. It is a tool for the awareness that the universe is not a universe that started with a Big Bang. The universe is a Double Torus, which uses + and - dark matter force to change quantum-gravity. The formula shows clearly a new view on the universe to be necessary. This paper shows theoretical predictions through calculations that match the experimental values of dark matter-accelerations in galaxies, the Pioneer-satellites (1 and 2) and produces the lowest limit for Newton-gravitational force. This Newton limit-acceleration matches the laboratory-experiments and moreover sets the limit to a slightly theoretical lower limit-acceleration of 2.8659 x 10^-14 [m/s2]. There is no other theory that predicts this limit. The conditions of the formula show, that quantum-gravity is related to a dark matter force. Dark energy-time from below the Planck-time is applied to the dark matter.
Category: Mathematical Physics

Gravitational Forces Are not Conservative

Authors: Florentino Muñiz Ania
Comments: 2 Pages. Spanish.

This article shows how the gravitational force is conservative only in ideal models of small size. It is shown as strictly mathematical generally is not conservative. From it you can get energy by asymmetric systems. Spanish: En este artículo se muestra como la fuerza gravitatoria sólo es conservativa en modelos ideales de pequeño tamaño. Se demuestra con rigor matemático como, en general, no es conservativa. De ella se puede obtener energía mediante sistemas asimétricos.
Category: Mathematical Physics

On the Söllinger, Weizsäcker Relations and Bošković’s Curve of Force (English Version)

Authors: Branko Zivlak
Comments: 7 Pages. 15 relations, 1 table, 1 figure

Abstract. In "The Code of Nature" [1] Helmut Söllinger gives his and Carl Friedrich von Weizsacker's (1912-2007) relationships between the fundamental physical constants. The first section examines Sollinger's the relationship between the masses of protons and electrons, and the fundamental physical constants. The second is about the Weizsacker's assumption of proportionality between the Planck length, Compton wavelength and radius of the Universe. In the third section, I try to explain, the previous relationships, in light of the attractive-repulsive forces of the Ruđer Bošković (1711-1787), the earliest founder of quantum theory.
Category: Mathematical Physics

Exact Solution of the Reduced Version of Pwe (Paraxial Wave Equation) in Bipolar Coordinate System

Authors: Sergey V. Ershkov
Comments: 5 Pages. Keywords: Helmholtz equation, PWE, paraxial approximation, bipolar coordinates

A new type of exact solution of the reduced 3 dimensional spatial PWE (paraxial wave equation) for the case of bipolar coordinates is presented here. First, we consider a self-similar representation of the solution in a bipolar coordinate system, the second we additionally reduce PWE under a proper paraxial assumption. Analyzing the structure of the final equation, we obtain the simple exact solution which is proved to satisfy to such an equation in bipolar coordinates. Besides, there is a limitation of the components of self-similar solution of a new type.
Category: Mathematical Physics

On the Söllinger, Weizsäcker Relations and Bošković’s Curve of Force

Authors: Branko Zivlak
Comments: pages 7+7, language Serbian+Italian, 15 formulas, 1Table, 1 picture

Abstract. In "The Code of Nature" [1] Helmut Söllinger gives his and Carl Friedrich von Weizsäcker's (1912-2007) the relationship between the fundamental physical constants. The first section examines the relationship between the mass of protons and electrons, and the fundamental physical constants. The second is about the Weizsacker's assumption of proportionality between the Planck length, Compton wavelength and radius of the Universe. In the third section, I try to explain, the previous relationships, in light of the attractive-repulsive forces of the Ruđer Bošković (1711-1787), the earliest founder of quantum theory. This version is written, in the lenguages, Bošković’s father and mother.
Category: Mathematical Physics

On the Orbits of the Magnetized Kepler Problems in Dimension $2k+1$

Authors: Zhanqiang Bai, Guowu Meng, Erxiao Wang
Comments: 13 Pages.

It is demonstrated that, for the recently introduced classical magnetized Kepler problems in dimension $2k+1$, the non-colliding orbits in the ``external configuration space" $\mathbb R^{2k+1}\setminus\{\mathbf 0\}$ are all conics, moreover, a conic orbit is an ellipse, a parabola, and a branch of a hyperbola according as the total energy is negative, zero, and positive. It is also demonstrated that the Lie group ${\mr {SO}}^+(1,2k+1)\times {\bb R}_+$ acts transitively on both the set of oriented elliptic orbits and the set of oriented parabolic orbits.
Category: Mathematical Physics

Robert Kiehn's Ideas About Falaco Solitons and Generation of Turbulent Wake from TGD Perspective

Authors: M. Pitkanen
Comments: 7 Pages.

I have been reading two highly interesting articles by Robert Kiehn. There are very many contacts on TGD inspired vision and its open interpretational problems. The notion of Falaco soliton has surprisingly close resemblance with K\"ahler magnetic flux tubes defining fundamental structures in TGD Universe. Fermionic strings are also fundamental structures of TGD accompanying magnetic flux tubes and this supports the vision that these string like objects could allow reduction of various condensed matter phenomena such as sound waves -usually regarded as emergent phenomena allowing only highly phenomenological description - to the fundamental microscopic level in TGD framework. This can be seen as the basic outcome of this article. Kiehn proposed a new description for the generation of various instability patterns of hydrodynamics flows (Kelvin-Helmholtz and Rayleigh-Taylor instabilities) in terms of hyperbolic dynamics so that a connection with wave phenomena like interference and diffraction would emerge. The role of characteristic surfaces as surfaces of tangential and also normal discontinuities is central for the approach. In TGD framework the characteristic surfaces have as analogs light-like wormhole throats at which the signature of the induced 4-metric changes and these surfaces indeed define boundaries of two phases and of material objects in general. This inspires a more detailed comparison of Kiehn's approach with TGD.
Category: Mathematical Physics

What P-Adic Icosahedron Could Mean? and What About P-Adic Manifold?

Authors: M. Pitkanen
Comments: 40 Pages.

The original focus of this article was p-adic icosahedron. The discussion of attempt to define this notion however leads to the challenge of defining the concept of p-adic sphere, and more generally, that of p-adic manifold, and this problem soon became the main target of attention since it is one of the key challenges of also TGD. There exists two basic philosophies concerning the construction of both real and p-adic manifolds: algebraic and topological approach. Also in TGD these approaches have been competing: algebraic approach relates real and p-adic space-time points by identifying the common rationals. Finite pinary cutoff is however required to achieve continuity and has interpretation in terms of finite measurement resolution. Canonical identification maps p-adics to reals and vice versa in a continuous manner but is not consistent with p-adic analyticity nor field equations unless one poses a pinary cutoff. It seems that pinary cutoff reflecting the notion of finite measurement resolution is necessary in both approaches. This represents a new notion from the point of view of mathematics. a) One can try to generalize the theory of real manifolds to p-adic context. The basic problem is that p-adic balls are either disjoint or nested so that the usual construction by gluing partially overlapping spheres fails. This leads to the notion of Berkovich disk obtained as a completion of p-adic disk having path connected topology (non-ultrametric) and containing p-adic disk as a dense subset. This plus the complexity of the construction is heavy price to be paid for path-connectedness. A related notion is Bruhat-Tits tree defining kind of skeleton making p-adic manifold path connected. The notion makes sense for the p-adic counterparts of projective spaces, which suggests that p-adic projective spaces (S² and CP₂ in TGD framework) are physically very special. b) Second approach is algebraic and restricts the consideration to algebraic varieties for which also topological invariants have algebraic counterparts. This approach looks very natural in TGD framework - at least for imbedding space. Preferred extremals of Kähler action can be characterized purely algebraically - even in a manner independent of the action principle - so that they might make sense also p-adically. Number theoretical universality is central element of TGD. Physical considerations force to generalize the number concept by gluing reals and various p-adic number fields along rationals and possible common algebraic numbers. This idea makes sense also at the level of space-time and of "world of classical worlds" (WCW). Algebraic continuation between different number fields is the key notion. Algebraic continuation between real and p-adic sectors takes place along their intersection which at the level of WCW correspond to surfaces allowing interpretation both as real and p-adic surfaces for some value(s) of prime p. The algebraic continuation from the intersection of real and p-adic WCWs is not possible for all p-adic number fields. For instance, real integrals as functions of parameters need not make sense for all p-adic number fields. This apparent mathematical weakness can be however turned to physical strength: real space-time surfaces assignable to elementary particles can correspond only some particular p-adic primes. This would explain why elementary particles are characterized by preferred p-adic primes. The p-adic prime determining the mass scale of the elementary particle could be fixed number theoretically rather than by some dynamical principle formulated in real context (number theoretic anatomy of rational number does not depend smoothly on its real magnitude!). Although Berkovich construction of p-adic disk does not look promising in TGD framework, it suggests that the difficulty posed by the total disconnectedness of p-adic topology is real. TGD in turn suggests that the difficulty could be overcome without the completion to a non-ultrametric topology. Two approaches emerge, which ought to be equivalent. a) The TGD inspired solution to the construction of path connected effective p-adic topology is based on the notion of canonical identification mapping reals to p-adics and vice versa in a continuous manner. The trivial but striking observation was that canonical identification satisfies triangle inequality and thus defines an Archimedean norm allowing to induce real topology to p-adic context. Canonical identification with finite measurement resolution defines chart maps from p-adics to reals and vice versa and preferred extremal property allows to complete the discrete image to hopefully space-time surface unique within finite measurement resolution so that topological and algebraic approach are combined. Finite resolution would become part of the manifold theory. p-Adic manifold theory would also have interpretation in terms of cognitive representations as maps between realities and p-adicities. b) One can ask whether the physical content of path connectedness could be also formulated as a quantum physical rather than primarily topological notion, and could boil down to the non-triviality of correlation functions for second quantized induced spinor fields essential for the formulation of WCW spinor structure. Fermion fields and their n-point functions could become part of a number theoretically universal definition of manifold in accordance with the TGD inspired vision that WCW geometry - and perhaps even space-time geometry - allow a formulation in terms of fermions. This option is a mere conjecture whereas the first one is on rigorous basis.
Category: Mathematical Physics

Exact Solution of Helmholtz Equation for the Case of Non-Paraxial Gaussian Beams

Authors: Sergey V. Ershkov
Comments: 7 Pages. Keywords: Helmholtz equation, paraxial approximation, Gaussian beam

A new type of exact solution of the full 3 dimensional spatial Helmholtz equation for the case of non-paraxial Gaussian beams is presented here. We consider appropriate representation of the solution for Gaussian beams in a spherical coordinate system, then implement it in the full 3 dimensional Helmholtz Eq. Analyzing the structure of the final equation, we obtain one of the simple exact solutions which is proved to satisfy to such an equation for Gaussian beams. Also the proper examples of implementing of the paraxial approximation for Gaussian beam could easily be obtained for a new type of exact solution of Helmholtz equation.
Category: Mathematical Physics

New Periodic Table of Elements 1-92

Authors: Chun-Xuan Jiang
Comments: 13 Pages.

We make the new periodic table of elements 1-92
Category: Mathematical Physics

Double Torus Hypothesis For The Universe In Perspective.

Authors: Dan Visser
Comments: 6 Pages.

The Double Torus, a new hypothesis for the universe, has been put in perspective and related to other theories and hypotheses. This ‘paper’ could be used by the press. The Double Torus hypothesis is theoretically and mathematically-physics-based. Examples of evidence might be available already.
Category: Mathematical Physics

Fine Structure Constant And Relations Between Dimensionless Constants

Authors: Branko Zivlak
Comments: 9 Pages. 25 relations

Abstract. The aim of this article is to determine dimensionless physical constants through mathematical constants and other dimensionless physical constants.
Category: Mathematical Physics

The Code of Nature

Authors: Helmut Söllinger
Comments: Pages. e-mail adress of the author: 64.soellinger@aon.at

The scope of the work described in this paper is a systematic investigation as to whether or not the mass of the proton and the electron can be represented by other fundamental constants. The author arrives at the conclusion that the mass of the proton and the electron can be expressed by a combination of five constants that occur in nature; namely, e, εo, h, c, G, plus a time-variable parameter. In this context, the author has studied more than 37,000 options using electronic support and powering the fundamental constants with natural numbers only. The simplest and most convincing formula the author has found is: me3 x mp3 = (e2 h/4p εo c G R)2 This equation results in the exact value of the mass of the proton and the electron. The beauty and simplicity of this equation give rise to the following question: What, if not this formula, is able to represent the mass of the two most important particles? The author’s conclusion is that either the electron and proton masses themselves are natural constants that cannot be represented by other constants of nature, or that – as shown in this paper – they can be perfectly well represented by five other fundamental constants, in addition to a time-variable parameter.
Category: Mathematical Physics

Pauli Matrices and Dirac Matrices in Geometric Algebra of Quarks

Authors: Martin Erik Horn
Comments: 8 Pages.

It is a historical accident that we describe Pauli matrices as (2 x 2) matrices and Dirac matrices as (4 x 4) matrices. As it will be shown in this paper we can use (3 x 3) matrices or (9 x 9) matrices for this purpose as well. This hopefully will enable us one day to construct a unified geometric algebra picture which includes Gell-Mann matrices in an appropriate manner.
Category: Mathematical Physics

Riemann Zeros Quantum Chaos Functional Determinants and Trace Formulae

Authors: Jose Javier garcia Moreta
Comments: 10 Pages.

ABSTRACT: We study the relation between the Guzwiller Trace for a dynamical system and the Riemann-Weil trace formula for the Riemann zeros, using the Bohr-Sommerfeld quantization condition and the fractional calculus we obtain a method to define implicitly a potential , we apply this method to define a Hamiltonian whose energies are the square of the Riemann zeros (imaginary part) , also we show that for big ‘x’ the potential is very close to an exponential function. In this paper and for simplicity we use units so • Keywords: = Riemann Hypothesis, WKB semiclassical approximation, Gutzwiller trace formula, Bohr-Sommerfeld quantization,exponential potential.
Category: Mathematical Physics

Universal Principles of Perfect Chaos

Authors: S. Kamenshchikov
Comments: 12 Pages. contact e-mail: kamphys@gmail.com. Author is looking for postdoc position.

The purpose of this work was to introduce strict, comprehensive definition of perfect chaos, to find out its basic properties in terms of phase transitions and give connections for uncertainties, lying in base of perfect chaos concept. Concept of perfect chaos as undetermined description was introduced basing on two formalized necessary and sufficient conditions: finite phase space resolution and instability of phase space trajectories. Properties of Kolmogorov system, including phase mixing, turned out to be consequences of chaotic state but not its comprehensive and sufficient conditions. Description relativity was defined as mandatory property of perfect chaos – the same areas of phase space may show regular and chaotic properties depending on description space - time accuracy. Herewith evolution of physical system in given generalized phase space can be represented by consequence of regular states and intermediate transitions. For chaotic state with uniform diffusion it was found out that nonlinear dispersion law is mandatory property. One in its turn necessarily leads to space – time instability of probability density and appearance of probability cavities in phase space - phase space attractors where particles density grows up. Case of chaotic state with fixed boundary and diffusion was considered. It turned out that Fourier decomposition allows to derive relations between coordinate – momentum and time - energy definition uncertainties. It was shown that chaos diffusion factor is the only parameter, limiting product of corresponding uncertainties.
Category: Mathematical Physics

The Mathematics Behind a New Dark Energy Force Related to Gravity and Anti-Gravity by Negative Mass Through a Dark Matter Force in Another Cosmology Named the ‘Double Torus Hypothesis’.

Authors: Dan Visser
Comments: 6 Pages.

I present the mathematics of a new ‘Dark Energy Force’ in replay with my former ‘papers’. The reason is particles that feel small gravity, and anti-particles that maybe feel anti-gravity, and the particle-cosmology I use, with negative mass, never have been exposed experimentally to General Relativity, in order to prove that a real anti-gravity exists. However, the mathematics in my frame work theoretically prove, that only dark matter-mass could have negative mass. This is in contradiction with a new theory of Entropy-Gravity, which theoretically proved gravity is not fundamental, but caused by entropy. My framework is also in contradiction with the Elementary Process theory, which also predicts gravity is not fundamental, but will cause anti-gravity by anti-matter with positive mass. Both these frameworks consider their theory in a Big Bang cosmology. So I replayed my mathematics to highlight again, that a new dark matter-force, embedded in a new cosmology named Double Torus Universe, is the only one that could cause real anti-gravity. My framework is independently developed from institutions and based on two extra-time arrows from below the Planck-scale. Additionally, and for the first time, I used a Feynman-diagram to express this dark matter-force in order to illustrate the existence of real anti-gravity theoretically.
Category: Mathematical Physics

Basic Blueprint for Making This Universe

Authors: Rodney Bartlett
Comments: 5 Pages.

Over 30 years of thinking, plus the insights and mistakes in my viXra articles, reveal the basic blueprint for making this universe. This article continues from where previous articles finished (throughout, I’ve provided links to prior contributions). I begin with explanation of quantum particles, forces and spin in terms of positioning of Mobius loops and the flow of the loops’ binary digits accounting for the interference between gravitation and electromagnetism – together with a link supporting the idea of an electronics-based universe and addressing the topics of hidden variables, quantum fluctuation and virtual particles. The next link speaks of the inverse-square law and infinity. I give Dr. Carl Sagan credit where credit is due - and conclude that, being years ahead of his time, he saw a fundamental truth about the universe’s nature which he decided to include in his book “Contact”. Then time travel into the past (via matrices and the figure-8 Klein bottle), before putting it all together and indulging in some speculation about how to make this universe we’re living in. I think it’s too simple to say “We don’t need to make the universe … it’s already here”. That statement relies on time being strictly linear (like a straight line, rectilinear). We know it isn’t, but is curvilinear and warped. It’s better to say the universe is here now because our future civilization did the following in the past –
Category: Mathematical Physics

Nonperturbational "Continued-Fraction" Spin-offs of Quantum Theory's Standard Perturbation Methods

Authors: Steven Kenneth Kauffmann
Comments: 8 Pages.

The inherently homogeneous stationary-state and time-dependent Schroedinger equations are often recast into inhomogeneous form in order to resolve their solution nonuniqueness. The inhomogeneous term can impose an initial condition or, for scattering, the preferred permitted asymptotic behavior. For bound states it provides sufficient focus to exclude all but one of the homogeneous version's solutions. Because of their unique solutions, such inhomogeneous versions of Schroedinger equations have long been the indispensable basis for a solution scheme of successive perturbational corrections which are anchored by their inhomogeneous term. Here it is noted that every such perturbational solution scheme for an inhomogeneous linear vector equation spins off a nonperturbational continued-fraction scheme. Unlike its representation-independent antecedent, the spin-off scheme only works in representations where all components of the equation's inhomogeneous term are nonzero. But that requirement seems to confer theoretical physics robustness heretofore unknown: for quantum fields the order of the perturbation places a bound on unperturbed particle number, the spin-off scheme contrariwise has only basis elements of unbounded unperturbed particle number. It furthermore is difficult to visualize such a continued-fraction spin-off scheme generating infinities, since its successive iterations always go into denominators.
Category: Mathematical Physics

Discrete Structure of Spacetime

Authors: Nicola D'Alfonso
Comments: 4 Pages.

In this paper, I introduce a particular discrete spacetime that should be seriously considered as part of physics because it allows to explain the characteristics of the motion properly, contrary to what happens with the continuous spacetime of the common conception.
Category: Mathematical Physics

The Force of Gravity Belongs to Another Cosmology.

Authors: Dan Visser
Comments: 4 Pages.

This article summarizes theoretical based evidence related to practice for the prediction the universe is not originated from a Big Bang. Instead cosmology could be based on a Double Torus Universe, as is published in my papers in the Vixra-archive. In a few website-articles I also express my vision on the revision of physics and cosmology within this framework. This paper in particular highlights how Gravity could violate General Relativity by a (new) dark energy force in the new Cosmology. This framework contains the connection of the Newton-Gravity force for tiny matter-particles to a dark matter force, producing “+” and “–“ mass-generation, both at scales of about 10^-22 meter. This can cause repulsive gravity in nature. This can open-up a new energy-source for travelling through space by non-relativistic scaling.
Category: Mathematical Physics

Mathematical Theory of Magnetic Field

Authors: Zafar Turakulov
Comments: 9 Pages. no comments

The study of magnetic fields produced by steady currents is a full-valued physical theory which like any other physical theory employs a certain mathematics. This theory has two limiting cases in which source of the field is confined on a surface or a curve. It turns out that mathematical methods to be used in these cases are completely different and differ from from that of the main of the main part of this theory, so, magnetostatics actually consists of three distinct theories. In this work, these three theories are discussed with special attention to the case current carried by a curve. In this case the source serves as a model of thin wire carrying direct current, therefore this theory can be termed magnetostatics of thin wires. The only mathematical method used in this theory till now, is the method of Green's functions. Critical analysis of this method completed in this work, shows that application of this method to the equation for vector potential of a given current density has no foundation and application of this method yields erroneous results
Category: Mathematical Physics

Symmetry-Nondependent Self-Gravitational Upper Bound on Static Local Energy from Use of a Nonperturbative Iteration Method for Lippmann-Schwinger Equations

Authors: Steven Kenneth Kauffmann
Comments: 5 Pages.

It has recently been shown that self-gravitation reduces static spherically-symmetric cumulative energy distributions below the value of their radii times the "Planck force", which is the inverse of G times the fourth power of c. In this article quantitative treatment of self-gravitation is extended to any static energy density that is nonnegative, smooth and globally integrable. The resulting dimensionless local gravitational energy-reduction factor (namely the inverse of the local gravitational time-dilation factor) is shown to satisfy the zero-momentum nonrelativistic Lippmann-Schwinger quantum scattering equation for a repulsive potential which is proportional (with a known coefficient) to that static energy density. Standard perturbative Born-type iteration of Lippmann-Schwinger equations can diverge for sufficiently strong potentials, which in the gravitational case correspond to sufficiently large static energy densities. We have been able, however, to devise an alternate, completely nonperturbative iteration method for Lippmann-Schwinger equations in coordinate representation. Every one of this nonperturbative method's successive approximations to the local gravitational energy-reduction factor turns out to be positive and less than or equal to unity. In consequence, the self-gravitationally corrected static energy contained in any sphere is bounded by that sphere's diameter times the "Planck force".
Category: Mathematical Physics

Mathematical Derivation of the Fine Structure Constant from Fundamental Properties of Natural Numbers

Authors: Otto G. Piringer
Comments: 12 Pages

Recent publications discussed a possible change with time of Sommerfeld's fine structure constant alpha, in which several of the fundamental constants of Nature are combined. The problem of a changing nature of alpha raises the question whether its value is ultimately a result of chance or reveals an objective law of nature. If the value of alpha is independent of human reason, a derivation of it may be possible from basic numbers, like e and pi, which appear in the logical development of mathematics[1]. In the following investigation a pure mathematical derivation of the fine structure constant is described, starting from a fundamental property of natural numbers. The constant alpha results as a limit value in an algorithm with exponential structures.
Category: Mathematical Physics

Riemann Zeros and an Exponential Potential

Authors: Jose Javier Garcia Moreta
Comments: 5 Pages.

ABSTRACT: We study a given exponential potential aebx on the Real half-line which is possible related to the imaginary part of the Riemann zeros. We study also how we could use the WKB method to recover the potential from the Eigenvalue Staircase for the Riemann
Category: Mathematical Physics

The Poisson Realization of $\mathfrak{so}(2, 2k+2)$ on Magnetic Leave

Authors: Guowu Meng
Comments: 13 Pages.

Let ${\mathbb R}^{2k+1}_*={\mathbb R}^{2k+1}\setminus\{\vec 0\}$ ($k\ge 1$) and $\pi$: ${\mathbb R}^{2k+1}_*\to \mathrm{S}^{2k}$ be the map sending $\vec r\in {\mathbb R}^{2k+1}_*$ to ${\vec r\over |\vec r|}\in \mathrm{S}^{2k}$. Denote by $P\to {\mathbb R}^{2k+1}_*$ the pullback by $\pi$ of the canonical principal $\mathrm{SO}(2k)$-bundle $\mathrm{SO}(2k+1)\to \mathrm{S}^{2k} $. Let $E_\sharp\to {\mathbb R}^{2k+1}_*$ be the associated co-adjoint bundle and $E^\sharp\to T^*{\mathbb R}^{2k+1}_*$ be the pullback bundle under projection map $T^*{\mathbb R}^{2k+1}_*\to {\mathbb R}^{2k+1}_*$. The canonical connection on $\mathrm{SO}(2k+1)\to \mathrm{S}^{2k} $ turns $E^\sharp$ into a Poisson manifold. The main result here is that the real Lie algebra $\mathfrak{so}(2, 2k+2)$ can be realized as a Lie subalgebra of the Poisson algebra $(C^\infty(\mathcal O^\sharp), \{, \})$, where $\mathcal O^\sharp$ is a symplectic leave of $E^\sharp$ of special kind. Consequently, in view of the earlier result of the author, an extension of the classical MICZ Kepler problems to dimension $2k+1$ is obtained. The hamiltonian, the angular momentum, the Lenz vector and the equation of motion for this extension are all explicitly worked out.
Category: Mathematical Physics

Zanaboni Theorem and Saint-Venant's Principle

Authors: Jian-zhong Zhao
Comments: 10 Pages.

Violating the law of energy conservation, Zanaboni Theorem is invalid and Zanaboni's proof is wrong. Zanaboni's mistake of " proof " is analyzed. Energy Theorem for Zanaboni Problem is suggested and proved. Equations and conditions are established in this paper for Zanaboni Problem, which are consistent with , equivalent or identical to each other. Zanaboni Theorem is, for its invalidity , not a mathematical formulation or proof of Saint-Venant's Principle. AMS Subject Classifications: 74-02, 74G50
Category: Mathematical Physics

Inconsistency of Magnetic Monopole

Authors: Ali R. Hadjesfandiari
Comments: 8 pages, 1 figure

It is simply proved that the Hamiltonian for an electric point charge interacting with a fixed magnetic monopole does not exist. This shows that the concept of magnetic monopole is inconsistent within the theory of electrodynamics.
Category: Mathematical Physics

The Symmetry Groups of Light

Authors: John A. Gowan
Comments: 4 Pages.

In the mathematical terms of Evariste Galois' "Group Theory", the "Tetrahedron Model" is a description of the symmetry group of light, including its destruction by asymmetric weak force decays (producing our matter-only Cosmos), and its on-going restoration in obedience to Noether's Theorem of symmetry conservation (as in the conversion of bound to free energy in stars). The usual symmetry group identified with light is that of local phase transformations, and it is designated as either SO(2) or U(1). However, I am suggesting here that light contains a very much larger (and more interesting) symmetry group associated with its transformation into particle-antiparticle pairs (and back again into light). I don't know what the formal designation of this group might be. For an expert's explanation of the formal aspects of symmetry and group theory, See: Keith Devlin The Language of Mathematics Chapt. 5 "The Mathematics of Beauty", 1998 W. H. Freeman & Co. (Holt Paperbacks); see also: Ian Stewart Why Beauty is Truth Chapt. 13 "The Five Dimensional Man", Basic Books 2007.
Category: Mathematical Physics

The Riemann Tesseract: a Geometrical and Topological Construction of Dual Interconnected 3-Cubes Equipped with Vertex Order Parameter Fields in 4D Space-Time

Authors: Nathan O. Schmidt
Comments: 8 pages and 2 figures

Our objective is to design and formulate an abstract, easy-to-visualize, and flexible geometric information structure in a 4D space-time with direct application to mathematics and physics. For this, we introduce and define the "Riemann tesseract": a 4-cube of dual interconnected 3-cubes which are inferred from a topological Riemannian circle that is isometrically embedded in a 1D Riemann surface. The Riemannian circle is a 2-sphere, stereographic superlense, Gedanken interferometer, and double-negative index meta-material that may be scaled to any size, and serves as a common 2D surface boundary between dual interconnected 3-branes: it is a time-like region that is simultaneously dual to both "micro" and "macro" space-like regions. From scratch, we geometrically and topologically construct the information structure in a generalized coordinate system that is compatible with the Schwarzschild metric. Complex order parameter fields are assigned to the tesseract's vertices to systematically encode its wavefunction state space using a relatively simplistic methodology. The number of distinct order parameter fields at each vertex is variable and is application-dependent; it is trivial to vary the complexity of the system by adding or subtracting representational degrees of freedom because the "core" tesseract structure remains unchanged. We mathematically prove that this general and configurable model is consistent with the CPT-theorem, and is fully-capable of representing the simultaneous breaking of multiple symmetries.
Category: Mathematical Physics

The Cuantifiplane (1)

Authors: Jose Miguel Hernandez Perez
Comments: 4 Pages. sorry for the grammatical errors

this text can explain where it all started
Category: Mathematical Physics

An Algebraic Approach to Systems with Dynamical Constraints

Authors: Jerzy Hanckowiak
Comments: 15 Pages.

Constraints imposed directly on accelerations of the system leading to the relation of constants of motion with appropriate local projectors occurring in the derived equations are considered. In this way a generalization of the Noether's theorem is obtained and constraints are also considered in the phase space.
Category: Mathematical Physics

Higgs Boson and Geometry

Authors: Markos Georgallides
Comments: 12 Pages.

This article explains how Geometry and geometrical constructions , [ Spaces , Anti-Spaces , Sub-Spaces ] are able to connect Properties , such as Mass , Energy , and the conservation laws of Physics with only one substance , that of Vector ds , which is ds = AB > 0 → < ∞ with Impulses P A + P B = 0 or ≠ 0 at edge points A , B . Since neutrinos may travel faster than light and also gravity instantly affects , a new explanation is needed . The present article < The Higgs Boson ( Particle ) and the Euclidean Geometry > describes the motion of one of the infinite Sub-Spaces in [ PNS ] , which are moving in Primary Neutral Space . It has been shown that , 1.. Primary point A , having not Position and consequently not any Space existing , therefore is the only Space , and to exist at a second point B somewhere else , point A must move at point B , where then A ≡ B . Point B is the Primary Anti-Space which Equilibrium point A [ PNS ] = [ A ≡ B ] . Since Primary point A is the only Space , then on it exists Principle of Virtual Displacements W = ∫ P.ds = 0 , or [ ds . ( PA + P B ) = 0 ] , i.e. for any ds > 0 Impulse P = ( PA + P B ) = 0 . All points may exist with P = 0 → ( PNS ) and also with P ≠ 0 , [ P A + P B = 0 for points in Spaces and Anti - Spaces ] , therefore [ PNS ] is self created , and because at each point may exist also P ≠ 0 , then [ PNS ] is a Field with infinite points which have a ± Charge with P = 0 → P → ∞ . Primary Neutral - Space [ PNS ] exists with the infinite Points N with three Spatial dimensions ( Xo , Yo , Zo ) and the infinite Impulses P , ( P xi , Pyi , Pzi ) , and i = 0 → ∞ . 2.. Since points A , B of [ PNS ] coincide with the infinite Points , of the infinite Spaces , Anti-Spaces and Sub-Spaces of [ PNS] , and since Motion may occur at all Bounded Sub-Spaces , then this Relative motion is happening between all points belonging to [ PNS ] and to those points belonging to the other Sub-Spaces ( A ≡ B ) . The Infinite points in [ PNS ] form infinite Units AiBi = ds , which equilibrium by Primary Anti – Space by an Inner Impulse ( P ) at edges A , B where PiA + Pi B ≠ 0 , and ds = 0 → N = ∞ . 3 .. Monads = Quantum = ds = AB / ( n = ∞ → 0 ) = [ a ± b.i ] = 0 → ∞ create Spaces ( S ) , Anti -Spaces ( A-S ) , and Sub-Spaces ( S-S ) of AB , which Sub-Spaces are Bounded Spaces , Anti-Spaces and Sub Spaces in it , and are not purely spatial because are Complex numbers which exist for all Spaces , since binomial dsⁿ is Complex number also . 4.. Monads ds ( dipole AB ) and according to their position in S , A-S , S-S , make the four types of matter , and the Combination of the four types of matter , creates all gauge magnitudes which is Mass and Energy , and so all types of Particles and Fields in universe .The 15 possible types of matter correspond to the , Visible , Invisible , Real and Imaginary Universes ( Visible and Invisible Dark matter and Energy ) . Dipole AB is composed of the two Elements → the [ Dipole AB = matter ≡ is the communicator] , and the Impulse [ P ] with the Bounded Impulses ( PA , P B ) . 5.. The difference of Impulses dP = P B – PA > = < 0 of Dipole AB (mass) , creates ± charge . For very small ds ( near zero) which happens at the boundaries of Spaces , ± charge may be point charge ( Spaces form open Strings while Sub-Spaces form closed Strings . Cosmic rays is a Quantum field from a , near zero ds Sub-Space ) , and in very large scales ds , is of ± charge and of the same Field , ( This because Particles , wave like , do not enter a Space smaller than their wavelength ) . 6.. A ray , say M - Ray , interact on the infinite Monads of [PNS] with infinite velocity and zero frequency . This Motion is Continuous and occurs on Dimensional Bounded Units , ds , and not on Points ( this happens only for very small ds near zero and at boundaries of the Spaces ) which are dimensionless . The difference of Impulse dP = P B-PA on points A , B and on points Pi A , Pi B is ± charge . All particles act as wave ( wave-particle duality ) because of the Total energy conservation law of Pythagoras in 2,3 Dim . 7.. The four Interaction of particles or Fields occur directly or Indirectly .
Category: Mathematical Physics

Can Differentiable Description of Physical Reality be Considered Complete? :toward a Complete Theory of Relativity

Authors: Xiong Wang
Comments: 15 Pages.

How to relate the physical \emph{real} reality with the logical \emph{true} abstract mathematics concepts is nothing but pure postulate. The most basic postulates of physics are by using what kind of mathematics to describe the most fundamental concepts of physics. Main point of relativity theories is to remove incorrect and simplify the assumptions about the nature of space-time. There are plentiful bonus of doing so, for example gravity emerges as natural consequence of curvature of spacetime. We argue that the Einstein version of general relativity is not complete, since it can't explain quantum phenomenon. If we want to reconcile quantum, we should give up one implicit assumption we tend to forget: the differentiability. What would be the benefits of these changes? It has many surprising consequences. We show that the weird uncertainty principle and non-commutativity become straightforward in the circumstances of non-differentiable functions. It's just the result of the divergence of usual definition of \emph{velocity}. All weirdness of quantum mechanics are due to we are trying to making sense of nonsense. Finally, we proposed a complete relativity theory in which the spacetime are non-differentiable manifold, and physical law takes the same mathematical form in all coordinate systems, under arbitrary differentiable or non-differentiable coordinate transformations. Quantum phenomenon emerges as natural consequence of non-differentiability of spacetime.
Category: Mathematical Physics

Extended Foundations of Stochastic Prediction

Authors: Sergey Kamenshchikov
Comments: 17 Pages.

The basic purpose of this work was to suggest universal quantitative description of ergodic system intermediate bifurcation and obligatory conditions of this transition. Conditions for existence of phase state and first order phase transition were introduced in terms of energy balance for system volume unit. Extended Fokker – Plank equation with time dependent diffusion factor was formulated. It turned out that for ergodic system with fixed boundary quantized energy spectrum of phase stable states exists. Obtained results may be applied for prediction of ergodic system behavior. If isolation condition is satisfied, phase spectrum quantization allows selecting proper control parameters for system stabilization. Information about current system coarsened energy allows predicting of future stochastic system behavior on the basis of extended Fokker – Plank model.
Category: Mathematical Physics

The Mass of the Electron-Neutrino Expressed by Known Physical Constants

Authors: Laszlo I. Orban
Comments: 5 Pages.

Many trials attempted to understand the neutrinos ever since Pauli theoretically concluded its existence from the conservation of energy calculations. The present paper demonstrates that commencing from two appropriately chosen measurement systems, the mass of the electron-neutrino can be calculated from the mass of the electron and the fine-structure constant. The mass of the neutrino can be determined by the theoretically derived expression (m_k=\alpha^3 m_e) (m_k is the mass of the neutrino, m_e is the mass of electron, alpha is the fine-structure constant).
Category: Mathematical Physics

What is Mass? Chapter One: Mass in Newtonian Mechanics and Lagrangian Mechanics

Authors: Xiong Wang
Comments: 13 Pages. author name: Xiong WANG Email:wangxiong8686@gmail.com

``To see a World in a Grain of Sand, And a Heaven in a Wild'' We will try to see the development and the whole picture of theoretical physic through the evolution of the very fundamental concept of mass. 1The inertial mass in Newtonian mechanics 2 The Newtonian gravitational mass 3 Mass in Lagrangian formulism 4 Mass in the special theory of relativity 5 $E = MC^2$ 6 Mass in quantum mechanics 7 Principle of equivalence and general relativity 8 The energy momentum tensor in general relativity 9 Mass in the standard model of particle physics 10 The higgs mechanism
Category: Mathematical Physics

Mathematical Follow-up for Dark Energy and Dark Matter in the Double Torus Universe.

Authors: Dan Visser
Comments: 7 Pages.

The main issue in this paper is my mathematics to be presented about the maximum of dark energy depending on the information-differences on the wall of any volume in the Double Torus. Secondly the expressions must be worked out further by invitation to them how are triggered by my ideas the universe has a Double Torus geometry. Thirdly I go deeper into details with dark matter, not only stating dark matter is a spatial particle that spins and gets its energy from its acceleration into a dark matter torus, but also pretending dark matter gets its mass from the vacuum energy. I lay out the conditions for understanding why the Big Bang dynamics is therefore a part of the Double Torus and how the dark flow in the universe emerge from the Double Torus dark energy equation. Fourthly I refer to the pretention neutrinos should be sensitive for the flow of dark matter particles expressed in the set of equations in a former paper. But extensively this paper amplifies this theoretical neutrino-evidence, despite all the confusion around the truth of neutrinos-faster-than-light. Fifthly I observe some dark energy and dark matter issues from some of my former papers.
Category: Mathematical Physics

Fractional Circuit Elements: Memristors, Memcapacitors, Meminductors and Beyond

Authors: Xiong Wang
Comments: 2 Pages.

Memristor was postulated by Chua in 1971 by analyzing mathematical relations between pairs of fundamental circuit variables and realized by HP laboratory in 2008. This relation can be generalized to include any class of two-terminal devices whose properties depend on the state and history of the system. These are called memristive systems, including current-voltage for the memristor, charge-voltage for the memcapacitor, and current-flux for the meminductor. This paper further enlarge the family of elementary circuit elements, in order to model many irregular and exotic nondifferentiable phenomena which are common and dominant to the nonlinear dynamics of many biological, molecular and nanodevices.
Category: Mathematical Physics

The Conceptual Basis for Multidimensional Physics

Authors: Alexander Egoyan
Comments: 10 Pages.

In this work a conceptual basis for multidimensional physics (MD physics) is proposed. The new physics is based on the elastic model of multidimensional geometry [1]. Reality may be considered as the process of time evolution of holistic macro objects - elastic membranes. An embedded membrane in this multidimensional world will look different for the external and internal observers: from the outside it will look like a material object with smooth infinitesimal geometry, while from the inside our Universe-like space-time fabric. When interacting with elementary particles and other membranes, a membrane will transform their energy into its elastic energy (a new form of energy) - the energy of stretching of the infinitesimal segments. For example, living organisms play the role of internal observers of the Universe, and at the same time they serve as external observers for 2D membranes embedded into our Universe. A new explanation of gravity and cosmological aspects are also discussed.
Category: Mathematical Physics

Uncertainty Principle of Mathematics

Authors: Mourici Shachter
Comments: 5 Pages.

This short paper prove that mathematically, Reality is not real . This short paper is not about Heisenberg's uncertainty principle of quantum physics. There is another uncertainty principle that depends solely on mathematical arguments and explains why our world can't be easily equated. Or more accurately can be describe in infinitely different ways all of those representations are mathematically correct. Which mean that the representation of any physical phenomena is not unique. . Given an example how to use it to solve complicate problems in engineering
Category: Mathematical Physics

The Unified Theory of Electrical Machines

Authors: Shachter Mourici
Comments: 23 Pages.

In this paper an entirely new approach is used to solve the old problem known as "The Unified Theory of Electrical Machines" Instead of solving an electric circuit with time dependent resistors and coils. I spent a lot of time in finding the appropriate coordinate system in which the problem becomes very simple. Instead of mathematical reasoning with innumerous number of mathematical equation I used pictorial reasoning This Unified Theory of Electrical Machines is the shortest and therefore can be used to teach student the principles of electrical machines in a one semester course The systematic procedure introduced in this paper is not limited to machine theory and can be applied in many physic and engineering problems.
Category: Mathematical Physics

A Functional Determinant Expression for the Riemann XI-Function

Authors: Jose Javier Garcia Moreta
Comments: 9 Pages.

• ABSTRACT: We give and interpretation of the Riemann Xi-function as the quotient of two functional determinants of an Hermitian Hamiltonian . To get the potential of this Hamiltonian we use the WKB method to approximate and evaluate the spectral Theta function over the Riemann zeros on the critical strip . Using the WKB method we manage to get the potential inside the Hamiltonian , also we evaluate the functional determinant by means of Zeta regularization, we discuss the similarity of our method to the method applied to get the Zeros of the Selberg Zeta function • Keywords: = Riemann Hypothesis, Functional determinant, WKB semiclassical Approximation , Trace formula ,Bolte’s law, Quantum chaos.
Category: Mathematical Physics

Unified Theory of Electrical Machines, a Philosophical Explanation to How a Universe Can be Created Just from Nothing

Authors: Mourici Shachter
Comments: 16 Pages.

It is well known that physics explains natural phenomena using mathematics, But what happens if we derive from a good physical experiment a wrong mathematical equation, use it for more than 100 years without knowing it is wrong. In this paper I develop again the equation of energy conversion in a general electrical machine and prove that the equations we use are wrong. One of Professor Einstein statements was that the laws of physics are the same for every observer so I made the electrical equation of the machine to look like the mechanical equation. What I found was astonishing and is valid for all other area in physics. It is found that in general any constant in physic can be represented by a superposition of time and phase dependent sinusoidal function. So may be what we think about speed of light and other physical constant is entirely wrong. In the last part of this issue, I use the consequences of that article to create a virtual universe.
Category: Mathematical Physics

A Technique for Cataloging Types of Particles and Types of Stuff

Authors: Thomas J. Buckholtz
Comments: 11 Pages.

We develop theory leading to an ability to catalog types of elementary particles. The resulting catalog provides for known interaction-mediating bosons, non-traditional interaction-carrying bosons, and fermions. Ratios of theoretical numbers of analogs of various types of particles match observed ratios of densities for baryonic matter, dark matter, and dark energy.
Category: Mathematical Physics

New Dark Matter Cosmology

Authors: Dan Visser
Comments: 10 Pages.

This paper presents a set of equations, as well as further derivations and calculations, to present dark matter in an alternative cosmology. The alternative is: The Universe did not start with the Big Bang, but is a recalculated Double Torus Universe of dark energy and dark matter. The calculations within this framework show a dark matter-particle, which has the shape of a smallest possible Double Torus, and has a constant diameter of R ≈ 0.712 x 10^-22 [m] at where it exists. At this length the lowest acceleration for Newton gravity becomes a dark matter-acceleration; the Newton force then changes into a lowest dark matter force. This paper shows both forces are embedded in a united ‘new dark energy force’. In the transition of Newton gravity to dark matter gravity, the new dark energy force has a value of 10^-116 kg^4s^2. Accordingly another calculation reveals this force is a ‘force to recalculate space-time itself’. Further calculations also enable to mark dark matter mass with a value of ≈ 2.8 keV/c^2. Again accordingly the dark matter-density (in kg/m) is ≈ 71 gram per 1 million-km (≈1/149 the distance to the sun, or more than three times the distance from Earth to Moon). The dark energy force uses extra time-arrows from an under-laying time-domain of conventional space-time’. This directly gives evidence to the perception gravity is not fundamental. In general this paper relates dark matter mass, new dark energy force and the level where Newton gravity and dark matter meet each other; specifically in that level space-time is recalculated. Moreover an alternative perception of the Higgs-field is given, related to this recalculation.
Category: Mathematical Physics

Metaphysics of the Free Fock Space with Local and Global Information

Authors: Jerzy Hanckowiak
Comments: 24 Pages.

A new interpretation of the basic vector of the free Fock space (FFS) and the FFS is proposed. The approximations to various equations with additional parameters, for n-point information (n-pi), are also considered in the case of non-polynomial nonlinearities. Key words: basic, generating and state vectors, local and global, Cuntz relations, perturbation and closure principles, homotopy analysis method, Axiom of Choice, consilience.
Category: Mathematical Physics

A Simple and Compact Approach to Hydrodynamic Using Geometric Algebra

Authors: Xiong Wang
Comments: 8 Pages.

A new simple and compact approach to hydrodynamic is presented using the formalism of geometric algebra (GA).
Category: Mathematical Physics

Cattaneo’s Projection Technique and Its Applications to a Charged Continuum

Authors: Elmo Benedetto
Comments: 10 Pages.

In this brief review, starting from the de…nition of an ordinary frame of reference in General Relativity, we consider the Cattaneo’s projection technique to write the intrinsic formulation of evolution equations of the electromagnetic matter.
Category: Mathematical Physics

Saint-Venant's Principe of the " Cavity in Cylinder " Problem

Authors: Jian-zhong Zhao
Comments: 16 Pages.

The problem of a cylinder with a small spherical cavity loaded by an equilibrium system of forces is suggested and discussed and its formulation of Saint-Venant's Principle is established. It is evident that finding solutions of boundary-value problems is a precise and pertinent approach to establish Saint-Venant type decay of elastic problems. Keywords : Saint-Venant’s Principe, proof, provability, solution, decay, formulation, cavity AMS Subject Classifications: 74-02, 74G50
Category: Mathematical Physics

Saint-Venant's Principe of the Problem of the Cylinder

Authors: Jian-zhong Zhao
Comments: 9 Pages.

The Statement of Modified Saint-Venant's Principle is suggested. The axisymmetrical deformation of the infinite circular cylinder loaded by an equilibrium system of forces on its near end is discussed and its formulation of Modified Saint-Venant's Principle is established. It is evident that finding solutions of boundary-value problems is a precise and pertinent approach to establish Saint-Venant type decay of elastic problems. AMS Subject Classifications: 74-02, 74G50
Category: Mathematical Physics

Proceedings of the Introduction to Neutrosophic Physics: Unmatter & Unparticle International Conference

Authors: editor Florentin Smarandache
Comments: 92 Pages.

Neutrosophic Physics. Let be a physical entity (i.e. concept, notion, object, space, field, idea, law, property, state, attribute, theorem, theory, etc.), be the opposite of , and be their neutral (i.e. neither nor , but in between). Neutrosophic Physics is a mixture of two or three of these entities , , and that hold together. Therefore, we can have neutrosophic fields, and neutrosophic objects, neutrosophic states, etc. Paradoxist Physics. Neutrosophic Physics is an extension of Paradoxist Physics, since Paradoxist Physics is a combination of physical contradictories and only that hold together, without referring to their neutrality . Paradoxist Physics describes collections of objects or states that are individually characterized by contradictory properties, or are characterized neither by a property nor by the opposite of that property, or are composed of contradictory sub-elements. Such objects or states are called paradoxist entities. These domains of research were set up by the editor in the 1998 within the frame of neutrosophy, neutrosophic logic/set/probability/statistics. This book includes papers by Larissa Borissova, Dmitri Rabounski, Indranu Suhendro, Florentin Smarandache, Thomas R. Love, and Ervin Goldfain. And Comments on Neutrosophic Physics by Dmitri Rabounski, Thomas R. Love, Ervin Goldfain, Diego Lucio Rapoport (Argentina), Armando Assis (Brasil), and Russell Bagdoo (Canada).
Category: Mathematical Physics

Memresistors and Non-Memristive Zero-Crossing Hysteresis Curves

Authors: Blaise Mouttet
Comments: 6 Pages.

It has been erroneously asserted by the circuit theorist Leon Chua that all zero-crossing pinched hysteresis curves define memristors. This claim has been used by Stan Williams of HPLabs to assert that all forms of RRAM and phase change memory are memristors. This paper demonstrates several examples of dynamic systems which fall outside of the constraints of memristive systems and yet also produce the same type of zero-crossing hysteresis curves claimed as a fingerprint for a memristor. This establishes that zero-crossing hysteresis serves as insufficient evidence for a memristor. Keywords- non-linear dynamic systems, memresistor, phase change memory, RRAM, ReRAM
Category: Mathematical Physics

Response to “Pinched Hysteresis Loops is the Fingerprint of Memristive Devices”

Authors: Blaise Mouttet
Comments: 2 Pages.

This is a short response to a recent paper by Kim et al. [1] which correctly notes that the zero-crossing pinched hysteresis loop of a memristor or memristive system must hold for all amplitudes, for all frequencies, and for all initial conditions, of any periodic testing waveform, such as sinusoidal or triangular signals, which assumes both positive and negative values over each period of the waveform. An example is noted from the literature indicating that TiO2 memory resistors might not be considered either memristors or memristive systems given this constraint. Keywords- non-linear dynamic systems, memresistor, RRAM, ReRAM
Category: Mathematical Physics

Pinched Hysteresis Loops Are a Fingerprint of Square Law Capacitors

Authors: Blaise Mouttet
Comments: 5 Pages.

It has been claimed that pinched hysteresis curves are the fingerprint of memristors. This paper demonstrates that a linear resistor in parallel with a nonlinear, square law capacitor also produces pinched hysteresis curves. Spice simulations are performed examining the current vs. voltage behavior of this circuitry under different amplitudes and frequencies of an input signal. Based on this finding a more generalized dynamic systems model is suggested for ReRAM and neuromorphic modeling to cover a broader range of pinched hysteresis curves. Keywords- non-linear circuit theory, RRAM, ReRAM, memristor, memristive systems, memadmittance systems, memresistor
Category: Mathematical Physics

About Material Equilibrium

Authors: Etkin V.A.
Comments: 9 Pages. English

It is shown, that change of image of heat and work in open systems entails necessity to revise of conditions of material equilibrium, found by Gibbs. Thus the chemical potential gives way to other potentials in conformity with conditions of unambiguity of researched processes.
Category: Mathematical Physics

The Resurfacing of Rindler's Coordinate Theory

Authors: sangwha Yi
Comments: 11 Pages. Thank you.

In the general relativity theory, in the Rindler’s coordinate theory, find the present accelerated theory’s problem, resurface Rindler’s coordinate theory that used the tetrad on the new method.The theory’s concept is that save Rindler’s coordinate transformation in the other way.
Category: Mathematical Physics

Duonistic Neutrinos Violate Relativity

Authors: Dan Visser
Comments: 10 Pages.

Neutrinos-faster-than-light? The science-battle is not yet over! ‘Yes’, said the OPERA-team in September 22 2011. ‘No’, said the ICARUS-team in February 23 2012. But this paper carries on that it is undoubtedly correct that neutrinos can go faster-than-light. Neutrinos can only do that in neutrino-pairs: I call these pairs Duonistic Neutrinos! This paper presents the set of equations to prove that. The smallest gravity-acceleration g’ appears to be prior to the trajectory of single packaged neutrinos. OPERA and ICARUS might be right both in the end.
Category: Mathematical Physics

"Physical Intuition" : What Is Wrong with It ?

Authors: Elemer E Rosinger
Comments: 13 Pages.

It appears not to be known that subjecting the axioms to certain conditions, such as for instance to be physically meaningful, may interfere with the logical essence of axiomatic systems, and do so in unforeseen ways, ways that should be carefully considered and accounted for. Consequently, the use of "physical intuition" in building up axiomatic systems for various theories of Physics may lead to situations which have so far not been carefully considered.
Category: Mathematical Physics

A Discinnect : Limitations of the Axiomatic Method in Physics

Authors: Elemer E Rosinger
Comments: 16 Pages.

This paper presents the phenomenon of disconnect in the axiomatic approach to theories of Physics, a phenomenon which appears due to the insistence on axioms which have a physical meaning. This insistence introduces a restriction which is foreign to the abstract nature of axiomatic systems as such. Consequently, it turns out to introduce as well the mentioned disconnect. The axiomatic approach in Physics has a longer tradition. It is there already in Newton's Principia. Recently for instance, a number of axiomatic approaches have been proposed in the literature related to Quantum Mechanics. Special Relativity, [2], had from its beginning in 1905 been built upon two axioms, namely, the Galilean Relativity and the Constancy of the Speed of Light in inertial reference frames. Hardly noticed in wider circles, the independence of these two axioms had quite early been subjected to scrutiny, [5,3], and that issue has on occasion been addressed ever since, see [8,4,24] and the literature cited there. Recently, [24], related to these two axioms in Special Relativity, the following phenomenon of wider importance in Physics was noted. As the example of axiomatization of Special Relativity shows it, it is possible to face a disconnect between a system of physically meaningful axioms, and on the other hand, one or another of the mathematical models used in the study of the axiomatized physical theory. The consequence is that, seemingly unknown so far, one faces in Physics the possibility that the axiomatic method has deeper, less obvious, and in fact not considered, or simply overlooked limitations. As there is no reason to believe that the system of the usual two axioms of Special Relativity is the only one subjected to such a disconnect, the various foundational ventures in modern Physics, related for instance to gravitation, quanta, or their bringing together in an overarching theory, may benefit from the study of the possible sources and reasons for such a disconnect. An attempt of such study is presented in this paper.
Category: Mathematical Physics

Quantum Adeles

Authors: Matti Pitkänen
Comments: 41 Pages.

A generalization of number concept is proposed. One can replace integer n with n-dimensional Hilbert space and sum + and product × with direct sum ⊕ and tensor product ⊗ and introduce their co-operations, the definition of which is highly non-trivial.

This procedure yields also Hilbert space variants of rationals, algebraic numbers, p-adic number fields, and even complex, quaternionic and octonionic algebraics. Also adeles can be replaced with their Hilbert space counterparts. Even more, one can replace the points of Hilbert spaces with Hilbert spaces and repeat this process, which is very similar to the construction of infinite primes having interpretation in terms of repeated second quantization. This process could be the counterpart for construction of n^th order logics and one might speak of Hilbert or quantum mathematics. The construction would also generalize the notion of algebraic holography and provide self-referential cognitive representation of mathematics.

This vision emerged from the connections with generalized Feynman diagrams, braids, and with the hierarchy of Planck constants realized in terms of coverings of the imbedding space. Hilbert space generalization of number concept seems to be extremely well suited for the purposes of TGD. For instance, generalized Feynman diagrams could be identifiable as arithmetic Feynman diagrams describing sequences of arithmetic operations and their co-operations. One could interpret ×_q and +_q and their co-algebra operations as 3-vertices for number theoretical Feynman diagrams describing algebraic identities X=Y having natural interpretation in zero energy ontology. The two vertices have direct counterparts as two kinds of basic topological vertices in quantum TGD (stringy vertices and vertices of Feynman diagrams). The definition of co-operations would characterize quantum dynamics. Physical states would correspond to the Hilbert space states assignable to numbers. One prediction is that all loops can be eliminated from generalized Feynman diagrams and diagrams are in projective sense invariant under permutations of incoming (outgoing legs).
Category: Mathematical Physics

About Absolute Galois Group

Authors: Matti Pitkänen
Comments: 18 Pages.

Absolute Galois Group defined as Galois group of algebraic numbers regarded as extension of rationals is very difficult concept to define. The goal of classical Langlands program is to understand the Galois group of algebraic numbers as algebraic extension of rationals - Absolute Galois Group (AGG) - through its representations. Invertible adeles -ideles - define Gl₁ which can be shown to be isomorphic with the Galois group of maximal Abelian extension of rationals (MAGG) and the Langlands conjecture is that the representations for algebraic groups with matrix elements replaced with adeles provide information about AGG and algebraic geometry.

I have asked already earlier whether AGG could act is symmetries of quantum TGD. The basis idea was that AGG could be identified as a permutation group for a braid having infinite number of strands. The notion of quantum adele leads to the interpretation of the analog of Galois group for quantum adeles in terms of permutation groups assignable to finite l braids. One can also assign to infinite primes braid structures and Galois groups have lift to braid groups.

Objects known as dessins d'enfant provide a geometric representation for AGG in terms of action on algebraic Riemann surfaces allowing interpretation also as algebraic surfaces in finite fields. This representation would make sense for algebraic partonic 2-surfaces, and could be important in the intersection of real and p-adic worlds assigned with living matter in TGD inspired quantum biology, and would allow to regard the quantum states of living matter as representations of AGG. Adeles would make these representations very concrete by bringing in cognition represented in terms of p-adics and there is also a generalization to Hilbert adeles.
Category: Mathematical Physics

The Universe Has 5 Dimensions

Authors: Jake Vlastos
Comments: 16 Pages.

our universe has 5 dimensions adn this project is the proof for this theory
Category: Mathematical Physics

Evidence for a Closed-Curved and Cyclic Double Torus Universe

Authors: Dan Visser
Comments: 7 Pages.

The evidence comes from an alternative calculation of deviations in the dipole fine-structure constant (α). The alternative calculation reveals α deviations spatially and timely connected to a curved dark flow that fits a closed-curved and cyclic Double Torus Universe. This opens-up a new perception the Big Bang is spinning inside another cosmological model. In reality this means the deviation in the dipole α represents the recalculation of the electromagnetic force. Hence, in terms of a cosmological completeness, with also the other forces involved, one could say reality is recalculated by new features of the Double Torus Universe. The features are described earlier in papers posted in the Vixra-archive. Specifically this paper gives the derivations and calculations to show the evidence α deviations are spatially and timely connected in the Double Torus Universe.
Category: Mathematical Physics

Is Indeed Information Physical ?

Authors: Elemer E Rosinger
Comments: 29 Pages.

Information being a relatively new concept in science, the likelihood is pointed out that we do not yet have a good enough grasp of its nature and relevance. This likelihood is further enhanced by the ubiquitous use of information which creates the perception of a manifest, yet in fact, rather superficial familiarity. The paper suggests several aspects which may be essential features of information, or on the contrary, may not be so. In this regard, further studies are obviously needed, studies which may have to avoid with care various temptations to reductionism, like for instance the one claiming that ``information is physical".
Category: Mathematical Physics

Comparison of Infinitely Large and Infinitely Small Quantities.

Authors: Victor Katyushchik
Comments: 4 Pages.

The three-dimensional space formatting allows to operate on infinitely large and infinitely small quantities confidently, without any contradictions, paradoxes and uncertainties.
Category: Mathematical Physics

Monopole Positioning in Closed Space, Trefoil Conversion & Energy Calculation.

Authors: Prakhar bhatnagar
Comments: 5 Pages.

Magnetic monopole is a hypothetical particle with a single pole. In this paper a new mathematical structure for Dirac string has been proposed .Quantitative aspects and qualitative aspects of a monopole represented as a Dirac string have been highlighted. Magnetic bundle has been defined in a complex form and pole has been defined as composition of that bundle for single pole the pole variable is single. The pole can be considered as a group of threads or a point lying on a monopole magnet bundle for one pole the thread is single and one dimensional. The function is in complex form and defines magnetic bundle.
Category: Mathematical Physics

New Dark Energy and Letter to the Nobel Committee.

Authors: Dan Visser
Comments: 10 Pages.

This paper describes what New Dark Energy is, as well as my comment “Nobel Nominations in a New World” to the Nobel Committee. These seemingly different looking subjects are importantly connected in a plead to change the way Nobel-nominations are performed in a changing world with internet and alternative archives, such as viXra. The point is, that institutional established physics and cosmology block new ideas from the public domain. Therefore an appendix has been added to this paper to summarize in a nutshell, relativity, space-expansion, accelerated space-expansion and the one time-direction in the Big Bang Cosmology, in order to explain better what new dark energy is and what its meaning is for physics and new cosmology. That message is meant for the Nobel Committee. Furthermore three-why’s are answered: Why are two extra time-directions needed in a new cosmology? Why is the introduction of dark mass, squared? Why would there be a Double Torus for New Cosmology? Also the basic-formulas are summarized, referring to my papers, such as the dark energy force formula from my thought-experiment, followed by its transformed-version: the force smaller than the smallest gravity. In the end New Dark Energy is formulated as a product of new dark mass (squared) and two extra time-directions from below the Planck scale. This marks that New Dark Energy could recalculate the established dual unitary-entangled quantum-spaces faster backwards in time than time goes forward in the Big Bang Cosmology ! This affects the established quantum dynamics and classical reality towards the past and future differently than Einstein’s General Relativity is prescribing. The letter to the Nobel Committee is send by me, because no other institutions is doing that for me.
Category: Mathematical Physics

Superluminal Physics & Instantaneous Physics as New Trends in Research

Authors: Florentin Smarandache
Comments: 4 Pages.

In a similar way as passing from Euclidean Geometry to Non-Euclidean Geometry, we can pass from Subluminal Physics to Superluminal Physics, and further to Instantaneous Physics. In the lights of two consecutive successful CERN experiments with superluminal particles in the Fall of 2011, we believe that these two new fields of research should begin developing.
Category: Mathematical Physics

Particle Mass Ratios

Authors: DT Froedge
Comments: 12 Pages.

This paper presents straightforward explanation of particle mass ratios, and the specific values for some well known particles. The additional nuclear modes postulated, are similar to the Schrödinger modes in the atom, and, though speculative, the mass ratios calculated for elementary particles are very close the observed mass ratios.
Category: Mathematical Physics

Higgs-Limited Boson Surface (Higgs-LBS) is a Mass-Surface – Thus not a Singular Higgs-Boson and is Related to Faster-Than-Light-Neutrinos.

Authors: Dan Visser
Comments: 4 Pages. Avalable only for arXiv and informing me by email.

In this paper I give a formulation for why a new Higgs-mass is considered otherwise than is usually done. This leads to an additional explanation of my earlier posted papers about neutrinos-faster-than-light-in-vacuum; I also refer to my earlier calculated new Higgs-mass. The addition is a deeper analysis, which does not change the result in the former paper. It is extending the Higgs-energy. The first prediction is the Higgs-particle is a Limited Boson-Surface, or Higgs-LBS. This means: It is not a singular Higgs-mass boson. The second prediction is: The occurrence of an Higgs-LBS may be about 60 nanoseconds before a proton-proton collision in an accelerator, such as the LHC in CERN. This is possible because the Higgs-energy is related to a new dark energy force squared. These predictions are supposed to fit in a new cosmological hypothesis, called the Double Torus of dark energy and dark matter, wherein a new dark energy force is defined as a force smaller than the smallest gravity, which could act beyond General Relativity.
Category: Mathematical Physics

A New Force Smaller Than The Smallest Gravity.

Authors: Dan Visser
Comments: 7 Pages.

In the formulations of this 'paper' speaks the existence of a force smaller than the smallest gravity. This is a new dark energy force, which affects neutrinos differently than is assumed according to current physics. The formulations also imply a different look on the Higgs-mass and dark matter-mass. The 'paper' is also is an overview of recent 'papers' [1] , which already described these issues, but a deeper analysis became important, because a new cosmological hypothesis is involved. The CERN-experiments on these issues are the falsification for my formulations, but until now my formulations withstand several experimental results, and in this case the match with the latest neutrino-faster-than-light experiments is very convincing (more attention might be given to this 'paper' towards institutional disciplines by the Arxiv or Nature).
Category: Mathematical Physics

Langlands Conjectures in TGD Framework

Authors: Matti Pitkänen
Comments: 24 pages.

The arguments of this article support the view that in TGD Universe number theoretic and geometric Langlands conjectures could be understood very naturally. The basic notions are following.

1. Zero energy ontology (ZEO) and the related notion of causal diamond CD (CD is short hand for the cartesian product of causal diamond of M⁴ and of CP₂). ZEO leads to the notion of partonic 2-surfaces at the light-like boundaries of CD and to the notion of string world sheet. These notions are central in the recent view about TGD. One can assign to the partonic 2-surfaces a conformal moduli space having as additional coordinates the positions of braid strand ends (punctures). By electric-magnetic duality this moduli space must correspond closely to the moduli space of string world sheets.

2. Electric-magnetic duality realized in terms of string world sheets and partonic 2-surfaces. The group G and its Langlands dual ^LG would correspond to the time-like and space-like braidings. Duality predicts that the moduli space of string world sheets is very closely related to that for the partonic 2-surfaces. The strong form of 4-D general coordinate invariance implying electric-magnetic duality and S-duality as well as strong form of holography indeed predicts that the collection of string world sheets is fixed once the collection of partonic 2-surfaces at light-like boundaries of CD and its sub-CDs is known.

3. The proposal is that finite measurement resolution is realized in terms of inclusions of hyperfinite factors of type II₁ at quantum level and represented in terms of confining effective gauge group. This effective gauge group could be some associate of G: gauge group, Kac-Moody group or its quantum counterpart, or so called twisted quantum Yangian strongly suggested by twistor considerations. At space-time level the finite measurement resolution would be represented in terms of braids at space-time level which come in two varieties correspond to braids assignable to space-like surfaces at the two light-like boundaries of CD and with light-like 3-surfaces at which the signature of the induced metric changes and which are identified as orbits of partonic 2-surfaces connecting the future and past boundaries of CDs.

   There are several steps leading from G to its twisted quantum Yangian. The first step replaces point like particles with partonic 2-surfaces: this brings in Kac-Moody character. The second step brings in finite measurement resolution meaning that Kac-Moody type algebra is replaced with its quantum version. The third step brings in zero energy ontology: one cannot treat single partonic surface or string world sheet as independent unit: always the collection of partonic 2-surfaces and corresponding string worlds sheets defines the geometric structure so that multilocality and therefore quantum Yangian algebra with multilocal generators is unavoidable.

   In finite measurement resolution geometric Langlands duality and number theoretic Langlands duality are very closely related since partonic 2-surface is effectively replaced with the punctures representing the ends of braid strands and the orbit of this set under a discrete subgroup of G defines effectively a collection of "rational" 2-surfaces. The number of the "rational" surfaces in geometric Langlands conjecture replaces the number of rational points of partonic 2-surface in its number theoretic variant. The ability to compute both these numbers is very relevant for quantum TGD.

4. The natural identification of the associate of G is as quantum Yangian of Kac-Moody type group associated with Minkowskian open string model assignable to string world sheet representing a string moving in the moduli space of partonic 2-surface. The dual group corresponds to Euclidian string model with partonic 2-surface representing string orbit in the moduli space of the string world sheets. The Kac-Moody algebra assigned with simply laced G is obtained using the standard tachyonic free field representation obtained as ordered exponentials of Cartan algebra generators identified as transversal parts of M⁴ coordinates for the braid strands. The importance of the free field representation generalizing to the case of non-simply laced groups in the realization of finite measurement resolution in terms of Kac-Moody algebra cannot be over-emphasized.

5. Langlands duality involves besides harmonic analysis side also the number theoretic side. Galois groups (collections of them) defined by infinite primes and integers having representation as symplectic flows defining braidings. I have earlier proposed that the hierarchy of these Galois groups define what might be regarded as a non-commutative homology and cohomology. Also G has this kind of representation which explains why the representations of these two kinds of groups are so intimately related. This relationship could be seen as a generalization of the MacKay correspondence between finite subgroups of SU(2) and simply laced Lie groups.

6. Symplectic group of the light-cone boundary acting as isometries of the WCW geometry kenociteallb/compl1 allowing to represent projectively both Galois groups and symmetry groups as symplectic flows so that the non-commutative cohomology would have braided representation. This leads to braided counterparts for both Galois group and effective symmetry group.

7. The moduli space for Higgs bundle playing central role in the approach of Witten and Kapustin to geometric Landlands program is in TGD framework replaced with the conformal moduli space for partonic 2-surfaces. It is not however possible to speak about Higgs field although moduli defined the analog of Higgs vacuum expectation value. Note that in TGD Universe the most natural assumption is that all Higgs like states are "eaten" by gauge bosons so that also photon and gluons become massive. This mechanism would be very general and mean that massless representations of Poincare group organize to massive ones via the formation of bound states. It might be however possible to see the contribution of p-adic thermodynamics depending on genus as analogous to Higgs contribution since the conformal moduli are analogous to vacuum expectation of Higgs field.

How Infinite Primes Relate to Other Views About Mathematical Infinity?

Authors: Matti Pitkänen
Comments: 18 pages.

Infinite primes is a purely TGD inspired notion. The notion of infinity is number theoretical and infinite primes have well defined divisibility properties. One can partially order them by the real norm. p-Adic norms of infinite primes are well defined and finite. The construction of infinite primes is a hierarchical procedure structurally equivalent to a repeated second quantization of a supersymmetric arithmetic quantum field theory. At the lowest level bosons and fermions are labelled by ordinary primes. At the next level one obtains free Fock states plus states having interpretation as bound many particle states. The many particle states of a given level become the single particle states of the next level and one can repeat the construction ad infinitum. The analogy with quantum theory is intriguing and I have proposed that the quantum states in TGD Universe correspond to octonionic generalizations of infinite primes. It is interesting to compare infinite primes (and integers) to the Cantorian view about infinite ordinals and cardinals. The basic problems of Cantor's approach which relate to the axiom of choice, continuum hypothesis, and Russell's antinomy: all these problems relate to the definition of ordinals as sets. In TGD framework infinite primes, integers, and rationals are defined purely algebraically so that these problems are avoided. It is not surprising that these approaches are not equivalent. For instance, sum and product for Cantorian ordinals are not commutative unlike for infinite integers defined in terms of infinite primes.

Set theory defines the foundations of modern mathematics. Set theory relies strongly on classical physics, and the obvious question is whether one should reconsider the foundations of mathematics in light of quantum physics. Is set theory really the correct approach to axiomatization?

1. Quantum view about consciousness and cognition leads to a proposal that p-adic physics serves as a correlate for cognition. Together with the notion of infinite primes this suggests that number theory should play a key role in the axiomatics.
2. Algebraic geometry allows algebraization of the set theory and this kind of approach suggests itself strongly in physics inspired approach to the foundations of mathematics. This means powerful limitations on the notion of set.
3. Finite measurement resolution and finite resolution of cognition could have implications also for the foundations of mathematics and relate directly to the fact that all numerical approaches reduce to an approximation using rationals with a cutoff on the number of binary digits.
4. The TGD inspired vision about consciousness implies evolution by quantum jumps meaning that also evolution of mathematics so that no fixed system of axioms can ever catch all the mathematical truths for the simple reason that mathematicians themselves evolve with mathematics.

Motives and Infinite Primes

Authors: Matti Pitkänen
Comments: 79 pages.

In this article the goal is to find whether the general mathematical structures associated with twistor approach, superstring models and M-theory could have a generalization or a modification in TGD framework. The contents of the chapter is an outcome of a rather spontaneous process, and represents rather unexpected new insights about TGD resulting as outcome of the comparisons.

1. Infinite primes, Galois groups, algebraic geometry, and TGD

In algebraic geometry the notion of variety defined by algebraic equation is very general: all number fields are allowed. One of the challenges is to define the counterparts of homology and cohomology groups for them. The notion of cohomology giving rise also to homology if Poincare duality holds true is central. The number of various cohomology theories has inflated and one of the basic challenges to find a sufficiently general approach allowing to interpret various cohomology theories as variations of the same motive as Grothendieck, who is the pioneer of the field responsible for many of the basic notions and visions, expressed it.

Cohomology requires a definition of integral for forms for all number fields. In p-adic context the lack of well-ordering of p-adic numbers implies difficulties both in homology and cohomology since the notion of boundary does not exist in topological sense. The notion of definite integral is problematic for the same reason. This has led to a proposal of reducing integration to Fourier analysis working for symmetric spaces but requiring algebraic extensions of p-adic numbers and an appropriate definition of the p-adic symmetric space. The definition is not unique and the interpretation is in terms of the varying measurement resolution.

The notion of infinite has gradually turned out to be more and more important for quantum TGD. Infinite primes, integers, and rationals form a hierarchy completely analogous to a hierarchy of second quantization for a super-symmetric arithmetic quantum field theory. The simplest infinite primes representing elementary particles at given level are in one-one correspondence with many-particle states of the previous level. More complex infinite primes have interpretation in terms of bound states.

1. What makes infinite primes interesting from the point of view of algebraic geometry is that infinite primes, integers and rationals at the n:th level of the hierarchy are in 1-1 correspondence with rational functions of n arguments. One can solve the roots of associated polynomials and perform a root decomposition of infinite primes at various levels of the hierarchy and assign to them Galois groups acting as automorphisms of the field extensions of polynomials defined by the roots coming as restrictions of the basic polynomial to planes x_n=0, x_n=x_n-1=0, etc...

2. These Galois groups are suggested to define non-commutative generalization of homotopy and homology theories and non-linear boundary operation for which a geometric interpretation in terms of the restriction to lower-dimensional plane is proposed. The Galois group G_k would be analogous to the relative homology group relative to the plane x_k-1=0 representing boundary and makes sense for all number fields also geometrically. One can ask whether the invariance of the complex of groups under the permutations of the orders of variables in the reduction process is necessary. Physical interpretation suggests that this is not the case and that all the groups obtained by the permutations are needed for a full description.

3. The algebraic counterpart of boundary map would map the elements of G_k identified as analog of homotopy group to the commutator group [G_k-2,G_k-2] and therefore to the unit element of the abelianized group defining cohomology group. In order to obtains something analogous to the ordinary homology and cohomology groups one must however replaces Galois groups by their group algebras with values in some field or ring. This allows to define the analogs of homotopy and homology groups as their abelianizations. Cohomotopy, and cohomology would emerge as duals of homotopy and homology in the dual of the group algebra.

4. That the algebraic representation of the boundary operation is not expected to be unique turns into blessing when on keeps the TGD as almost topological QFT vision as the guide line. One can include all boundary homomorphisms subject to the condition that the anticommutator δⁱ_kδ^j_k-1+δ^j_kδⁱ_k-1 maps to the group algebra of the commutator group [G_k-2,G_k-2]. By adding dual generators one obtains what looks like a generalization of anticommutative fermionic algebra and what comes in mind is the spectrum of quantum states of a SUSY algebra spanned by bosonic states realized as group algebra elements and fermionic states realized in terms of homotopy and cohomotopy and in abelianized version in terms of homology and cohomology. Galois group action allows to organize quantum states into multiplets of Galois groups acting as symmetry groups of physics. Poincare duality would map the analogs of fermionic creation operators to annihilation operators and vice versa and the counterpart of pairing of k:th and n-k:th homology groups would be inner product analogous to that given by Grassmann integration. The interpretation in terms of fermions turns however to be wrong and the more appropriate interpretation is in terms of Dolbeault cohomology applying to forms with homomorphic and antiholomorphic indices.

5. The intuitive idea that the Galois group is analogous to 1-D homotopy group which is the only non-commutative homotopy group, the structure of infinite primes analogous to the braids of braids of braids of ... structure, the fact that Galois group is a subgroup of permutation group, and the possibility to lift permutation group to a braid group suggests a representation as flows of 2-D plane with punctures giving a direct connection with topological quantum field theories for braids, knots and links. The natural assumption is that the flows are induced from transformations of the symplectic group acting on δ M²_+/-× CP₂ representing quantum fluctuating degrees of freedom associated with WCW ("world of classical worlds"). Discretization of WCW and cutoff in the number of modes would be due to the finite measurement resolution. The outcome would be rather far reaching: finite measurement resolution would allow to construct WCW spinor fields explicitly using the machinery of number theory and algebraic geometry.

6. A connection with operads is highly suggestive. What is nice from TGD perspective is that the non-commutative generalization homology and homotopy has direct connection to the basic structure of quantum TGD almost topological quantum theory where braids are basic objects and also to hyper-finite factors of type II₁. This notion of Galois group makes sense only for the algebraic varieties for which coefficient field is algebraic extension of some number field. Braid group approach however allows to generalize the approach to completely general polynomials since the braid group make sense also when the ends points for the braid are not algebraic points (roots of the polynomial).

This construction would realize the number theoretical, algebraic geometrical, and topological content in the construction of quantum states in TGD framework in accordance with TGD as almost TQFT philosophy, TGD as infinite-D geometry, and TGD as generalized number theory visions.

2. p-Adic integration and cohomology

This picture leads also to a proposal how p-adic integrals could be defined in TGD framework.

1. The calculation of twistorial amplitudes reduces to multi-dimensional residue calculus. Motivic integration gives excellent hopes for the p-adic existence of this calculus and braid representation would give space-time representation for the residue integrals in terms of the braid points representing poles of the integrand: this would conform with quantum classical correspondence. The power of 2π appearing in multiple residue integral is problematic unless it disappears from scattering amplitudes. Otherwise one must allow an extension of p-adic numbers to a ring containing powers of 2π.

2. Weak form of electric-magnetic duality and the general solution ansatz for preferred extremals reduce the Kähler action defining the Kähler function for WCW to the integral of Chern-Simons 3-form. Hence the reduction to cohomology takes places at space-time level and since p-adic cohomology exists there are excellent hopes about the existence of p-adic variant of Kähler action. The existence of the exponent of Kähler gives additional powerful constraints on the value of the Kähler fuction in the intersection of real and p-adic worlds consisting of algebraic partonic 2-surfaces and allows to guess the general form of the Kähler action in p-adic context.

3. One also should define p-adic integration for vacuum functional at the level of WCW. p-Adic thermodynamics serves as a guideline leading to the condition that in p-adic sector exponent of Kähler action is of form (m/n)^r, where m/n is divisible by a positive power of p-adic prime p. This implies that one has sum over contributions coming as powers of p and the challenge is to calculate the integral for K= constant surfaces using the integration measure defined by an infinite power of Kähler form of WCW reducing the integral to cohomology which should make sense also p-adically. The p-adicization of the WCW integrals has been discussed already earlier using an approach based on harmonic analysis in symmetric spaces and these two approaches should be equivalent. One could also consider a more general quantization of Kähler action as sum K=K₁+K₂ where K₁=rlog(m/n) and K₂=n, with n divisible by p since exp(n) exists in this case and one has exp(K)= (m/n)^r × exp(n). Also transcendental extensions of p-adic numbers involving n+p-2 powers of e^1/n can be considered.

4. If the Galois group algebras indeed define a representation for WCW spinor fields in finite measurement resolution, also WCW integration would reduce to summations over the Galois groups involved so that integrals would be well-defined in all number fields.

3. Floer homology, Gromov-Witten invariants, and TGD

Floer homology defines a generalization of Morse theory allowing to deduce symplectic homology groups by studying Morse theory in loop space of the symplectic manifold. Since the symplectic transformations of the boundary of δ M⁴_+/-× CP₂ define isometry group of WCW, it is very natural to expect that Kähler action defines a generalization of the Floer homology allowing to understand the symplectic aspects of quantum TGD. The hierarchy of Planck constants implied by the one-to-many correspondence between canonical momentum densities and time derivatives of the imbedding space coordinates leads naturally to singular coverings of the imbedding space and the resulting symplectic Morse theory could characterize the homology of these coverings.

One ends up to a more precise definition of vacuum functional: Kähler action reduces Chern-Simons terms (imaginary in Minkowskian regions and real in Euclidian regions) so that it has both phase and real exponent which makes the functional integral well-defined. Both the phase factor and its conjugate must be allowed and the resulting degeneracy of ground state could allow to understand qualitatively the delicacies of CP breaking and its sensitivity to the parameters of the system. The critical points with respect to zero modes correspond to those for Kähler function. The critical points with respect to complex coordinates associated with quantum fluctuating degrees of freedom are not allowed by the positive definiteness of Kähler metric of WCW. One can say that Kähler and Morse functions define the real and imaginary parts of the exponent of vacuum functional.

The generalization of Floer homology inspires several new insights. In particular, space-time surface as hyper-quaternionic surface could define the 4-D counterpart for pseudo-holomorphic 2-surfaces in Floer homology. Holomorphic partonic 2-surfaces could in turn correspond to the extrema of Kähler function with respect to zero modes and holomorphy would be accompanied by super-symmetry.

Gromov-Witten invariants appear in Floer homology and topological string theories and this inspires the attempt to build an overall view about their role in TGD. Generalization of topological string theories of type A and B to TGD framework is proposed. The TGD counterpart of the mirror symmetry would be the equivalence of formulations of TGD in H=M⁴× CP₂ and in CP₃× CP₃ with space-time surfaces replaced with 6-D sphere bundles.

4. K-theory, branes, and TGD

K-theory and its generalizations play a fundamental role in super-string models and M-theory since they allow a topological classification of branes. After representing some physical objections against the notion of brane more technical problems of this approach are discussed briefly and it is proposed how TGD allows to overcome these problems. A more precise formulation of the weak form of electric-magnetic duality emerges: the original formulation was not quite correct for space-time regions with Euclidian signature of the induced metric. The question about possible TGD counterparts of R-R and NS-NS fields and S, T, and U dualities is discussed.

5. p-Adic space-time sheets as correlates for Boolean cognition

p-Adic physics is interpreted as physical correlate for cognition. The so called Stone spaces are in one-one correspondence with Boolean algebras and have typically 2-adic topologies. A generalization to p-adic case with the interpretation of p pinary digits as physically representable Boolean statements of a Boolean algebra with 2ⁿ>p>p^n-1 statements is encouraged by p-adic length scale hypothesis. Stone spaces are synonymous with profinite spaces about which both finite and infinite Galois groups represent basic examples. This provides a strong support for the connection between Boolean cognition and p-adic space-time physics. The Stone space character of Galois groups suggests also a deep connection between number theory and cognition and some arguments providing support for this vision are discussed.

Could One Generalize Braid Invariant Defined by Vacuum Expecation of Wilson Loop to and Invariant of Braid Cobordisms and of 2-Knots?

Authors: Matti Pitkänen
Comments: 17 pages.

Witten was awarded by Fields medal from a construction recipe of Jones polynomial based on topological QFT assigned with braids and based on Chern-Simons action. Recently Witten has been working with an attempt to understand in terms of quantum theory the so called Khovanov polynomial associated with a much more abstract link invariant whose interpretation and real understanding remains still open.

The attempts to understand Witten's thoughts lead to a series of questions unavoidably culminating to the frustrating "Why I do not have the brain of Witten making perhaps possible to answer these questions?". This one must just accept. In this article I summarize some thoughts inspired by the associations of the talk of Witten with quantum TGD and with the model of DNA as topological quantum computer. In my own childish manner I dare believe that these associations are interesting and dare also hope that some more brainy individual might take them seriously.

An idea inspired by TGD approach which also main streamer might find interesting is that the Jones invariant defined as vacuum expectation for a Wilson loop in 2+1-D space-time generalizes to a vacuum expectation for a collection of Wilson loops in 2+2-D space-time and could define an invariant for 2-D knots and for cobordisms of braids analogous to Jones polynomial. As a matter fact, it turns out that a generalization of gauge field known as gerbe is needed and that in TGD framework classical color gauge fields defined the gauge potentials of this field. Also topological string theory in 4-D space-time could define this kind of invariants. Of course, it might well be that this kind of ideas have been already discussed in literature.

Could the Notion of Hyperdeterminant be Useful in TGD Framework?

Authors: Matti Pitkänen
Comments: 4 pages.

The vanishing of ordinary determinant tells that a group of linear equations possesses non-trivial solutions. Hyperdeterminant generalizes this notion to a situation in which one has homogenous multilinear equations. The notion has applications to the description of quantum entanglement and has stimulated interest in physics blogs. Hyperdeterminant applies to hyper-matrices with n matrix indices defined for an n-fold tensor power of vector space - or more generally - for a tensor product of vector spaces with varying dimensions. Hyper determinant is an n-linear function of the arguments in the tensor factors with the property that all partial derivatives of the hyper determinant vanish at the point, which corresponds to a non-trivial solution of the equation. A simple example is potential function of n arguments linear in each argument.

Why the notion of hyperdeterminant- or rather its infinite-dimensional generalization- might be interesting in TGD framework relates to the quantum criticality of TGD stating that TGD Universe involves a fractal hierarchy of criticalities: phase transitions inside phase transitions inside... At classical level the lowest order criticality means that the extremal of Kähler action possesses non-trivial second variations for which the action is not affected. The system is critical. In QFT context one speaks about zero modes. The vanishing of the so called Gaussian (of functional) determinant associated with second variations is the condition for the existence of critical deformations. In QFT context this situation corresponds to the presence of zero modes.

The simplest physical model for a critical system is cusp catastrophe defined by a potential function V(x) which is fourth order polynomial. At the edges of cusp two extrema of potential function stable and unstable extrema co-incide and the rank of the matrix defined by the potential function vanishes. This means vanishing of its determinant. At the tip of the cusp the also the third derivative vanishes of potential function vanishes. This situation is however not describable in terms of hyperdeterminant since it is genuinely non-linear rather than only multilinear.

In a complete analogy, one can consider also the vanishing of n:th variations in TGD framework as higher order criticality so that the vanishing of hyperdeterminant might serve as a criterion for the higher order critical point and occurrence of phase transition. Why multilinearity might replace non-linearity in TGD framework could be due to the non-locality. Multilinearty with respect to imbedding space-coordinates at different space-time points would imply also the vanishing of the standard local divergences of quantum field theory known to be absent in TGD framework on basis of very general arguments. In this article an attempt to concretize this idea is made. The challenge is highly non-trivial since in finite measurement resolution one must work with infinite-dimensional system.

What Could be the Generalization of Yangian Symmetry of N=4 Susy in TGD Framework?

Authors: Matti Pitkänen
Comments: 45 pages.

There have been impressive steps in the understanding of N=4 maximally sypersymmetric YM theory possessing 4-D super-conformal symmetry. This theory is related by AdS/CFT duality to certain string theory in AdS₅× S⁵ background. Second stringy representation was discovered by Witten and is based on 6-D Calabi-Yau manifold defined by twistors. The unifying proposal is that so called Yangian symmetry is behind the mathematical miracles involved.

In the following I will discuss briefly the notion of Yangian symmetry and suggest its generalization in TGD framework by replacing conformal algebra with appropriate super-conformal algebras. Also a possible realization of twistor approach and the construction of scattering amplitudes in terms of Yangian invariants defined by Grassmannian integrals is considered in TGD framework and based on the idea that in zero energy ontology one can represent massive states as bound states of massless particles. There is also a proposal for a physical interpretation of the Cartan algebra of Yangian algebra allowing to understand at the fundamental level how the mass spectrum of n-particle bound states could be understood in terms of the n-local charges of the Yangian algebra.

Twistors were originally introduced by Penrose to characterize the solutions of Maxwell's equations. Kähler action is Maxwell action for the induced Kähler form of CP₂. The preferred extremals allow a very concrete interpretation in terms of modes of massless non-linear field. Both conformally compactified Minkowski space identifiable as so called causal diamond and CP₂ allow a description in terms of twistors. These observations inspire the proposal that a generalization of Witten's twistor string theory relying on the identification of twistor string world sheets with certain holomorphic surfaces assigned with Feynman diagrams could allow a formulation of quantum TGD in terms of 3-dimensional holomorphic surfaces of CP₃× CP₃ mapped to 6-surfaces dual CP₃× CP₃, which are sphere bundles so that they are projected in a natural manner to 4-D space-time surfaces. Very general physical and mathematical arguments lead to a highly unique proposal for the holomorphic differential equations defining the complex 3-surfaces conjectured to correspond to the preferred extremals of Kähler action.

A Possible Explanation for Shnoll Effect

Authors: Matti Pitkänen
Comments: 17 pages.

Shnoll and collaborators have discovered strange repeating patterns of random fluctuations of physical observables such as the number n of nuclear decays in a given time interval. Periodically occurring peaks for the distribution of the number N(n) of measurements producing n events in a series of measurements as a function of n is observed instead of a single peak. The positions of the peaks are not random and the patterns depend on position and time varying periodically in time scales possibly assignable to Earth-Sun and Earth-Moon gravitational interaction.

These observations suggest a modification of the expected probability distributions but it is very difficult to imagine any physical mechanism in the standard physics framework. Rather, a universal deformation of predicted probability distributions would be in question requiring something analogous to the transition from classical physics to quantum physics.

The hint about the nature of the modification comes from the TGD inspired quantum measurement theory proposing a description of the notion of finite measurement resolution in terms of inclusions of so called hyper-finite factors of type II₁ (HFFs) and closely related quantum groups. Also p-adic physics -another key element of TGD- is expected to be involved. A modification of a given probability distribution P(nkenovert λ_i) for a positive integer valued variable n characterized by rational-valued parameters λ_i is obtained by replacing n and the integers characterizing λ_i with so called quantum integers depending on the quantum phase q_m=exp(i2π/m). Quantum integer n_q must be defined as the product of quantum counterparts p_q of the primes p appearing in the prime decomposition of n. One has p_q= sin(2π p/m)/sin(2π/m) for p≠ P and p_q=P for p=P. m must satisfy m≥ 3, m≠ p, and m≠ 2p.

The quantum counterparts of positive integers can be negative. Therefore quantum distribution is defined first as p-adic valued distribution and then mapped by so called canonical identification I to a real distribution by the map taking p-adic -1 to P and powers Pⁿ to P^-n and other quantum primes to themselves and requiring that the mean value of n is for distribution and its quantum variant. The map I satisfies I(∑ P_n)=∑ I(P_n). The resulting distribution has peaks located periodically with periods coming as powers of P. Also periodicities with peaks corresponding to n=n⁺n^-, n⁺_q>0 with fixed n^-_q<0, are predicted. These predictions are universal and easily testable. The prime P and integer m characterizing the quantum variant of distribution can be identified from data. The shapes of the distributions obtained are qualitatively consistent with the findings of Shnoll but detailed tests are required to see whether the number theoretic predictions are correct.

The periodic dependence of the distributions would be most naturally assignable to the gravitational interaction of Earth with Sun and Moon and therefore to the periodic variation of Earth-Sun and Earth-Moon distances. The TGD inspired proposal is that the p-dic prime P and integer m characterizing the quantum distribution are determined by a process analogous to a state function reduction and their most probably values depend on the deviation of the distance R through the formulas Δ p/p≈ k_pΔ R/R and Δ m/m≈ k_mΔ R/R. The p-adic primes assignable to elementary particles are very large unlike the primes which could characterize the empirical distributions. The hierarchy of Planck constants allows the gravitational Planck constant assignable to the space-time sheets mediating gravitational interactions to have gigantic values and this allows p-adicity with small values of the p-adic prime P.

A New Dark Energy Force Theoretically Calculates Faster-than-light-neutrinos.

Authors: Dan Visser
Comments: 7 pages

A theoretical calculation with a new dark energy force formula discloses the correctness of the experimental faster-than-light-neutrinos in the CERN-San Grasso experiment. The formulation in this paper theoretically confirms that Einstein's Relativity could be violated. This introduces the obligation to accept a new cosmological model, called the Double Torus hypothesis . The theoretical calculation in this paper is based on a new momentum of dark energy force, formulated by its new force and two extra time dimensions below the Planck scale. In detail this completely new perspective shows that the dark energy force starts to dominate the lowest limit of the Newton-force-acceleration under specific conditions of neutrino-oscillations. This paper theoretically calculates 62.8 nanosecond for the experimental detected early-arrival of muon-neutrinos related to how light-in-vacuum would have arrived. This is a marvelous close match compared to the ((60.7 ± 6.9 (stat.) ± 7.4 (sys.)) nanosecond found during the 'neutrino-flight path' from CERN to San Grasso.
Category: Mathematical Physics

Zeno's Paradoxe and the Nature of Points in Quantized Euclidean Universe

Authors: Markos Georgallides
Comments: 9 pages

This article explains the correlation between Euclidean Geometry , Complex Numbers and Physics . A Straight line AB is continuous in Points between A and B [ i.e. all points between AB are the elements which fill AB ] , which Points are also , Nothing , Everything , and maybe Anywhere , without any Dimension , and one has to pass the infinite points between A and B . A point C is on line AB only when exists CA+ CB = AB , or the whole AB is equal to the parts CA , CB , and this is an equation , which differentiates geometries . Since points have not any dimension and since only AB has dimension ( the length AB and for ÃC the length AC ) and since on ÃB exist infinite AC → AB , which have infinite Spaces , Anti-Spaces and Sub-Spaces , then 1. Straight line AB is continuous with points as filling ( Infinitively divisible ) . 2. Straight line AB is discontinuous (discrete) with dimensional Units , ds =AB as filling ( that is made up of finite divisible or indivisible parts the Monads ds ) or ds → AB / n , where n = 1 , 2 , → ∞ ) , and for n = ∞ then ds = 0 . 3. Straight line AB is discontinuous (discrete) with dimensional Units ds , or ds = quantum = AB / n [ where n = 1,2,3 → ∞ , = ( a + b.i ) / n , Infinitively divisible and keeping always the conservation of properties at end points A , B ] as filling , and continuous with points as filling ( for n = ∞ then ds = 0 i.e. a point ) . This is the Vector relation of Monads , ds , ( or , as Complex Numbers in their general form , ds = a + b. i ) , which is the Dual Nature of lines AB , ( discrete and continuous ) . So travelling on Points ( ds = 0 ) between AB one never comes to B , on the contrary travelling with ds > 0 one comes in finite time . 4 . Achilles has to pass every point of line AB which is then as passing from the starting point A , ds =0 , where Velocity of Achilles is v(A) = ds/dt = 0 . The same happens for Tortoise at point B where Velocity v(T) = ds/dt = 0 . On the contrary , Achilles passing AB on dimensional Units , ds , then Achilles velocity v(A) = ds/dt(A) is greater than that of Tortoise v(T) = ds / dt(T) . Since in PNS , v = ∞ , T = 0 , meaning infinite velocity and Time not existing , then Arrow AB in [PNS] is constant because AB = ds = Constant = u . 0 = ∞ . 0 Straight line AB is discontinuous (discrete) with dimensional Units ds = AB / n where n = 1 → ∞ and continuous with points [ n = ∞ ] . Continuously on AB happens also with all discrete ds , ( This is the Dual Nature of lines ( Geometry ), discrete and continuous ) . Monads ds = 0 → ∞ are Simultaneously , actual infinite ( because for n = ∞ then ds = [ AB / n = ∞ ] = 0 i.e. a point ) , and potential infinite , ( because for n = 0 then ds = [ AB / n=0 ] = ∞ i.e. the straight line through AB .
Category: Mathematical Physics

Can Electromagnetic Scalar Waves be Radiated by a Metal Sphere?

Authors: Giuliano Bettini
Comments: 24 pages.

There is a lot of chattering on the Internet about Tesla waves, vacuum energy, scalar waves and so on. Professor Meyl says he has a complete theory, experimental evidence and apparatus on these waves. In a theoretical paper Van Vlaenderen introduced a generalization of classical electrodynamics for the prediction of scalar field effects. It is said the Monstein has demonstrated the physical existence of such scalar waves. NASA in a report seems to consider such waves as a promising item to be studied. Some other papers appeared in arXiv. I've already showed that such waves are a consequence of "generalized" Maxwell fields which simply mean space time analytic functions not limited by the Lorenz gauge condition, but accepted instead in a wide sense. In this paper I remember my ideas on these waves, together with my doubts about their physical existence. In fact, the deduction of the scalar waves equations, together with their physical interpretation, in my opinion demonstrates nothing about the physical existence of scalar waves. I discuss the experiment of Monstein, and suggest some other experiment. Obviously I think that the lack of demonstration of the existence doesn't mean the demonstration of inexistence.
Category: Mathematical Physics

New Value for the Higgs Mass.

Authors: Dan Visser
Comments: 4 pages. Publication is free for any institute or magazine

The Higgs mass is approximately 0.5 TeV/c2 . This is a new predicted value. This mass value for Higgs is calculated with a new dark energy force formula, which is performing in the hypothesis of the Double Torus, a new cosmological "model" for the universe. This "model" embeds the Big Bang framework. The new Higgs mass is theoretically disclosed by implementing the dimensional features of relativistic spacetime into the dimensions of the dark energy force formula and an equalization to the lowest limit for acceleration in Newton force, being a transition where Newton breaks down and the dark energy force takes over. The dark energy force generates gravitational movement and thus mass. This enables to calculate a Higgs mass differently than in currently used theories.
Category: Mathematical Physics

Electrostatics and Fluid Flow with 3D Analytic Functions

Authors: Giuliano Bettini
Comments: 21 pages.

I present examples of applications of 3-dimensional analytic functions to electrostatics and potential flows, mainly devoted to engineers and physicists. Of course, the paper only suggests areas of future development, despite that a persistent idea, from Sommerfeld on, seems to be "The powerful tool of the theory of complex functions cannot be used in three-dimensional potential theory" (Sommerfeld, "Mechanics of Deformable Bodies", Academic Press, 1950) I summarize here unpublished manuscripts dated 1994.
Category: Mathematical Physics

Majorana Neutrino: Chirality and Helicity

Authors: Valeriy V. Dvoeglazov
Comments: 19 Pages

We introduce the Majorana spinors in the momentum representation. They obey the Dirac-like equation with eight components, which has been first introduced by Markov. Thus, the Fock space for corresponding quantum fields is doubled (as shown by Ziino). Particular attention has been paid to the questions of chirality and helicity (two concepts which frequently are confused in the literature) for Dirac and Majorana states.
Category: Mathematical Physics

Black Hole vs. Variable Rest Mass Neutron Star

Authors: D.T. Froedge
Comments: 14 pages

In a previous paper we have discussed the conjecture of a variable particle rest mass, as a function of gravitational potential (Scalar Gravitational Theory with Variable Rest Mass) This paper discuses the implications of that theory in regard to a large neutron star, and contrast the difference between the predicted phenomena, and Black Hole theory as put fourth by standard GR. The validity of this theory would be established by the finding of a neutron star having a mass greater than allowed by Black Hole theory.
Category: Mathematical Physics

Scalar Gravitational Theory with Variable Rest Mass

Authors: D.T. Froedge
Comments: 16 pages

The purpose of this paper is to present a non-tensor theory of gravitation that provides the equivalent equations of motion, but does not result in the issue of black holes, non-localizable energy, or spacetime singularities. The prime assumption is the notion that the rest mass of a particle entering a gravitational potential is reduced in proportion to the energy gained by the velocity increases. One could designate this development as a "catalytic" theory in that gravitation is a vector catalyst, that converts rest energy into kinetic energy. The total mass energy will be considered localized with the individual mass particles, and defined relative to a given observer. No energy will be ascribed to the field, thus there is no stress energy tensor. We will develop the energy mass relation, and show that it can result in the proper orbital precession as demonstrated by GR. The rest mass is not significantly different from that of a particle defined in a stationary asymptotically flat GR space-time, when the defining point particles via the Komar mass. Since rest mass of a particle goes to zero on approaching a Schwarzschild boundary, the formation of black holes becomes problematic.
Category: Mathematical Physics

Matter Over Anti-Matter Explained in a Double Torus.

Authors: Dan Visser
Comments: 4 pages

September 1 2009 a new cosmological hypothesis was proposed to imagine the universe otherwise. Many institutional scientists still try to implement their results into the commonly accepted Big Bang theory. I am one of the two scientists who propose a Double Torus as a new shape for the universe. Several 'papers' are hosted at 'vixra' about this subject. This paper adds an explanaition to these afore published papers and in particular explains the 'dominance of matter over anti-matter', deduced from the earlier derived 'dark energy force formula' and the perspective of the Double Torus.
Category: Mathematical Physics

Recalculation-mechanism of the Big Bang in a Double Torus Universe.

Authors: Dan Visser
Comments: 5 pages

Here in this 'paper' a deeper analysis of the 'dark energy force formula' in the Double Torus Cosmology is made. Earlier 'papers' were posted as 'pre-papers' in the vixra-archive. These are subject to a new cosmological hypothesis, being further described in higher order mathematics in the near future by Christopher Forbes. I have found evidence through an unexpected interpretation of CMB results, that the 'dark energy force formula' appears to be a functional 'recalculation-mechanism' for quantum gravity.
Category: Mathematical Physics

The Gauge Theory's Expansion in the Electro-Magnetic Field

Authors: Sangwha Yi
Comments: 6 pages.

In the special relativity theory, study the gauge theory in the Electro-magnetic field theory.Using that the Electro-magnetic potential is 4-vector, treat invariant potential. And the Electro-Magnetic field theory's the gauge theory expand.
Category: Mathematical Physics

Exact Solution of Viscous-Plastic Flow Equations for Glacier Dynamics in 2-Dimensional Case.

Authors: Sergey V. Ershkov
Comments: 6 pages

Here is presented a new exact solution of Ice dynamics in Glaciers in terms of viscousplastic theory of movements, for 2-dimensional case: x (t) = y (t). In general case, 2-D solution of Ice dynamics could be classified as Riccati's type. Due to a very special character of Riccati's type equation, it's general solution is proved to have a proper gap of components of such a solution.
Category: Mathematical Physics

Discussion Needed About Three (New) Cosmological Models Based on Mathematics and Physics

Authors: Dan Visser
Comments: 5 pages

Three new cosmological models are 'circling' the science-community: a 'Bouncing Universe', a 'Conformal Cyclic Cosmology and a 'Double Torus Cosmology'. All new ideas, commented anonymously and discussed institutionally. This paper wants more cosmologists and physicists to involve the discussion openly in the media, without walking save roads. The authors and readers are invited to involve this discussion. Which model is true and can the mathematics be matched?
Category: Mathematical Physics

Hard Theoretical Evidence for the Dark Energy Force Formula in a Double Torus Universe.

Authors: Dan Visser
Comments: 6 pages

This paper shows how a 'dark energy force formula' emerges five more space- and two more time-dimensions in nature. The 'formula' is earlier described in vixra-papers, announcing the universe is a double torus of dark energy and dark matter. The 'formula' is a completely different force than the cosmological constant of Einstein used to explain accelerated expansion in the big bang. With this in mind, two independent experimental investigations has given additional proof, that the 'dark energy force formula' correlates the independent investigations, because: 1) Five extra space-dimensions have to exist, according to how electrons behave in graphene-experiments. 2) Computer-simulation shows a double torus geometry, that emerges from two colliding blackholes involving a third torus during pulsation. These results match with the postulate that the universe exists of a double torus of dark energy and dark matter, including a 'dark energy force formula'.
Category: Mathematical Physics

A Few Implications of the Laws of Transactions, from the Abstraction Theory.

Authors: Subhajit Ganguly
Comments: 10 pages.

Considering transport of light through space-time, following the laws of physical transactions (viXra:1101.0035) it may be said that there must be a spreading effect on it. Over suitable distances from a source of light, an observer's perception is bound to be affected due to this spreading. In the following paper, these effects on the reception of a signal, due to the spreading of light are studied. Experimental set-ups are desired to verify the actual angles of spread with their theoretical values. An experiment regarding the minimum distance between two disturbances for them to be distinguishable is also carried out. The energy quantum is also studied in a new light.
Category: Mathematical Physics

Dark Matter Formula For Fundamental Calculation Of Satelite Flybys In Hyperbolic Orbits.

Authors: Dan Visser
Comments: 8 pages

For the first time an announcement is made in this paper to have found fundamental evidence for the flyby-anomalies of six satelites earlier investigated by John Anderson and coworkers of the Jet Propulsion Laboratory (JPL) in Pasadena USA. The central part of this theoretical evidence exists of a 'dark matter impuls flow' being the cause of a velocity-change for satelites during their 'flyby' along the earth. A formula is given to calculate the velocity-change caused by dark matter. Also a dark matter constant is suggested. The origin of the evidence is related to a 'dark energy force formula', which is a new force in a new proposed cosmological model describing dark energy and dark matter in a double torus geometry (TTM). Originally the 'dark energy force formula' is discovered by an independent cosmologist and E-ingenieur, Dan Visser from Almere, the Netherlands. Then afterwards a few 'papers' have been published in the vixra-archive since September 1 2009 in colaboration with mathematician and physicist Christopher Forbes (UK). These 'papers' could be considered as 'pre-exercises' in awaiting for a more robust mathematical framework for the new proposed double torus cosmological model.
Category: Mathematical Physics

A Note on the Action at a Distance

Authors: José Francisco García Juliá
Comments: 3 pages

We consider that the action at a distance is carried out by virtual carriers of the force, which are created and annihilated in the vacuum by the field in a time less than that of the Heisenberg's uncertainty.
Category: Mathematical Physics

The Finite Element Method (Fem)to Finding the Reverberation Times of Irregular Rooms

Authors: Jalil Olia, Vahid Afshinmehr
Comments: 20 pages

In this paper we applied a finite element method to finding the effects on the reverberation times of common irregularities like curved surfaces, non-parallel walls and large open-walled ante-rooms, found in auditoria. The number of modes having a reverberation time in a specified time interval is expressed as a function of the total allowed degrees of freedom and it is shown that even when the number of degrees of freedom of the model is large there is, in general, no one dominant group. Curved surfaces in particular lead to a situation where some modes have very long reverberation times, leading to bad acoustics. In such situations a knowledge of the offending mode shapes give an indication on where to position absorptive material for optimum effect.
Category: Mathematical Physics

Double Torus Cosmology Reveals Cosmic Backround to Measure Dark Energy.

Authors: Dan Visser, Chris Forbes
Comments: 7 pages.

Particularly this paper announces dark energy could be measured as a cosmic backround (CMB)-frame related to a specific quantumstate of dark energy and dark matter conform a double torus cosmology (TTM[1,2,3,4,5,6,7]). In addition this paper also refers to a planned dark energy interferometerproject expected to be operational in 2014[8]. Both aspects can be combined in order to get a better expectation and interpretation of the detection of dark energy. This paper shows dark energy to interact differently than in the planned experiments of the dark energy interferometer-project. That is the motive to publish this paper. Benchmark is dark energy and dark matter are not considered in a big bang cosmology, but in a double torus universe of one torus of dark energy, which encloses and intertwines a second torus of dark matter (TTM-cosmology). In derivations is shown that dark energy will affect falling (super positioned) Cesium atoms in the dark energy interferometerproject unexpectedly: 1. It will touch the super-positioned atoms twice! ; the cause will be the torus-geometry of the TTM-derived CMB-frame. 2. The 'Twice-touching' will vary subsequently; the cause will be a dependency on 'expansion or contraction' of the CMB-torus geometry through dynamics that are caused by the "+" and "-" strength of the dark energy force in the TTM and which is produced by the dark energy torus. This has motivated me to calculate a specific value for TTM-dark energy on about 4 x 10^-114 [X.s] in 6.4 x 10^-48 [m²] as a new sort of spin-quantumstate [X.s], which drives the expansion of big bang cosmology analogue to how elementary spin is a 'generator for rotation' in conventional quantumphysics. Probably this paper might be of interest to the dark energy interferometer-project at least.
Category: Mathematical Physics

A Beautiful Theory of Everything: How Simplexity Leads to Reality!

Authors: Ayind T Mahamba
Comments: 8 Pages. Submissions for FQXi essay contest.

The quest to explain the true nature of reality is one of the great scientific goals. In fact, this essay contest asks: is Nature fundamentally continuous or discrete and how can these two different but very useful concepts be fully reconciled? Physical science is vast, complex and remains mysterious [10]. Since long ago, the great thinkers and scholars have dedicated their lives to the attempted comprehension1 of this reality that has become so abstract. Throughout the centuries and through experimentation, they have established numerous laws, concepts, theories, and principles concerning the fundamental notions of reality (centered on matter-energy and spacetime). I propose a central theory (MIT), based on the information of, and compatible with, the contemporary scientific knowledge; the existing fundamental relation between the "physical entities" passes through the determined quantitative transmission (quantity) of this preserved transcendent greatness (quality). In addition to a "formal" relationship (existence) which creates an informal description of what is real, there is a causal relationship between "phenomena" (relativity). My informational approach has been productive in several domains where many enigma persist; solutions for these problems must be envisaged globally, using ideas and concepts from numerous different fields, with a coherent schema... The "Theory of Universal Relativity" (TUR as a ToE) proposed here lays bridges between domains which, at first glance, have nothing to do with each other; it also provides insight into how we can improve our knowledge by understanding the interplay of complexity and simplicity. Therefore emerging from simplexity (contraction of simplicity and complexity), reality is both digital and analogue (and between) and also more! We know there is a strange and mysterious world that surrounds us, a world largely hidden from our senses with extra dimensions and as a mathematical concept of reality, MIT may confirm that we are part of a cosmic hologram (a paradigm shift). My theory has the advantage of being extremely simple, not limited to scientists because everyone can understand it (I = 1 ± i). So, in this essay, I will try to explain why and how [1][13][48][51].
Category: Mathematical Physics

Quintessence-Momentum as Link Between Mass and Charge

Authors: Malcolm Macleod
Comments: 3 pages.

This paper suggests a 'quantity of momentum', a square root of Planck momentum, here referred to as Quintessence-momentum, as a natural unit that is common to both mass and charge. In terms of this Quintessence momentum Q, alpha (Sommerfeld fine structure constant) and c; geometrical formulas for the natural physical constants and the electron mass are proposed. Results are consistent with CODATA 2006.
Category: Mathematical Physics

Scale Dimension as the Fifth Dimension of Spacetime

Authors: Sergey G. Fedosin
Comments: 5 pages. In Russian

The scale dimension which is discovered in the theory of infinite nesting of matter is studied from the perspective of the physical implementation of well-studied four-and n-dimensional geometric objects. Adding of the scale dimension to Minkowski space means the need to use the five-dimensional spacetime.
Category: Mathematical Physics

On the Stability of Linear Systems

Authors: Daniele Sasso
Comments: 7 pages, 5 figures.

The criteria of stability defined in the standard theory of linear systems aren't exhaustive and show some inconsistencies. In this article we define new criteria of stability more consistent with real physical situations. In particular we distinguish between static stability and dynamic stability in order to analyse the stability of systems in the time domain and in Laplace's equivalent domain. Let introduce then the frequency stability in order to analyse the stability of systems in the Fourier domain.
Category: Mathematical Physics

Entanglement Related to Cosmology-TTM

Authors: Dan Visser, Christopher Forbes
Comments: 10 pages

This paper postulates a theoretical structure within entangled photons. The postulate is introduced within the framework of the cosmological Twin-Tori Model (TTM). Related papers are to be found in viXra[1,2,3,4,5,6]. After generally derived equations and interpretations, a mass (m_t) per 2π is calculated on ~ 2.6 x 10^-34 [(J.s) m² / s]. Such a tiny spinning-forward surface per second (torus geometry) has an energy much smaller than the Planckenergy in Joule, suggesting a subdivision of 10⁴³ surfaces below the Planckscale: If one photon changes spin, the entangled photon could follow by means of the spinning-forward tiny surface-structure within the torus geometry (per 2π), being an information-flux for entanglement in general below the Planckscale.
Category: Mathematical Physics

Hidden Mathematical Symmetries in the 32 Crystal Point Groups?

Authors: Giuliano Bettini
Comments: 10 pages

In a preceding paper we introduced a conjecture: the classification of the 32 crystal classes with 5 bits. In the present paper we will review our preceding result, and continue showing some further interesting issues. In the paper, it is argued that bits should be identified with five basic unknown symmetries generating these 32 groups. Probably it is not merely a coincidence that 32 means 5 bits. And probably is it not merely a coincidence that each complete subset of bits (properties) means the holohedry of a crystal system; and each new bit means a new crystal system. The purpose of this article was of course not to draw a conclusive theory, but to suggest ideas that, we hope, will be useful for researchers in mathematics, group theory and crystallography.
Category: Mathematical Physics

Fine Structure Constant α ~ 1/137.036 and Blackbody Radiation Constant α_R ~ 1/157.555

Authors: Ke Xiao
Comments: 5 pages

The fine structure constant α = e²/hc ~ 1/137.036 and the blackbody radiation constant α_R = e²(a_R/k⁴_B)^1/3 ~ 1/157.555 are linked by prime numbers. The blackbody radiation constant is a new method to measure the fine structure constant. It also links the fine structure constant to the Boltzmann constant.
Category: Mathematical Physics

Abstraction In Theory - Laws of Physical Transactions

Authors: Subhajit Ganguly
Comments: 15 pages

Considering transport or tendency ... (see paper)
Category: Mathematical Physics

32 Point Groups of Three Dimensional Crystal Cells Described by 5 Bits

Authors: Giuliano Bettini
Comments: 9 pages, in Italian

There are 32 possible combinations of symmetry operations that define the external symmetry of crystals. These 32 possible combinations result in the 32 crystal classes. But for a radar engineer it is inevitable to associate "32" to "5 bits". I submit a tentative classification of the 32 crystal classes with a 5 bit classification, obviously with a (tentative) physical meaning of each bit. Each bit means a physical property.
Category: Mathematical Physics

Further on Non-Cartesian Systems

Authors: Elemér E Rosinger
Comments: 9 pages

A class of non-Cartesian physical systems, [7], are those whose composite state spaces are given by significantly extended tensor products. A more detailed presentation of the way such extended tensor products are constructed is offered, based on a step by step comparison with the construction of usual tensor products. This presentation clarifies the extent to which the extended tensor products are indeed more general than the usual ones.
Category: Mathematical Physics

Non-Cartesian Systems : an Open Problem

Authors: Elemer E Rosinger
Comments: 6 pages

The following open problem is presented and motivated : Are there physical systems whose state spaces do not compose according to either the Cartesian product, as classical systems do, or the usual tensor product, as quantum systems do ?
Category: Mathematical Physics

Four Departures in Mathematics and Physics

Authors: Elemer E Rosinger
Comments: 28 pages

Much of Mathematics, and therefore Physics as well, have been limited by four rather consequential restrictions. Two of them are ancient taboos, one is an ancient and no longer felt as such bondage, and the fourth is a surprising omission in Algebra. The paper brings to the attention of those interested these four restrictions, as well as the fact that each of them has by now ways, even if hardly yet known ones, to overcome them.
Category: Mathematical Physics

Some Comments on Projective Quadrics Subordinate to Pseudo-Hermitian Spaces

Authors: Arkadiusz Jadczyk
Comments: 7 pages, To appear in Advances in Applied Clifford Algebras

We study in some detail the structure of the projective quadric Q' obtained by taking the quotient of the isotropic cone in a standard pseudohermitian space H_p,q with respect to the positive real numbers R⁺ and, further, by taking the quotient ~Q = Q'/U(1). The case of signature (1. 1) serves as an illustration. ~Q is studied as a compactification of RxH_p-1,q-1
Category: Mathematical Physics

Some Orbital and Other Properties of the 'Special Gravitating Annulus'

Authors: Guy Moore, Richard Moore
Comments: 40 pages

Our obtaining the analytical equations for the gravitation of a particular type of mathematical annulus, which we called a 'Special Gravitating Annulus' (SGA), greatly facilitates studying its orbital properties by computer programming. This includes isomorphism, periodic and chaotic polar orbits, and orbits in three dimensions. We provide further insights into the gravitational properties of this annulus and describe our computer algorithms and programs. We study a number of periodic orbits, giving them names to aid identification. 'Ellipses extraordinaires' which are bisected by the annulus, have no gravitating matter at either focus and represent a fundamental departure from the normal association of elliptical orbits with Keplerian motion. We describe how we came across this type of orbit and the analysis we performed. We present the simultaneous differential equations of motion of 'ellipses extraordinaires' and other orbits as a mathematical challenge. The 'St.Louis Gateway Arch' orbit contains two 'instantaneous static points' (ISP). Polar elliptical orbits can wander considerably without tending to form other kinds of orbit. If this type of orbit is favoured then this gives a similarity to spiral galaxies containing polar orbiting material. Annular oscillatory orbits and rotating polar elliptical orbits are computed in isometric projection. A 'daisy' orbit is computed in stereo-isometric projection. The singularity at the centre of the SGA is discussed in relation to mechanics and computing, and it appears mathematically different from a black hole. In the Appendix, we prove by a mathematical method that a thin plane self-gravitating Newtonian annulus, free from external influence, exhibiting radial gravitation that varies inversely with the radius in the annular plane, must have an area mass density which also varies inversely with the radius and this exact solution is the only exact solution.
Category: Mathematical Physics

Dark Matter and Visible Matter Fundamentally Related in New Cosmological Model and Recalculated.

Authors: Dan Visser
Comments: 7 pages

A new surface energy-value for dark matter is calculated, derivated from a perspective of a "higher order universe". A universe of dark energy, dark- and visible matter, and a dark energy force. A fundamental connection between dark matter and visible matter is related to dark energy (viXra-paper 1010.0014 in particular is the reference for this novum [5]. The surface energy-density of dark matter seems to be a factor 5 to 20 times higher than earlier predicted value-ranges by the CDMS-project and the Fermi-satelite. Also the produced energies through particle-collissions by LHC CERN will not be enough to achieve the dark matter surface energy-value. The "fact" some of these projects have announced some vaque "bliebs" might be due to a new phenomenon in the search of dark matter. It could be caused by "three dimensional time", which is embedded as a "new duality" in the new model, the "Twin-Tori cosmological Model (TTM)"[1,2,3,4]. The "three dimensional time" might cause dark matter taking unknown paths before detection. This paper has calculated the surface energy-density value for dark matter on 1 TeV in a surface of 6.4 x 10 ^-48 [m2].
Category: Mathematical Physics

On Using Quaternion Operators in Quantum and Microphysics.

Authors: Debayan Dasgupta
Comments: 10 pages

The main purpose of this paper is to show the application of mathematical system of quaternions in physics. We show a basic way of generating a quaternion operator and a method of obtaining the Schrodinger's equation using this operator matrix. Then we investigate the various probable uses of the coefficient matrix to scale relativity and spacetime quantization.
Category: Mathematical Physics

Deeper Properties Through Dark and Visible-Matter in a New Cosmological Twin-Tori Model (TTM).

Authors: Dan Visser
Comments: 5 pages

A new cosmological model, named the Twin-Tori Model (TTM)[1], postulates a dark energy force F_de , which empowers the dynamic of a lower order universe, well known as the big bang. In this paper I introduce the 1st derivative F'_de of this dark energy force to reveal deeper properties of the TTM, such as: why quantummechanics exists in the big bang, why dark matter and visible matter are equally responsible for gravity in galaxies for 1/4 of the density of dark matter at a specific length, why the big bang universe is recalculated by subquantumlevel-information below the Plancklength, and why the impression of space-expansion is due to the higher order cosmological model TTM.
Category: Mathematical Physics

A Multiple Particle System Equation Underlying the Klein-Gordon-Dirac-Schrödinger Equations

Authors: D.T. Froedge
Comments: 16 pages 38 equations 98kb

The purpose of this paper is to illustrate a fundamental, multiple particle, system equation for which the Klein-Gordon-Dirac-Schrödinger equations are single particle special cases. In the same manner that eigenvalues of the Schrödinger equation represents energy levels of an interacting atomic system, eigenvalues represent particle energies in an interacting system of particles. An equation is proposed that has vector solutions defined in Dirac, or Clifford algebra, that treats a collection of particles as a single system..The proposed solution is a descriptor of a symmetric, light speed expanding group of interacting particles having real, as well as the familiar QM constituents.
Category: Mathematical Physics

A New Generic Class of Beltrami "Force-Free" Fields. Part-I: Theoretical Considerations

Authors: T. E. Raptis
Comments: 13 pages.

We report on a new general class of solutions of the Beltrami equation, with special characteristics. We also provide examples of solutions that also satisfy Maxwell equations. A subset of these solutions can be isolated which corresponds to "gauge" fields. A special projective geometry of vacuum fields is also revealed and discussed.
Category: Mathematical Physics

Causal Set Theory and the Origin of Mass-ratio

Authors: Carey R Carlson
Comments: 16 pages.

Quantum theory is reconstructed using standalone causal sets. The frequency ratios inherent in causal sets are used to define energy-ratios, implicating the causal link as the quantum of action. Space-time and its particle-like sequences are then constructed from causal links. A 4-D time-lattice structure is defined and then used to model neutrinos and electron clouds, which together constitute a 4-D manifold. A 6-D time-lattice is used to model the nucleons. The integration of the nucleus with its electron cloud affords calculation of the mass-ratio of the proton (or the neutron) with respect to the electron. Arrow diagrams, along with several ball-and-stick models, are used to streamline the presentation.
Category: Mathematical Physics

Handbook of Functions Errata

Authors: Fredy Zypman
Comments: 2 pages

Formulas connecting toroidal functions and elliptical functions are useful in various areas of physics. In solving a problem in electrostatics we run across an error in the Handbook of mathematical functions of Abramowitz and Stegun. In this paper we report the details.
Category: Mathematical Physics

The Geometry of CP₂ and its Relationship to Standard Model

Authors: Matti Pitkänen
Comments: 13 Pages.

This appendix contains basic facts about CP₂ as a symmetric space and Kähler manifold. The coding of the standard model symmetries to the geometry of CP₂, the physical interpretation of the induced spinor connection in terms of electro-weak gauge potentials, and basic facts about induced gauge fields are discussed
Category: Mathematical Physics

Could the Dynamics of Kähler Action Predict the Hierarchy of Planck Constants?

Authors: Matti Pitkänen
Comments: 5 Pages.

The original justification for the hierarchy of Planck constants came from the indications that Planck constant could have large values in both astrophysical systems involving dark matter and also in biology. The realization of the hierarchy in terms of the singular coverings and possibly also factor spaces of CD and CP₂ emerged from consistency conditions. It however seems that TGD actually predicts this hierarchy of covering spaces. The extreme non-linearity of the field equations defined by Kähler action means that the correspondence between canonical momentum densities and time derivatives of the imbedding space coordinates is 1-to-many. This leads naturally to the introduction of the covering space of CD x CP₂, where CD denotes causal diamond defined as intersection of future and past directed light-cones.
Category: Mathematical Physics

Weak Form of Electric-Magnetic Duality, Electroweak Massivation, and Color Confinement

Authors: Matti Pitkänen
Comments: 13 Pages.

The notion of electric magnetic duality emerged already two decades ago in the attempts to formulate the Kähler geometry of the "world of classical worlds". Quite recently a considerable step of progress took place in the understanding of this notion. This concept leads to the identification of the physical particles as string like objects defined by magnetic charged wormhole throats connected by magnetic ux tubes. The second end of the string contains particle having electroweak isospin neutralizing that of elementary fermion and the size scale of the string is electro-weak scale would be in question. Hence the screening of electro-weak force takes place via weak confinement. This picture generalizes to magnetic color confinement. Electric-magnetic duality leads also to a detailed understanding of how TGD reduces to almost topological quantum field theory. A surprising outcome is the necessity to replace CP₂ Kähler form in Kähler action with its sum with S² Kähler form.
Category: Mathematical Physics

How to Define Generalized Feynman Diagrams?

Authors: Matti Pitkänen
Comments: 16 Pages.

Generalized Feynman diagrams have become the central notion of quantum TGD and one might even say that space-time surfaces can be identified as generalized Feynman diagrams. The challenge is to assign a precise mathematical content for this notion, show their mathematical existence, and develop a machinery for calculating them. Zero energy ontology has led to a dramatic progress in the understanding of generalized Feynman diagrams at the level of fermionic degrees of freedom. In particular, manifest finiteness in these degrees of freedom follows trivially from the basic identifications as does also unitarity and non-trivial coupling constant evolution. There are however several formidable looking challenges left.

1. One should perform the functional integral over WCW degrees of freedom for fixed values of on mass shell momenta appearing in the internal lines. After this one must perform integral or summation over loop momenta.
2. One must define the functional integral also in the p-adic context. p-Adic Fourier analysis relying on algebraic continuation raises hopes in this respect. p-Adicity suggests strongly that the loop momenta are discretized and ZEO predicts this kind of discretization naturally.

Physics as Generalized Number Theory: Infinite Primes

Authors: Matti Pitkänen
Comments: 37 Pages.

Physics as a generalized number theory program involves three threads: various p-adic physics and their fusion together with real number based physics to a larger structure, the attempt to understand basic physics in terms of classical number fields, and infinite primes discussed in this article. The construction of infinite primes is formally analogous to a repeated second quantization of an arithmetic quantum field theory by taking the many particle states of previous level elementary particles at the new level. Besides free many particle states also the analogs of bound states appear. In the representation in terms of polynomials the free states correspond to products of first order polynomials with rational zeros. Bound states correspond to nth order polynomials with non-rational but algebraic zeros. The construction can be generalized to classical number fields and their complexifications obtained by adding a commuting imaginary unit. Special class corresponds to hyper-octonionic primes for which the imaginary part of ordinary octonion is multiplied by the commuting imaginary unit so that one obtains a sub-space M⁸ with Minkowski signature of metric. Also in this case the basic construction reduces to that for rational or complex rational primes and more complex primes are obtained by acting using elements of the octonionic automorphism group which preserve the complex octonionic integer property. Can one map infinite primes/integers/rationals to quantum states? Do they have space-time surfaces as correlates? Quantum classical correspondence realized in terms of modified Dirac operator implies that if infinite rationals can be mapped to quantum states then the mapping of quantum states to space-time surfaces automatically gives the map to space-time surfaces. The question is therefore whether the mapping to quantum states defined by WCW spinor fields is possible. A natural hypothesis is that number theoretic fermions can be mapped to real fermions and number theoretic bosons to WCW ("world of classical worlds") Hamiltonians. The crucial observation is that one can construct infinite hierarchy of hyper-octonionic units by forming ratios of infinite integers such that their ratio equals to one in real sense: the integers have interpretation as positive and negative energy parts of zero energy states. One can construct also sums of these units with complex coefficients using commuting imaginary unit and these sums can be normalized to unity and have interpretation as states in Hilbert space. These units can be assumed to possess well defined standard model quantum numbers. It is possible to map the quantum number combinations of WCW spinor fields to these states. Hence the points of M⁸ can be said to have infinitely complex number theoretic anatomy so that quantum states of the universe can be mapped to this anatomy. One could talk about algebraic holography or number theoretic Brahman=Atman identity. One can also ask how infinite primes relate to the p-adicization program and to the hierarchy of Planck constants. The key observation is that infinite primes are in one-one correspondence with rational numbers at the lower level of hierarchy. At the first level of hierarchy the p-adic norm with respect to p-adic prime for this rational gives power p^-n so that one has two powers of p - pⁿ⁺ and p^n- since two infinite primes corresponding to fermionic vacua X±1, where X is the product of all primes at given level of hierarchy, characterize the partonic 2-surface. The proposal inspired by the p-adicization program is that Δφ = 2π/pⁿ defines angle measurement resolution crucial in the construction of p-adic variants of WCW ("world of classical world") as a union of symmetric coset spaces by starting from discrete variants of the real counterpart of symmetric space having common points with tis p-adic variant. The two measurement resolutions correspond to CD and CP₂ degrees of freedom. The hierarchy of Planck constants generalizes imbedding space to a book like structure with pages identified in terms of singular coverings and factor spaces of CD and CP₂. There are good arguments suggesting that only coverings characterized by integers n_a and n_bare realized. The identifications na = n₊ and nb = n- lead to highly non-trivial physical predictions and allow sharpen the view about the hierarchy of Planck constants. Therefore the notion of finite measurement resolution becomes the common denominator for the three threads of the number theoretic vision and give also a connection with the physics as infinite-dimensional geometry program and with the inclusions of hyper-finite factors defined by WCW spinor fields and proposed to characterize finite measurement resolution at quantum level.
Category: Mathematical Physics

Physics as Generalized Number Theory: Classical Number Fields

Authors: Matti Pitkänen
Comments: 28 Pages.

Physics as a generalized number theory program involves three threads: various p-adic physics and their fusion together with real number based physics to a larger structure, the attempt to understand basic physics in terms of classical number fields discussed in this article, and infinite primes whose construction is formally analogous to a repeated second quantization of an arithmetic quantum field theory. In this article the connection between standard model symmetries and classical number fields is discussed. The basis vision is that the geometry of the infinite-dimensional WCW ("world of classical worlds") is unique from its mere existence. This leads to its identification as union of symmetric spaces whose Kähler geometries are fixed by generalized conformal symmetries. This fixes space-time dimension and the decomposition M⁴ x S and the idea is that the symmetries of the Kähler manifold S make it somehow unique. The motivating observations are that the dimensions of classical number fields are the dimensions of partonic 2-surfaces, space-time surfaces, and imbedding space and M⁸ can be identified as hyper-octonions- a sub-space of complexified octonions obtained by adding a commuting imaginary unit. This stimulates some questions. Could one understand S = CP₂ number theoretically in the sense that M⁸ and H = M⁴ x CP₂ be in some deep sense equivalent ("number theoretical compactification" or M⁸ - H duality)? Could associativity define the fundamental dynamical principle so that space-time surfaces could be regarded as associative or co-associative (defined properly) sub-manifolds of M⁸ or equivalently of H. One can indeed define the associativite (co-associative) 4-surfaces using octonionic representation of gamma matrices of 8-D spaces as surfaces for which the modified gamma matrices span an associate (co-associative) sub-space at each point of space-time surface. Also M⁸ - H duality holds true if one assumes that this associative sub-space at each point contains preferred plane of M⁸ identifiable as a preferred commutative or co-commutative plane (this condition generalizes to an integral distribution of commutative planes in M⁸). These planes are parametrized by CP₂ and this leads to M⁸ - H duality. WCW itself can be identified as the space of 4-D local sub-algebras of the local Clifford algebra of M⁸ or H which are associative or co-associative. An open conjecture is that this characterization of the space-time surfaces is equivalent with the preferred extremal property of Kähler action with preferred extremal identified as a critical extremal allowing infinite-dimensional algebra of vanishing second variations.
Category: Mathematical Physics

Physics as Generalized Number Theory: P-Adic Physics and Number Theoretic Universality

Authors: Matti Pitkänen
Comments: 51 Pages.

Physics as a generalized number theory program involves three threads: various p-adic physics and their fusion together with real number based physics to a larger structure, the attempt to understand basic physics in terms of classical number fields (in particular, identifying associativity condition as the basic dynamical principle), and infinite primes whose construction is formally analogous to a repeated second quantization of an arithmetic quantum field theory. In this article p-adic physics and the technical problems relates to the fusion of p-adic physics and real physics to a larger structure are discussed. The basic technical problems relate to the notion of definite integral both at space-time level, imbedding space level and the level of WCW (the "world of classical worlds"). The expressibility of WCW as a union of symmetric spacesleads to a proposal that harmonic analysis of symmetric spaces can be used to define various integrals as sums over Fourier components. This leads to the proposal the p-adic variant of symmetric space is obtained by a algebraic continuation through a common intersection of these spaces, which basically reduces to an algebraic variant of coset space involving algebraic extension of rationals by roots of unity. This brings in the notion of angle measurement resolution coming as Δφ = 2π/pⁿ for given p-adic prime p. Also a proposal how one can complete the discrete version of symmetric space to a continuous p-adic versions emerges and means that each point is effectively replaced with the p-adic variant of the symmetric space identifiable as a p-adic counterpart of the real discretization volume so that a fractal p-adic variant of symmetric space results. If the Kähler geometry of WCW is expressible in terms of rational or algebraic functions, it can in principle be continued the p-adic context. One can however consider the possibility that that the integrals over partonic 2-surfaces defining ux Hamiltonians exist p-adically as Riemann sums. This requires that the geometries of the partonic 2-surfaces effectively reduce to finite sub-manifold geometries in the discretized version of δM₊⁴. If Kähler action is required to exist p-adically same kind of condition applies to the space-time surfaces themselves. These strong conditions might make sense in the intersection of the real and p-adic worlds assumed to characterized living matter.
Category: Mathematical Physics

Construction of Configuration Space Spinor Structure

Authors: Matti Pitkänen
Comments: 34 Pages.

There are three separate approaches to the challenge of constructing WCW Kähler geometry and spinor structure. The first approach relies on a direct guess of Kähler function. Second approach relies on the construction of Kähler form and metric utilizing the huge symmetries of the geometry needed to guarantee the mathematical existence of Riemann connection. The third approach discussed in this article relies on the construction of spinor structure based on the hypothesis that complexified WCW gamma matrices are representable as linear combinations of fermionic oscillator operator for the second quantized free spinor fields at space-time surface and on the geometrization of super-conformal symmetries in terms of spinor structure. This implies a geometrization of fermionic statistics. The basic philosophy is that at fundamental level the construction of WCW geometry reduces to the second quantization of the induced spinor fields using Dirac action. This assumption is parallel with the bosonic emergence stating that all gauge bosons are pairs of fermion and antifermion at opposite throats of wormhole contact. Vacuum function is identified as Dirac determinant and the conjecture is that it reduces to the exponent of Kähler function. In order to achieve internal consistency induced gamma matrices appearing in Dirac operator must be replaced by the modified gamma matrices defined uniquely by Kähler action and one must also assume that extremals of Kähler action are in question so that the classical space-time dynamics reduces to a consistency condition. This implies also super-symmetries and the fermionic oscillator algebra at partonic 2-surfaces has intepretation as N = 1 generalization of space-time supersymmetry algebra different however from standard SUSY algebra in that Majorana spinors are not needed. This algebra serves as a building brick of various super-conformal algebras involved. The requirement that there exist deformations giving rise to conserved Noether charges requires that the preferred extremals are critical in the sense that the second variation of the Kähler action vanishes for these deformations. Thus Bohr orbit property could correspond to criticality or at least involve it. Quantum classical correspondence demands that quantum numbers are coded to the properties of the preferred extremals given by the Dirac determinant and this requires a linear coupling to the conserved quantum charges in Cartan algebra. Effective 2-dimensionality allows a measurement interaction term only in 3-D Chern-Simons Dirac action assignable to the wormhole throats and the ends of the space-time surfaces at the boundaries of CD. This allows also to have physical propagators reducing to Dirac propagator not possible without the measurement interaction term. An essential point is that the measurement interaction corresponds formally to a gauge transformation for the induced Kähler gauge potential. If one accepts the weak form of electric-magnetic duality Kähler function reduces to a generalized Chern-Simons term and the effect of measurement interaction term to Kähler function reduces effectively to the same gauge transformation. The basic vision is that WCW gamma matrices are expressible as super-symplectic charges at the boundaries of CD. The basic building brick of WCW is the product of infinite-D symmetric spaces assignable to the ends of the propagator line of the generalized Feynman diagram. WCW Kähler metric has in this case "kinetic" parts associated with the ends and "interaction" part between the ends. General expressions for the super-counterparts of WCW ux Hamiltoniansand for the matrix elements of WCW metric in terms of their anticommutators are proposed on basis of this picture.
Category: Mathematical Physics

Construction of Configuration Space Geometry from Symmetry Principles

Authors: Matti Pitkänen
Comments: 26 Pages.

There are three separate approaches to the challenge of constructing WCW Kähler geometry and spinor structure. The first one relies on a direct guess of Kähler function. Second approach relies on the construction of Kähler form and metric utilizing the huge symmetries of the geometry needed to guarantee the mathematical existence of Riemann connection. The third approach relies on the construction of spinor structure assuming that complexified WCW gamma matrices are representable as linear combinations of fermionic oscillator operator for the second quantized free spinor fields at space-time surface and on the geometrization of super-conformal symmetries in terms of spinor structure. In this article the construction of Kähler form and metric based on symmetries is discussed. The basic vision is that WCW can be regarded as the space of generalized Feynman diagrams with lines thickned to light-like 3-surfaces and vertices identified as partonic 2-surfaces. In zero energy ontology the strong form of General Coordinate Invariance (GCI) implies effective 2-dimensionality and the basic objects are pairs partonic 2-surfaces X² at opposite light-like boundaries of causal diamonds (CDs). The hypothesis is that WCW can be regarded as a union of infinite-dimensional symmetric spaces G/H labeled by zero modes having an interpretation as classical, non-quantum uctuating variables. A crucial role is played by the metric 2-dimensionality of the light-cone boundary δM₊⁴ + and of light-like 3-surfaces implying a generalization of conformal invariance. The group G acting as isometries of WCW is tentatively identified as the symplectic group of δM₊⁴ x CP₂ localized with respect to X². H is identified as Kac-Moody type group associated with isometries of H = M₊⁴ x CP₂ acting on light-like 3-surfaces and thus on X². An explicit construction for the Hamiltonians of WCW isometry algebra as so called ux Hamiltonians is proposed and also the elements of Kähler form can be constructed in terms of these. Explicit expressions for WCW ux Hamiltonians as functionals of complex coordinates of the Cartesisian product of the infinite-dimensional symmetric spaces having as points the partonic 2-surfaces defining the ends of the the light 3-surface (line of generalized Feynman diagram) are proposed.
Category: Mathematical Physics

Identification of the Configuration Space Kähler Function

Authors: Matti Pitkänen
Comments: 38 Pages.

There are two basic approaches to quantum TGD. The first approach, which is discussed in this article, is a generalization of Einstein's geometrization program of physics to an infinitedimensional context. Second approach is based on the identification of physics as a generalized number theory. The first approach relies on the vision of quantum physics as infinite-dimensional Kähler geometry for the "world of classical worlds" (WCW) identified as the space of 3-surfaces in in certain 8-dimensional space. There are three separate approaches to the challenge of constructing WCW Kähler geometry and spinor structure. The first approach relies on direct guess of Kähler function. Second approach relies on the construction of Kähler form and metric utilizing the huge symmetries of the geometry needed to guarantee the mathematical existence of Riemann connection. The third approach relies on the construction of spinor structure based on the hypothesis that complexified WCW gamma matrices are representable as linear combinations of fermionic oscillator operator for second quantized free spinor fields at space-time surface and on the geometrization of super-conformal symmetries in terms of WCW spinor structure. In this article the proposal for Kähler function based on the requirement of 4-dimensional General Coordinate Invariance implying that its definition must assign to a given 3-surface a unique space-time surface. Quantum classical correspondence requires that this surface is a preferred extremal of some some general coordinate invariant action, and so called Kähler action is a unique candidate in this respect. The preferred extremal has intepretation as an analog of Bohr orbit so that classical physics becomes and exact part of WCW geometry and therefore also quantum physics. The basic challenge is the explicit identification of WCW Kähler function K. Two assumptions lead to the identification of K as a sum of Chern-Simons type terms associated with the ends of causal diamond and with the light-like wormhole throats at which the signature of the induced metric changes. The first assumption is the weak form of electric magnetic duality. Second assumption is that the Kähler current for preferred extremals satisfies the condition jK ^ djK = 0 implying that the ow parameter of the ow lines of jK defines a global space-time coordinate. This would mean that the vision about reduction to almost topological QFT would be realized. Second challenge is the understanding of the space-time correlates of quantum criticality. Electric-magnetic duality helps considerably here. The realization that the hierarchy of Planck constant realized in terms of coverings of the imbedding space follows from basic quantum TGD leads to a further understanding. The extreme non-linearity of canonical momentum densities as functions of time derivatives of the imbedding space coordinates implies that the correspondence between these two variables is not 1-1 so that it is natural to introduce coverings of CD x CP₂. This leads also to a precise geometric characterization of the criticality of the preferred extremals.
Category: Mathematical Physics

Physics as Infinite-dimensional Geometry and Generalized Number Theory: Basic Visions

Authors: Matti Pitkänen
Comments: 32 Pages.

There are two basic approaches to the construction of quantum TGD. The first approach relies on the vision of quantum physics as infinite-dimensional Kähler geometry for the "world of classical worlds" identified as the space of 3-surfaces in in certain 8-dimensional space. Essentially a generalization of the Einstein's geometrization of physics program is in question. The second vision is the identification of physics as a generalized number theory. This program involves three threads: various p-adic physics and their fusion together with real number based physics to a larger structure, the attempt to understand basic physics in terms of classical number fields (in particular, identifying associativity condition as the basic dynamical principle), and infinite primes whose construction is formally analogous to a repeated second quantization of an arithmetic quantum field theory. In this article brief summaries of physics as infinite-dimensional geometry and generalized number theory are given to be followed by more detailed articles.
Category: Mathematical Physics

Unfolding the Labyrinth: Open Problems in Physics, Mathematics, Astrophysics, and Other Areas of Science

Authors: Florentin Smarandache, V. Christianto, Fu Yuhua, Radi I. Khrapko, J. Hutchison
Comments: 147 pages

The reader will find herein a collection of unsolved problems in mathematics and the physical sciences. Theoretical and experimental domains have each been given consideration. The authors have taken a liberal approach in their selection of problems and questions, and have not shied away from what might otherwise be called speculative, in order to enhance the opportunities for scientific discovery. Progress and development in our knowledge of the structure, form and function of the Universe, in the true sense of the word, its beauty and power, and its timeless presence and mystery, before which even the greatest intellect is awed and humbled, can spring forth only from an unshackled mind combined with a willingness to imagine beyond the boundaries imposed by that ossified authority by which science inevitably becomes, as history teaches us, barren and decrepit. Revealing the secrets of Nature, so that we truly see 'the sunlit plains extended, and at night the wondrous glory of the everlasting stars', requires far more than mere technical ability and mechanical dexterity learnt form books and consensus. The dustbin of scientific history is replete with discredited consensus and the grand reputations of erudite reactionaries. Only by boldly asking questions, fearlessly, despite opposition, and searching for answers where most have not looked for want of courage and independence of thought, can one hope to discover for one's self. From nothing else can creativity blossom and grow, and without which the garden of science can only aspire to an overpopulation of weeds.
Category: Mathematical Physics

A 'Planck-like' Characterization of Exponential Function

Authors: Constantinos Ragazas
Comments: 8 pages

We derive a characterization of simple exponential functions that has the exact mathematical form to Planck's Formula for blackbody radiation in Quantum Physics.
Category: Mathematical Physics

Mathematical Model of Information

Authors: Elemér E Rosinger
Comments: 17 pages.

A simple and rather general mathematical model of the phenomenon of information is presented, followed by several examples and comments.
Category: Mathematical Physics

Invariants Relative to Change of Value of the Independent Variable and Their Role in the Physics.

Authors: Vladimir I. Smirnov
Comments: 34 pages, Russian and English versions included

It is identified the new class of invariants which values are constant at change of value of an inde-pendent variable. Their properties and a deriving method are shown on already known and still un-known instances, concerning to various areas of physics. In particular, new invariants (50), (55) and (57) for the straight lines intersected in one point on a plane have been discovered. Besides, the re-quest for detection of the third (not dependent on two already known) an invariant (31) electromag-netic fields for a special case of the special theory of relativity is made.
Category: Mathematical Physics

Mathematical and Phenomenological Elements of the Twin-Tori Model of Physics and Cosmology.

Authors: Chris Forbes
Comments: 10 pages

In this, a follow up to a previous paper 'A Short Article On A Newly Proposed Model Of Cosmology' (viXra:0909.0005), some of the basic mathematical structures to be used in the formulation of the model are shown, and several advantages are discussed. The paper then takes a more phenomenological approach and several simple (1+1) dimensional models are explored.
Category: Mathematical Physics

A First Order Singular Perturbation Solution to a Simple One-Phase Stefan Problem with Finite Neumann Boundary Conditions

Authors: Bruce Rout
Comments: 13 pages

This paper examines the difference between infinite and finite domains of a Stefan Problem. It is pointed out that attributes of solutions to the Diffusion Equation suggest assumptions of an infinite domain are invalid during initial times for finite domain Stefan Problems. The paper provides a solution for initial and early times from an analytical approach using a perturbation. This solution can then easily be applied to numerical models for later times. The differences of the two domains are examined and discussed.
Category: Mathematical Physics

Cylindrical Wave, Wave Equation, and Method of Separation of Variables

Authors: Hamid V. Ansari
Comments: 7 pages

It is shown that the wave equation cannot be solved for the general spreading of the cylindrical wave using the method of separation of variables. But an equation is presented in case of its solving the above act will have occurred. Also using this equation the above-mentioned general spreading of the cylindrical wave for large distances is obtained which contrary to what is believed consists of arbitrary functions.
Category: Mathematical Physics

New Calculuses

Authors: Vladislav Konovalov
Comments: 12 pages

In the article the new calculuses are offered similar differential and integral, but differing, that in them the analysis of the previous and subsequent values of a function is made. The new calculuses allow to decide problems, the solution which one with usage customary differential and integral calculus is impossible.
Category: Mathematical Physics

3x3 Unitary to Magic Matrix Transformations

Authors: Philip Gibbs
Comments: 5 pages, published in Prespacetime Journal, V5

We prove that any 3x3 unitary matrix can be transformed to a magic matrix by multiplying its rows and columns by phase factors. A magic matrix is defined as one for which the sum of the elements in any row or column add to the same value. This result is relevant to recent observations on particle mixing matrices.
Category: Mathematical Physics

Topological Maxwell Field Theory and Symmetry Breaking.

Authors: R. M. Kiehn
Comments: recovered from sciprint.org

Finally, I have found time to think about, and the incentive to study, how the field theory of Topological thermodynamics, electrodynamics, and hydrodynamics can be compared to field theory concepts that have been developed by Lagrangian methods, for both the classic and quantum mechanical varieties. For more than 30 years I have known that Cartan's topological methods could be applied to dissipative systems; the methods based on diffeomorphic-invariant Lagrangian field theories can not. The incentive came when I realized that the topological methods of Cartan gave dynamical results that can explain "symmetry breaking" and quantization in terms of continuous topological evolution.
Category: Mathematical Physics

A Chern-Simons E8 Gauge Theory of Gravity in D = 15, Grand Unification and Generalized Gravity in Clifford Spaces

Authors: Carlos Castro
Comments: recovered from sciprint.org

A novel Chern-Simons E8 gauge theory of Gravity in D = 15 based on an octic E8 invariant expression in D = 16 (recently constructed by Cederwall and Palmkvist) is developed. A grand unification model of gravity with the other forces is very plausible within the framework of a supersymmetric extension (to incorporate spacetime fermions) of this Chern-Simons E8 gauge theory. We review the construction showing why the ordinary 11D Chern-Simons Gravity theory (based on the Anti de Sitter group) can be embedded into a Clifford-algebra valued gauge theory and that an E8 Yang-Mills field theory is a small sector of a Clifford (16) algebra gauge theory. An E8 gauge bundle formulation was instrumental in the understanding the topological part of the 11-dim M-theory partition function. The nature of this 11-dim E8 gauge theory remains unknown. We hope that the Chern-Simons E8 gauge theory of Gravity in D = 15 advanced in this work may shed some light into solving this problem after a dimensional reduction.
Category: Mathematical Physics

Quantization in Dynamic Smarandache Multi-Space

Authors: Fu Yuhua
Comments: recovered from sciprint.org

Discussing the applications of Dynamic Smarandache Multi-Space (DSMS) Theory. Supposing for the n different dynamic spaces (n is a dynamic positive integer and the function of time) the different equations have been established, as these n different dynamic spaces synthesize the DSMS, and they are mutually affected, some new coupled equations need to establish in the DSMS to replace some equations in the original dynamic spaces, as well as supply other equations to process the contact, boundary conditions and so on. For the unified processing of all equations in the DSMS, this paper proposes to run the quantization processing to all the variables and all the equations and establish the unified variational principle of quantization with the collocation method based on the method of weighted residuals, and simultaneously solve all the equations in the DSMS with the optimization method. Thus by using the unified variational principle of quantization in the DSMS and the fractal quantization method, will pave the way for the unified processing of the theory of relativity and the quantum mechanics, and the unified processing of the four foundational interactions. Finally the coupled solution for the problem of relativity and quantum mechanics is discussed.
Category: Mathematical Physics

Smarandache Stepped Functions

Authors: Mircea Eugen Selariu
Comments: recovered from sciprint.org

The discovery of mathematical complements, assembled under the name of the eccentric mathematics, gave the opportunity for a series of applications, amongst which, in this article, are presented the impulse, step, and unitary ramp functions. The difference, in comparison with the same classic functions, from the distributions theory, is that those eccentric are periodical with a 2π period. By combining these between them, new mathematical functions have been defined; united under the name Smarandache stepped functions.
Category: Mathematical Physics

The Generalizations of the First Noether Theorem.

Authors: Vyacheslav Telnin
Comments: 7 Pages.

This paper deals with the generalizations of the First Noether theorem. It takes into account not only the first derivatives of the fields by the coordinates in Lagrangian, but also the second. And this theorem is generalized on the curved spaces. And also it's generalized on asymmetric metric tensors.
Category: Mathematical Physics

Gravitational Forces Are not Conservative

Authors: Florentino Muñiz Ania
Comments: 2 Pages. Sólo hablo español.

This article shows how the gravitational force is conservative only in ideal models of small size. It is shown as strictly mathematical generally is not conservative. From it you can get energy by asymmetric systems. Spanish: En este artículo se muestra como la fuerza gravitatoria sólo es conservativa en modelos ideales de pequeño tamaño. Se demuestra con rigor matemático como, en general, no es conservativa. De ella se puede obtener energía mediante sistemas asimétricos. vixra.org@gmail.com
Category: Mathematical Physics

Fractional Lagrangian and Hamiltonian Formulations in Field Theory

Authors: Hosein Nasrolahpour
Comments: 5 Pages.

The fractional variational principle represents an important part of fractional calculus and has found many applications. There are several versions of fractional variational principles and so different kinds of fractional Euler-Lagrange equations. In this paper, we propose the fractional sine-Gordon Lagrangian density. Then using the fractional Euler-Lagrange equations we obtain fractional sine-Gordon equation.
Category: Mathematical Physics

Riemann Zeros Quantum Chaos Functional Determinants and Trace Formulae

Authors: Jose Javier Garcia
Comments: 11 Pages.

We study the relation between the Guzwiller Trace for a dynamical system and the Riemann-Weil trace formula for the Riemann zeros, using the Bohr-Sommerfeld quantization condition and the fractional calculus we obtain a method to define implicitly a potential , we apply this method to define a Hamiltonian whose energies are the square of the Riemann zeros (imaginary part) , also we show that for big ‘x’ the potential is very close to an exponential function. In this paper and for simplicity we use units so • Keywords: = Riemann Hypothesis, WKB semiclassical approximation, Gutzwiller trace formula, Bohr-Sommerfeld quantization,exponential potential.
Category: Mathematical Physics

Riemann Zeros Quantum Chaos Functional Determinants and Trace Formulae

Authors: Jose Javier Garcia
Comments: 11 Pages.

We study the relation between the Guzwiller Trace for a dynamical system and the Riemann-Weil trace formula for the Riemann zeros, using the Bohr-Sommerfeld quantization condition and the fractional calculus we obtain a method to define implicitly a potential , we apply this method to define a Hamiltonian whose energies are the square of the Riemann zeros (imaginary part) , also we show that for big ‘x’ the potential is very close to an exponential function. In this paper and for simplicity we use units so • Keywords: = Riemann Hypothesis, WKB semiclassical approximation, Gutzwiller trace formula, Bohr-Sommerfeld quantization,exponential potential.
Category: Mathematical Physics

Universal Principles of Perfect Chaos

Authors: Sergey Kamenshchikov
Comments: 12 Pages. Author is looking for postdoc position. kamphys@gmail.com

The purpose of this work was to introduce strict, comprehensive definition of perfect chaos, to find out its basic properties in terms of phase transitions and give connections for uncertainties, lying in base of perfect chaos concept. Concept of perfect chaos as undetermined description was introduced basing on two formalized necessary and sufficient conditions: finite phase space resolution and instability of phase space trajectories. Properties of Kolmogorov system, including phase mixing, turned out to be consequences of chaotic state but not its comprehensive and sufficient conditions.Description relativity was defined as mandatory property of perfect chaos. The same areas of phase space may show regular or chaotic properties depending on space - time description accuracy. Herewith evolution of physical system in given generalized phase space can be represented by consequence of regular states and intermediate transitions. For chaotic state with uniform diffusion it was found out that nonlinear dispersion law is mandatory property.One in its turn necessarily leads to space - time instability of probability density and appearance of probability cavities in phase space - phase space attractors where particles density grows up. Case of chaotic state with fixed boundary and diffusion was considered. It turned out that Fourier decomposition allows to derive relations between coordinate - momentum and time - energy definition uncertainties. It was shown that chaos diffusion factor is the only parameter, limiting product of corresponding uncertainties.
Category: Mathematical Physics

Extended Foundations of Stochastic Prediction

Authors: Sergey Kamenshchikov
Comments: 17 Pages. Author is looking for postdoc position. kamphys@gmail.com

The basic purpose of this work was to suggest universal quantitative description of ergodic system intermediate bifurcation and obligatory conditions of this transition. Conditions for existence of phase state and first order phase transition were introduced in terms of energy balance for system volume unit. Extended Fokker – Plank equation with time dependent diffusion factor was formulated. It turned out that for ergodic system with fixed boundary quantized energy spectrum of phase stable states exists. Obtained results may be applied for prediction of ergodic system behavior. If isolation condition is satisfied, phase spectrum quantization allows selecting proper control parameters for system stabilization. Information about current system coarsened energy allows predicting of future stochastic system behavior on the basis of extended Fokker – Plank model.
Category: Mathematical Physics

Unified Integro-Differential Equation for Relaxation and Oscillation

Authors: Hosein Nasrolahpour
Comments: 3 Pages. Another short report on " Fractional Classical Mechanics" :Prespacetime Journal| November 2012 | Volume 3| Issue 13 | pp. 1247-1250

In this paper we discuss some important consequences of application of fractional operators in physics. Also we present a unified integro-differential equation for relaxation and oscillation. We focus on time fractional formalism whose derivative is in Caputo sense.
Category: Mathematical Physics

Riemann Zeros and an Exponential Potential

Authors: Jose Javier Garcia
Comments: 8 Pages.

ABSTRACT: We study a given exponential potential aebx on the Real half-line which is possible related to the imaginary part of the Riemann zeros. We extend alsostudy also our WKB method to recover the potential from the Eigenvalue Staircase for the Riemann zeros, this eigenvalue
Category: Mathematical Physics

Riemann Zeros and an Exponential Potential

Authors: Jose Javier Garcia
Comments: 8 Pages.

ABSTRACT: We study a given exponential potential aebx on the Real half-line which is possible related to the imaginary part of the Riemann zeros. We extend alsostudy also our WKB method to recover the potential from the Eigenvalue Staircase for the Riemann zeros, this eigenvalue staircase includes the oscillatory and smooth part of the Number of Riemann zeros. In this paper and for simplicity we use units so 2m =1= h · Keywords: = Riemann
Category: Mathematical Physics

Riemann Zeros and an Exponential Potential

Authors: Jose Javier Garcia Moreta
Comments: 7 Pages.

ABSTRACT: We study a given exponential potential aebx on the Real half-line which is possible related to the imaginary part of the Riemann zeros. We extend alsostudy also our WKB method to recover the potential from the Eigenvalue Staircase for the Riemann zeros, this eigenvalue staircase includes the oscillatory and smooth part of the Number of Riemann
Category: Mathematical Physics

A Note on Fractional Electrodynamics

Authors: Hosein Nasrolahpour
Comments: 7 Pages. A few formulas and references added.

We investigate the time evolution of the fractional electromagnetic waves by using the time fractional Maxwell's equations. We show that electromagnetic plane wave has amplitude which exhibits an algebraic decay, at asymptotically long times.
Category: Mathematical Physics

The Symmetry Groups of Light

Authors: John A. Gowan
Comments: 6 Pages.

In the mathematical terms of Evariste Galois' "Group Theory", the "Tetrahedron Model" is a description of the symmetry group of light, including its destruction by asymmetric weak force decays (producing our matter-only Cosmos), and its on-going restoration in obedience to Noether's Theorem of symmetry conservation (as in the conversion of bound to free energy in stars). The usual symmetry group identified with light is that of local phase transformations, and it is designated as either SO(2) or U(1). However, I am suggesting here that light contains a very much larger (and more interesting) symmetry group associated with its transformation into particle-antiparticle pairs (and back again into light). I don't know what the formal designation of this group might be. For an expert's explanation of the formal aspects of symmetry and group theory, See: Keith Devlin The Language of Mathematics Chap. 5 "The Mathematics of Beauty", 1998 W. H. Freeman & Co. (Holt Paperbacks); see also: Ian Stewart Why Beauty is Truth Chapt. 13 "The Five Dimensional Man", Basic Books 2007.
Category: Mathematical Physics

The Symmetry Groups of Light

Authors: John A. Gowan
Comments: 5 Pages.

A symmetry group consists (for one example) of a collection of figures that can be transformed into one another without changing the original. The symmetry group of an equilateral triangle (say) consists of all the triangles that can be created from an original by means of rotation, translation, reflection, etc. - provided the transformed articles are indistinguishable from the original. How do we apply this notion to the case of light? In what sense is there a symmetry group associated with (consisting of) transformations of light (free electromagnetic radiation)? Beyond the simple phase transformations of the electromagnetic field, the examples of interest here are the particle-antiparticle pairs of the Dirac/Heisenberg "vacuum" of spacetime. These particle-antiparticle pairs are constantly produced from borrowed energy and instantaneously annihilate each other in an endless cycle of creation and destruction alternating between light and virtual particles, a cycle which has been ongoing throughout spacetime since its beginning in the "Big Bang". Since they are "virtual" rather than "real" particles we do not notice them even though they are everywhere around us. Essentially, we do not notice them because their symmetry is so complete. We only notice the asymmetries which surround (and comprise) us.
Category: Mathematical Physics

The Symmetry Groups of Light

Authors: John A. Gowan
Comments: 5 Pages. A table is added to the original paper

A symmetry group consists (for one example) of a collection of figures that can be transformed into one another without changing the original. The symmetry group of an equilateral triangle (say) consists of all the triangles that can be created from an original by means of rotation, translation, reflection, etc. - provided the transformed articles are indistinguishable from the original. How do we apply this notion to the case of light? In what sense is there a symmetry group associated with (consisting of) transformations of light (free electromagnetic radiation)? Beyond the simple phase transformations of the electromagnetic field, the examples of interest here are the particle-antiparticle pairs of the Dirac/Heisenberg "vacuum" of spacetime. These particle-antiparticle pairs are constantly produced from borrowed energy and instantaneously annihilate each other in an endless cycle of creation and destruction alternating between light and virtual particles, a cycle which has been ongoing throughout spacetime since its beginning in the "Big Bang". Since they are "virtual" rather than "real" particles we do not notice them even though they are everywhere around us. Essentially, we do not notice them because their symmetry is so complete. We only notice the asymmetries which surround (and comprise) us.
Category: Mathematical Physics

The Cuantifiplane

Authors: Jose M. Hernandez
Comments: 4 Pages.

is fine
Category: Mathematical Physics

The Cuantifiplane (1)

Authors: Jose M Hernandez
Comments: 4 Pages. view

is fine
Category: Mathematical Physics

Extended Foundations of Stochastic Prediction

Authors: Sergey Kamenshchikov
Comments: 17 Pages. Author is looking for postdoc position. kamphys@gmail.com

The basic purpose of this work was to suggest universal quantitative description of ergodic system intermediate bifurcation and obligatory conditions of this transition. Conditions for existence of phase state and first order phase transition were introduced in terms of energy balance for system volume unit. Extended Fokker – Plank equation with time dependent diffusion factor was formulated. It turned out that for ergodic system with fixed boundary quantized energy spectrum of phase stable states exists. Obtained results may be applied for prediction of ergodic system behavior. If isolation condition is satisfied, phase spectrum quantization allows selecting proper control parameters for system stabilization. Information about current system coarsened energy allows predicting of future stochastic system behavior on the basis of extended Fokker – Plank model.
Category: Mathematical Physics

Electrical Maxwell Demon and Szilard Engine Utilizing Johnson Noise, Measurement, Logic and Control

Authors: Laszlo B. Kish, Claes-Göran Granqvist
Comments: 17 Pages. Accepted for publication in PLOS ONE (September 6, 2012)

We introduce a purely electrical version of Maxwell’s demon which does not involve mechanically moving parts such as trapdoors, etc. It consists of a capacitor, resistors, amplifiers, logic circuitry and electronically controlled switches and uses thermal noise in resistors (Johnson noise) to pump heat. The only types of energy of importance in this demon are electrical energy and heat. We also demonstrate an entirely electrical version of Szilard’s engine, i.e., an information-controlled device that can produce work by employing thermal fluctuations. The only moving part is a piston that executes work, and the engine has purely electronic controls and it is free of the major weakness of the original Szilard engine in not requiring removal and repositioning the piston at the end of the cycle. For both devices, the energy dissipation in the memory and other binary informatics components are insignificant compared to the exponentially large energy dissipation in the analog part responsible for creating new information by measurement and decision. This result contradicts the view that the energy dissipation in the memory during erasure is the most essential dissipation process in a demon. Nevertheless the dissipation in the memory and information processing parts is sufficient to secure the Second Law of Thermodynamics.
Category: Mathematical Physics

A Functional Determinant Expression for the Riemann XI Function

Authors: Jose Javier Garcia Moreta
Comments: 13 Pages.

• ABSTRACT: We give and interpretation of the Riemann Xi-function as the quotient of two functional determinants of an Hermitian Hamiltonian . To get the potential of this Hamiltonian we use the WKB method to approximate and evaluate the spectral Theta function over the Riemann zeros on the critical strip . Using the WKB method we manage to get the potential inside the Hamiltonian , also we evaluate the functional determinant by means of Zeta regularization, we discuss the similarity of our method to the method applied to get the Zeros of the Selberg Zeta function. In this paper and for simplicity we use units so • Keywords: = Riemann Hypothesis, Functional determinant, WKB semiclassical Approximation , Trace formula ,Bolte’s law, Quantum chaos.
Category: Mathematical Physics

A Functional Determinant Expression for the Riemann XI-Function

Authors: Jose Javier Garcia Moreta
Comments: 15 Pages.

• ABSTRACT: We give and interpretation of the Riemann Xi-function as the quotient of two functional determinants of an Hermitian Hamiltonian . To get the potential of this Hamiltonian we use the WKB method to approximate and evaluate the spectral Theta function over the Riemann zeros on the critical strip . Using the WKB method we manage to get the potential inside the Hamiltonian , also we evaluate the functional determinant by means of Zeta regularization, we discuss the similarity of our method to the method applied to get the Zeros of the Selberg Zeta function. In this paper and for simplicity we use units so • Keywords: = Riemann Hypothesis, Functional determinant, WKB semiclassical Approximation , Trace formula ,Bolte’s law, Quantum chaos.
Category: Mathematical Physics

A Functional Determinant Expression for the Riemann XI Function

Authors: Jose Javier Garcia Moreta
Comments: 14 Pages. upadted with more equations, corrected several errors

• ABSTRACT: We give and interpretation of the Riemann Xi-function as the quotient of two functional determinants of an Hermitian Hamiltonian . To get the potential of this Hamiltonian we use the WKB method to approximate and evaluate the spectral Theta function over the Riemann zeros on the critical strip . Using the WKB method we manage to get the potential inside the Hamiltonian , also we evaluate the functional determinant by means of Zeta regularization, we discuss the similarity of our method to the method applied to get the Zeros of the Selberg Zeta function • Keywords: = Riemann Hypothesis, Functional determinant, WKB semiclassical Approximation , Trace formula ,Bolte’s law, Quantum chaos.
Category: Mathematical Physics

A Functional Determinant Expression for the Riemann XI Function

Authors: Jose Javier Garcia Moreta
Comments: 12 Pages.

• ABSTRACT: We give and interpretation of the Riemann Xi-function as the quotient of two functional determinants of an Hermitian Hamiltonian . To get the potential of this Hamiltonian we use the WKB method to approximate and evaluate the spectral Theta function over the Riemann zeros on the critical strip . Using the WKB method we manage to get the potential inside the Hamiltonian , also we evaluate the functional determinant by means of Zeta regularization, we discuss the similarity of our method to the method applied to get the Zeros of the Selberg Zeta function • Keywords: = Riemann Hypothesis, Functional determinant, WKB semiclassical Approximation , Trace formula ,Bolte’s law, Quantum chaos.
Category: Mathematical Physics

Time Fractional Formalism: Classical and Quantum Phenomena

Authors: Hosein Nasrolahpour
Comments: 10 Pages.

In this review, we present some fundamental classical and quantum phenomena in view of time fractional formalism. Time fractional formalism is a very useful tool in describing systems with memory and delay. We hope that this study can provide a deeper understanding of the physical interpretations of fractional derivative.
Category: Mathematical Physics

A Technique for Cataloging Types of Particles and Types of Stuff

Authors: Thomas J. Buckholtz
Comments: 15 Pages.

We develop theory leading to an ability to catalog types of elementary particles. The resulting catalog provides for known interaction-mediating bosons, non-traditional interaction-carrying bosons, and fermions. Some ratios of numbers of instances of various types of particles are 1:6:24:48. Potentially, the actual ratios of densities of baryonic matter, dark matter, dark energy are 1:5:18 and do not depend on time, even though interpretations of data provide ratios that vary with the age of the universe. Potentially there is another type of stuff. Here, 5=6-1, 18=24-6, and there could be 24=48-24 units of the other type of stuff.
Category: Mathematical Physics

A Technique for Cataloging Types of Particles and Types of Stuff

Authors: Thomas J. Buckholtz
Comments: 15 Pages.

We develop theory leading to an ability to catalog types of elementary particles. The resulting catalog provides for known interaction-mediating bosons, non-traditional interaction-carrying bosons, and fermions. Ratios of theoretical numbers of analogs of various types of particles may be consistent with observed ratios of densities for baryonic matter, dark matter, and dark energy.
Category: Mathematical Physics

Duonistic Neutrinos Violate Relativity.

Authors: Dan Visser
Comments: 10 Pages.

Neutrinos-faster-than-light? The science-battle is not yet over! ‘Yes’, said the OPERA-team in September 22 2011. ‘No’, said the ICARUS-team in February 23 2012. But this paper carries on that it is undoubtedly correct that neutrinos can go faster-than-light. Neutrinos can only do that in neutrino-pairs: I call these pairs Duonistic Neutrinos! This paper presents the set of equations to prove that. The smallest gravity-acceleration g’ appears to be prior to the trajectory of single packaged neutrinos. OPERA and ICARUS might be right both in the end.
Category: Mathematical Physics

In Search Of A Variant Kaluza Theory

Authors: Robert Watson
Comments: 34 Pages. ...looking better now...

Kaluza's 1921 theory of gravity and electromagnetism using a fifth wrapped-up spatial dimension is at the root of many modern attempts to develop new physical theories. Lacking non-null electromagnetic fields however the theory is incomplete. Variants of the theory are explored to find ways to introduce non-null solutions by making the fifth dimension more physical, using alternative, weaker cylinder conditions. The Lorentz force law is investigated starting with a non-Maxwellian definition of charge, this is assumed to be related to Maxwellian charge by ansatz. Order of magnitude methods are used. Kaluza theory remains inadequate to support electromagnetism in full, non-null solutions are not readily shown to be admitted. An argument is made in favour of torsion resolving this issue. Postulates are derived from the argument for a variant theory. The charge ansatz is shown to follow from the postulates. It is concluded that Kaluza's 5D space and torsion need to go together in a Kaluza-Cartan theory. Tentatively, generalized Bel super-energy is hypothesized to be a conserved quantity.
Category: Mathematical Physics

Particle Mass Ratios

Authors: DT Froedge
Comments: 15 Pages. This should be the final. Accuracies calculated in parts per billion

Based on the developments in a previous paper [1], this paper presents straightforward explanation of particle mass ratios, yielding specific values for some well studied particles. The additional nuclear modes postulated, are similar to the Schrodinger modes in the atom, and the mass ratios calculated for these particles have been calculated within the experimental values, with accuracies up to 2.4 parts per billion. The examined particles are the Proton, Neutron, Muon, Tauon, and Pion+/-/0.
Category: Mathematical Physics

Particle Mass Ratios

Authors: DT Froedge
Comments: 14 Pages. Finally, very precise values.

Based on the developments in a previous paper [1], this paper presents straightforward explanation of particle mass ratios, and the specific values for some well known particles. The additional nuclear modes postulated, are similar to the Schrödinger modes in the atom, and the mass ratios calculated for elementary particles have been calculated within the experimental values. Those particles are the Proton, Neutron, Muon, Tauon, and Pion+/-/0
Category: Mathematical Physics

Particle Mass Ratios

Authors: DT Froedge
Comments: 13 Pages. This worked out

Based on the developments in a previous paper [1], this paper presents straightforward explanation of particle mass ratios, and the specific values for some well known particles. The additional nuclear modes postulated, are similar to the Schrödinger modes in the atom, and the mass ratios calculated for elementary particles have been calculated within the experimental values. Those particles are the Proton, Neutron, Muon, Tauon, and Pion+/-/0
Category: Mathematical Physics

Particle Mass Ratios

Authors: DT Froedge
Comments: 14 Pages. finally figured it out,

Based on the developments in a previous paper, this paper presents straightforward explanation of particle mass ratios, and the specific values for some well known particles. The additional nuclear modes postulated, are similar to the Schrödinger modes in the atom, and the mass ratios calculated for elementary particles have been calculated within the experimental values.Those particles are the Proton, Neutron, Muon, Tauon, and Pion+/-/0.
Category: Mathematical Physics

Particle Mass Ratios

Authors: DT Froedge
Comments: 10 Pages. Major improvement, ongoing project.

Based on the developments in a previous paper [1], this paper presents straightforward explanation of particle mass ratios, and the specific values for some well known particles. The additional nuclear modes postulated, are similar to the Schrödinger modes in the atom, and the mass ratios calculated for elementary particles are very close the observed mass ratios. Integral amplitudes for the Pauli and quark matrix are shown to produce values that are far more accurate than those that could be produced by random coincidence.
Category: Mathematical Physics

Particle Mass Ratios

Authors: DT Froedge
Comments: 13 Pages. A little progress.

Based on the developments in a previous paper, this paper presents straightforward explanation of particle mass ratios, and the specific values for some well known particles. The additional nuclear modes postulated, are similar to the Schrödinger modes in the atom, and, though speculative, the mass ratios calculated for elementary particles are very close the observed mass ratios.
Category: Mathematical Physics

Particle Mass Ratios

Authors: DT Froedge
Comments: 14 Pages. Minor Corrections

Based on the developments in a previous paper [1], this paper presents straightforward explanation of particle mass ratios, and the specific values for some well known particles. The additional nuclear modes postulated, are similar to the Schrödinger modes in the atom, and, though speculative, the mass ratios calculated for elementary particles are very close the observed mass ratios.
Category: Mathematical Physics

Particle Mass Ratios

Authors: DT Froedge
Comments: 12 Pages. This is the 6Th and final revision (v3-11-12),

Based on the developments in a previous paper [1], this paper presents straightforward explanation of particle mass ratios, and the specific values for some well known particles. The additional nuclear modes postulated, are similar to the Schrödinger modes in the atom, and, though speculative, the mass ratios calculated for elementary particles are very close the observed mass ratios.
Category: Mathematical Physics

A New Force Smaller Than The Smallest Gravity.

Authors: Dan Visser
Comments: 7 Pages.

In this version2 the formulations are given for the existence of a force smaller than the smallest gravity. This is a new dark energy force, which affects neutrinos differently than is assumed according to current physics. The formulations also imply a different look on the Higgs-mass and dark matter-mass. A deeper analysis became important, because a new cosmological hypothesis is involved. The CERN-experiments on these issues are far from criticism. My set of equations mentioned in my paper “A New Dark Energy Force Theoretically Calculates Faster-than-light-neutrinos” and “Duonistic Neutrinos Violate Relativity”, reveal such a criticism. However, until now my formulations withstand a hurricane, even after a Director of the OPERA-project had to resign. My set of equations theoretically proves the neutrino-faster-than-light experiments had to be investigated to the bottom.
Category: Mathematical Physics

A Hamiltonian Whose Energies Are the Zeros of the Riemann XI Function

Authors: Jose Javier Garcia Moreta
Comments: 12 Pages.

· ABSTRACT: We give a possible interpretation of the Xi-function of Riemann as the Functional determinant det ( E - H ) for a certain Hamiltonian quantum operator in one dimension 2 2 d V (x) dx - + for a real-valued function V(x) , this potential V is related to the half-integral of the logarithmic derivative for the Riemann Xi-function, through the paper we will assume that the reduced Planck constant is defined in units where h =1 and that the mass is 2m =1.In this case the Energies of the Hamiltonian operator will be the square of the imaginary part of the Riemann Zeros 2 n n E =g Also trhough this paper we may refer to the Hamiltonian Operator whose Energies are the square of the imaginary part of the Riemann Zeros as H or 2 H (square) in the same case we will refer to the potential inside this Hamiltonian either as 2 V (x) or V (x) to simplify notation. · Keywords: = Riemann Hypothesis, Functional determinant,
Category: Mathematical Physics

A Hamiltonian Operator Whose Energies Are the Roots of the Riemann XI-Function

Authors: Jose Javier Garicia Moreta
Comments: 25 Pages.

· ABSTRACT: We give a possible interpretation of the Xi-function of Riemann as the Functional determinant det ( E - H ) for a certain Hamiltonian quantum operator in one dimension 2 2 d V (x) dx - + for a real-valued function V(x) , this potential V is related to the half-integral of the logarithmic derivative for the Riemann Xi-function, through the paper we will assume that the reduced Planck constant is defined in units where h =1 and that the mass is 2m =1.In this case the Energies of the Hamiltonian operator will be the square of the imaginary part of the Riemann Zeros 2 n n E =g Also trhough this paper we may refer to the Hamiltonian Operator whose Energies are the square of the imaginary part of the Riemann Zeros as H or 2 H (square) in the same case we will refer to the potential inside this Hamiltonian either as 2 V (x) or V (x) to simplify notation. · Keywords: = Riemann Hypothesis, Functional determinant,
Category: Mathematical Physics

A Hamiltonian Operator Whose Energies Are the Roots of the Riemann XI-Function 1 2

Authors: Jose Javier Garcia Moreta
Comments: 25 Pages.

ABSTRACT: We give a possible interpretation of the Xi-function of Riemann as the Functional determinant det ( E - H ) for a certain Hamiltonian quantum operator in one dimension 2 2 d V (x) dx - + for a real-valued function V(x) , this potential V is related to the half-integral of the logarithmic derivative for the Riemann Xi-function, through the paper we will assume that the reduced Planck constant is defined in units where h =1 and that the mass is 2m =1.In this case the Energies of the Hamiltonian operator will be the square of the imaginary part of the Riemann Zeros 2 n n E =g Also trhough this paper we may refer to the Hamiltonian Operator whose Energies are the square of the imaginary part of the Riemann Zeros as H or 2 H (square) in the same case we will refer to the potential inside this Hamiltonian either as 2 V (x) or V (x) to simplify notation. · Keywords: = Riemann Hypothesis, Functional determinant, WKB semiclassical Approximation , Trace formula , Quantum chaos.
Category: Mathematical Physics

A Hamiltonian Operator Whose Energies Are the Roots of the Riemann XI-Function

Authors: Jose Javier garcia
Comments: 25 Pages.

We give a possible interpretation of the Xi-function of Riemann as the Functional determinant for a certain Hamiltonian quantum operator in one dimension for a real-valued function V(x) , this potential V is related to the half-integral of the logarithmic derivative for the Riemann Xi-function, through the paper we will assume that the reduced Planck constant is defined in units where and that the mass is .In this case the Energies of the Hamiltonian operator will be the square of the imaginary part of the Riemann Zeros Also trhough this paper we may refer to the Hamiltonian Operator whose Energies are the square of the imaginary part of the Riemann Zeros as or (square) in the same case we will refer to the potential inside this Hamiltonian either as or to simplify notation. Keywords: = Riemann Hypothesis, Functional determinant, WKB semiclassical Approximation , Trace formula , Quantum chaos.
Category: Mathematical Physics

A Hamiltonian Operator Whose Roots Are the Roots of the Riemann XI-Function

Authors: Jose Javier Garcia Moreta
Comments: 23 Pages. there is an ERROR, whenever it says 'Whose zeros' should we put 'Whose Energies' (A Hamiltonian has no zeros)H

We give a possible interpretation of the Xi-function of Riemann as the Functional determinant for a certain Hamiltonian quantum operator in one dimension for a real-valued function V(x) , this potential V is related to the half-integral of the logarithmic derivative for the Riemann Xi-function, through the paper we will assume that the reduced Planck constant is defined in units where and that the mass is .In this case the Energies of the Hamiltonian operator will be the square of the imaginary part of the Riemann Zeros Also trhough this paper we may refer to the Hamiltonian Operator whose Energies are the square of the imaginary part of the Riemann Zeros as or (square) in the same case we will refer to the potential inside this Hamiltonian either as or to simplify notation.
Category: Mathematical Physics

A Hamiltonian Operator Whose Zeros Are the Roots of the Riemann XI-Function

Authors: Jose Javier Garcia Moreta
Comments: 23 Pages.

We give a possible interpretation of the Xi-function of Riemann as the Functional determinant for a certain Hamiltonian quantum operator in one dimension for a real-valued function V(x) , this potential V is related to the half-integral of the logarithmic derivative for the Riemann Xi-function, through the paper we will assume that the reduced Planck constant is defined in units where and that the mass is
Category: Mathematical Physics

Langlands Conjectures in TGD Framework

Authors: Matti Pitkänen
Comments: 24 Pages.

The arguments of this article support the view that in TGD Universe number theoretic and geometric Langlands conjectures could be understood very naturally. The basic notions are following.

1. Zero energy ontology (ZEO) and the related notion of causal diamond CD (CD is short hand for the cartesian product of causal diamond of M⁴ and of CP₂). ZEO leads to the notion of partonic 2-surfaces at the light-like boundaries of CD and to the notion of string world sheet. These notions are central in the recent view about TGD. One can assign to the partonic 2-surfaces a conformal moduli space having as additional coordinates the positions of braid strand ends (punctures). By electric-magnetic duality this moduli space must correspond closely to the moduli space of string world sheets.

2. Electric-magnetic duality realized in terms of string world sheets and partonic 2-surfaces. The group G and its Langlands dual ^LG would correspond to the time-like and space-like braidings. Duality predicts that the moduli space of string world sheets is very closely related to that for the partonic 2-surfaces. The strong form of 4-D general coordinate invariance implying electric-magnetic duality and S-duality as well as strong form of holography indeed predicts that the collection of string world sheets is fixed once the collection of partonic 2-surfaces at light-like boundaries of CD and its sub-CDs is known.

3. The proposal is that finite measurement resolution is realized in terms of inclusions of hyperfinite factors of type II₁ at quantum level and represented in terms of confining effective gauge group. This effective gauge group could be some associate of G: gauge group, Kac-Moody group or its quantum counterpart, or so called twisted quantum Yangian strongly suggested by twistor considerations. At space-time level the finite measurement resolution would be represented in terms of braids at space-time level which come in two varieties correspond to braids assignable to space-like surfaces at the two light-like boundaries of CD and with light-like 3-surfaces at which the signature of the induced metric changes and which are identified as orbits of partonic 2-surfaces connecting the future and past boundaries of CDs.

   There are several steps leading from G to its twisted quantum Yangian. The first step replaces point like particles with partonic 2-surfaces: this brings in Kac-Moody character. The second step brings in finite measurement resolution meaning that Kac-Moody type algebra is replaced with its quantum version. The third step brings in zero energy ontology: one cannot treat single partonic surface or string world sheet as independent unit: always the collection of partonic 2-surfaces and corresponding string worlds sheets defines the geometric structure so that multilocality and therefore quantum Yangian algebra with multilocal generators is unavoidable.

   In finite measurement resolution geometric Langlands duality and number theoretic Langlands duality are very closely related since partonic 2-surface is effectively replaced with the punctures representing the ends of braid strands and the orbit of this set under a discrete subgroup of G defines effectively a collection of "rational" 2-surfaces. The number of the "rational" surfaces in geometric Langlands conjecture replaces the number of rational points of partonic 2-surface in its number theoretic variant. The ability to compute both these numbers is very relevant for quantum TGD.

4. The natural identification of the associate of G is as quantum Yangian of Kac-Moody type group associated with Minkowskian open string model assignable to string world sheet representing a string moving in the moduli space of partonic 2-surface. The dual group corresponds to Euclidian string model with partonic 2-surface representing string orbit in the moduli space of the string world sheets. The Kac-Moody algebra assigned with simply laced G is obtained using the standard tachyonic free field representation obtained as ordered exponentials of Cartan algebra generators identified as transversal parts of M⁴ coordinates for the braid strands. The importance of the free field representation generalizing to the case of non-simply laced groups in the realization of finite measurement resolution in terms of Kac-Moody algebra cannot be over-emphasized.

5. Langlands duality involves besides harmonic analysis side also the number theoretic side. Galois groups (collections of them) defined by infinite primes and integers having representation as symplectic flows defining braidings. I have earlier proposed that the hierarchy of these Galois groups define what might be regarded as a non-commutative homology and cohomology. Also G has this kind of representation which explains why the representations of these two kinds of groups are so intimately related. This relationship could be seen as a generalization of the MacKay correspondence between finite subgroups of SU(2) and simply laced Lie groups.

6. Symplectic group of the light-cone boundary acting as isometries of the WCW geometry kenociteallb/compl1 allowing to represent projectively both Galois groups and symmetry groups as symplectic flows so that the non-commutative cohomology would have braided representation. This leads to braided counterparts for both Galois group and effective symmetry group.

7. The moduli space for Higgs bundle playing central role in the approach of Witten and Kapustin to geometric Landlands program is in TGD framework replaced with the conformal moduli space for partonic 2-surfaces. It is not however possible to speak about Higgs field although moduli defined the analog of Higgs vacuum expectation value. Note that in TGD Universe the most natural assumption is that all Higgs like states are "eaten" by gauge bosons so that also photon and gluons become massive. This mechanism would be very general and mean that massless representations of Poincare group organize to massive ones via the formation of bound states. It might be however possible to see the contribution of p-adic thermodynamics depending on genus as analogous to Higgs contribution since the conformal moduli are analogous to vacuum expectation of Higgs field.

How Infinite Primes Relate to Other Views About Mathematical Infinity?

Authors: Matti Pitkänen
Comments: 16 Pages.

Infinite primes is a purely TGD inspired notion. The notion of infinity is number theoretical and infinite primes have well defined divisibility properties. One can partially order them by the real norm. p-Adic norms of infinite primes are well defined and finite. The construction of infinite primes is a hierarchical procedure structurally equivalent to a repeated second quantization of a supersymmetric arithmetic quantum field theory. At the lowest level bosons and fermions are labelled by ordinary primes. At the next level one obtains free Fock states plus states having interpretation as bound many particle states. The many particle states of a given level become the single particle states of the next level and one can repeat the construction ad infinitum. The analogy with quantum theory is intriguing and I have proposed that the quantum states in TGD Universe correspond to octonionic generalizations of infinite primes. It is interesting to compare infinite primes (and integers) to the Cantorian view about infinite ordinals and cardinals. The basic problems of Cantor's approach which relate to the axiom of choice, continuum hypothesis, and Russell's antinomy: all these problems relate to the definition of ordinals as sets. In TGD framework infinite primes, integers, and rationals are defined purely algebraically so that these problems are avoided. It is not surprising that these approaches are not equivalent. For instance, sum and product for Cantorian ordinals are not commutative unlike for infinite integers defined in terms of infinite primes.

Set theory defines the foundations of modern mathematics. Set theory relies strongly on classical physics, and the obvious question is whether one should reconsider the foundations of mathematics in light of quantum physics. Is set theory really the correct approach to axiomatization?

1. Quantum view about consciousness and cognition leads to a proposal that p-adic physics serves as a correlate for cognition. Together with the notion of infinite primes this suggests that number theory should play a key role in the axiomatics.
2. Algebraic geometry allows algebraization of the set theory and this kind of approach suggests itself strongly in physics inspired approach to the foundations of mathematics. This means powerful limitations on the notion of set.
3. Finite measurement resolution and finite resolution of cognition could have implications also for the foundations of mathematics and relate directly to the fact that all numerical approaches reduce to an approximation using rationals with a cutoff on the number of binary digits.
4. The TGD inspired vision about consciousness implies evolution by quantum jumps meaning that also evolution of mathematics so that no fixed system of axioms can ever catch all the mathematical truths for the simple reason that mathematicians themselves evolve with mathematics.

Motives and Infinite Primes

Authors: Matti Pitkänen
Comments: 80 Pages.

In this article the goal is to find whether the general mathematical structures associated with twistor approach, superstring models and M-theory could have a generalization or a modification in TGD framework. The contents of the chapter is an outcome of a rather spontaneous process, and represents rather unexpected new insights about TGD resulting as outcome of the comparisons.

1. Infinite primes, Galois groups, algebraic geometry, and TGD

In algebraic geometry the notion of variety defined by algebraic equation is very general: all number fields are allowed. One of the challenges is to define the counterparts of homology and cohomology groups for them. The notion of cohomology giving rise also to homology if Poincare duality holds true is central. The number of various cohomology theories has inflated and one of the basic challenges to find a sufficiently general approach allowing to interpret various cohomology theories as variations of the same motive as Grothendieck, who is the pioneer of the field responsible for many of the basic notions and visions, expressed it.

Cohomology requires a definition of integral for forms for all number fields. In p-adic context the lack of well-ordering of p-adic numbers implies difficulties both in homology and cohomology since the notion of boundary does not exist in topological sense. The notion of definite integral is problematic for the same reason. This has led to a proposal of reducing integration to Fourier analysis working for symmetric spaces but requiring algebraic extensions of p-adic numbers and an appropriate definition of the p-adic symmetric space. The definition is not unique and the interpretation is in terms of the varying measurement resolution.

The notion of infinite has gradually turned out to be more and more important for quantum TGD. Infinite primes, integers, and rationals form a hierarchy completely analogous to a hierarchy of second quantization for a super-symmetric arithmetic quantum field theory. The simplest infinite primes representing elementary particles at given level are in one-one correspondence with many-particle states of the previous level. More complex infinite primes have interpretation in terms of bound states.

1. What makes infinite primes interesting from the point of view of algebraic geometry is that infinite primes, integers and rationals at the n:th level of the hierarchy are in 1-1 correspondence with rational functions of n arguments. One can solve the roots of associated polynomials and perform a root decomposition of infinite primes at various levels of the hierarchy and assign to them Galois groups acting as automorphisms of the field extensions of polynomials defined by the roots coming as restrictions of the basic polynomial to planes x_n=0, x_n=x_n-1=0, etc...

2. These Galois groups are suggested to define non-commutative generalization of homotopy and homology theories and non-linear boundary operation for which a geometric interpretation in terms of the restriction to lower-dimensional plane is proposed. The Galois group G_k would be analogous to the relative homology group relative to the plane x_k-1=0 representing boundary and makes sense for all number fields also geometrically. One can ask whether the invariance of the complex of groups under the permutations of the orders of variables in the reduction process is necessary. Physical interpretation suggests that this is not the case and that all the groups obtained by the permutations are needed for a full description.

3. The algebraic counterpart of boundary map would map the elements of G_k identified as analog of homotopy group to the commutator group [G_k-2,G_k-2] and therefore to the unit element of the abelianized group defining cohomology group. In order to obtains something analogous to the ordinary homology and cohomology groups one must however replaces Galois groups by their group algebras with values in some field or ring. This allows to define the analogs of homotopy and homology groups as their abelianizations. Cohomotopy, and cohomology would emerge as duals of homotopy and homology in the dual of the group algebra.

4. That the algebraic representation of the boundary operation is not expected to be unique turns into blessing when on keeps the TGD as almost topological QFT vision as the guide line. One can include all boundary homomorphisms subject to the condition that the anticommutator δⁱ_kδ^j_k-1+δ^j_kδⁱ_k-1 maps to the group algebra of the commutator group [G_k-2,G_k-2]. By adding dual generators one obtains what looks like a generalization of anticommutative fermionic algebra and what comes in mind is the spectrum of quantum states of a SUSY algebra spanned by bosonic states realized as group algebra elements and fermionic states realized in terms of homotopy and cohomotopy and in abelianized version in terms of homology and cohomology. Galois group action allows to organize quantum states into multiplets of Galois groups acting as symmetry groups of physics. Poincare duality would map the analogs of fermionic creation operators to annihilation operators and vice versa and the counterpart of pairing of k:th and n-k:th homology groups would be inner product analogous to that given by Grassmann integration. The interpretation in terms of fermions turns however to be wrong and the more appropriate interpretation is in terms of Dolbeault cohomology applying to forms with homomorphic and antiholomorphic indices.

5. The intuitive idea that the Galois group is analogous to 1-D homotopy group which is the only non-commutative homotopy group, the structure of infinite primes analogous to the braids of braids of braids of ... structure, the fact that Galois group is a subgroup of permutation group, and the possibility to lift permutation group to a braid group suggests a representation as flows of 2-D plane with punctures giving a direct connection with topological quantum field theories for braids, knots and links. The natural assumption is that the flows are induced from transformations of the symplectic group acting on δ M²_+/-× CP₂ representing quantum fluctuating degrees of freedom associated with WCW ("world of classical worlds"). Discretization of WCW and cutoff in the number of modes would be due to the finite measurement resolution. The outcome would be rather far reaching: finite measurement resolution would allow to construct WCW spinor fields explicitly using the machinery of number theory and algebraic geometry.

6. A connection with operads is highly suggestive. What is nice from TGD perspective is that the non-commutative generalization homology and homotopy has direct connection to the basic structure of quantum TGD almost topological quantum theory where braids are basic objects and also to hyper-finite factors of type II₁. This notion of Galois group makes sense only for the algebraic varieties for which coefficient field is algebraic extension of some number field. Braid group approach however allows to generalize the approach to completely general polynomials since the braid group make sense also when the ends points for the braid are not algebraic points (roots of the polynomial).

This construction would realize the number theoretical, algebraic geometrical, and topological content in the construction of quantum states in TGD framework in accordance with TGD as almost TQFT philosophy, TGD as infinite-D geometry, and TGD as generalized number theory visions.

2. p-Adic integration and cohomology

This picture leads also to a proposal how p-adic integrals could be defined in TGD framework.

1. The calculation of twistorial amplitudes reduces to multi-dimensional residue calculus. Motivic integration gives excellent hopes for the p-adic existence of this calculus and braid representation would give space-time representation for the residue integrals in terms of the braid points representing poles of the integrand: this would conform with quantum classical correspondence. The power of 2π appearing in multiple residue integral is problematic unless it disappears from scattering amplitudes. Otherwise one must allow an extension of p-adic numbers to a ring containing powers of 2π.

2. Weak form of electric-magnetic duality and the general solution ansatz for preferred extremals reduce the Kähler action defining the Kähler function for WCW to the integral of Chern-Simons 3-form. Hence the reduction to cohomology takes places at space-time level and since p-adic cohomology exists there are excellent hopes about the existence of p-adic variant of Kähler action. The existence of the exponent of Kähler gives additional powerful constraints on the value of the Kähler fuction in the intersection of real and p-adic worlds consisting of algebraic partonic 2-surfaces and allows to guess the general form of the Kähler action in p-adic context.

3. One also should define p-adic integration for vacuum functional at the level of WCW. p-Adic thermodynamics serves as a guideline leading to the condition that in p-adic sector exponent of Kähler action is of form (m/n)^r, where m/n is divisible by a positive power of p-adic prime p. This implies that one has sum over contributions coming as powers of p and the challenge is to calculate the integral for K= constant surfaces using the integration measure defined by an infinite power of Kähler form of WCW reducing the integral to cohomology which should make sense also p-adically. The p-adicization of the WCW integrals has been discussed already earlier using an approach based on harmonic analysis in symmetric spaces and these two approaches should be equivalent. One could also consider a more general quantization of Kähler action as sum K=K₁+K₂ where K₁=rlog(m/n) and K₂=n, with n divisible by p since exp(n) exists in this case and one has exp(K)= (m/n)^r × exp(n). Also transcendental extensions of p-adic numbers involving n+p-2 powers of e^1/n can be considered.

4. If the Galois group algebras indeed define a representation for WCW spinor fields in finite measurement resolution, also WCW integration would reduce to summations over the Galois groups involved so that integrals would be well-defined in all number fields.

3. Floer homology, Gromov-Witten invariants, and TGD

Floer homology defines a generalization of Morse theory allowing to deduce symplectic homology groups by studying Morse theory in loop space of the symplectic manifold. Since the symplectic transformations of the boundary of δ M⁴_+/-× CP₂ define isometry group of WCW, it is very natural to expect that Kähler action defines a generalization of the Floer homology allowing to understand the symplectic aspects of quantum TGD. The hierarchy of Planck constants implied by the one-to-many correspondence between canonical momentum densities and time derivatives of the imbedding space coordinates leads naturally to singular coverings of the imbedding space and the resulting symplectic Morse theory could characterize the homology of these coverings.

One ends up to a more precise definition of vacuum functional: Kähler action reduces Chern-Simons terms (imaginary in Minkowskian regions and real in Euclidian regions) so that it has both phase and real exponent which makes the functional integral well-defined. Both the phase factor and its conjugate must be allowed and the resulting degeneracy of ground state could allow to understand qualitatively the delicacies of CP breaking and its sensitivity to the parameters of the system. The critical points with respect to zero modes correspond to those for Kähler function. The critical points with respect to complex coordinates associated with quantum fluctuating degrees of freedom are not allowed by the positive definiteness of Kähler metric of WCW. One can say that Kähler and Morse functions define the real and imaginary parts of the exponent of vacuum functional.

The generalization of Floer homology inspires several new insights. In particular, space-time surface as hyper-quaternionic surface could define the 4-D counterpart for pseudo-holomorphic 2-surfaces in Floer homology. Holomorphic partonic 2-surfaces could in turn correspond to the extrema of Kähler function with respect to zero modes and holomorphy would be accompanied by super-symmetry.

Gromov-Witten invariants appear in Floer homology and topological string theories and this inspires the attempt to build an overall view about their role in TGD. Generalization of topological string theories of type A and B to TGD framework is proposed. The TGD counterpart of the mirror symmetry would be the equivalence of formulations of TGD in H=M⁴× CP₂ and in CP₃× CP₃ with space-time surfaces replaced with 6-D sphere bundles.

4. K-theory, branes, and TGD

K-theory and its generalizations play a fundamental role in super-string models and M-theory since they allow a topological classification of branes. After representing some physical objections against the notion of brane more technical problems of this approach are discussed briefly and it is proposed how TGD allows to overcome these problems. A more precise formulation of the weak form of electric-magnetic duality emerges: the original formulation was not quite correct for space-time regions with Euclidian signature of the induced metric. The question about possible TGD counterparts of R-R and NS-NS fields and S, T, and U dualities is discussed.

5. p-Adic space-time sheets as correlates for Boolean cognition

p-Adic physics is interpreted as physical correlate for cognition. The so called Stone spaces are in one-one correspondence with Boolean algebras and have typically 2-adic topologies. A generalization to p-adic case with the interpretation of p pinary digits as physically representable Boolean statements of a Boolean algebra with 2ⁿ>p>p^n-1 statements is encouraged by p-adic length scale hypothesis. Stone spaces are synonymous with profinite spaces about which both finite and infinite Galois groups represent basic examples. This provides a strong support for the connection between Boolean cognition and p-adic space-time physics. The Stone space character of Galois groups suggests also a deep connection between number theory and cognition and some arguments providing support for this vision are discussed.

Could One Generalize Braid Invariant Defined by Vacuum Expecation of Wilson Loop to and Invariant of Braid Cobordisms and of 2-Knots?

Authors: Matti Pitkänen
Comments: 17 Pages.

Witten was awarded by Fields medal from a construction recipe of Jones polynomial based on topological QFT assigned with braids and based on Chern-Simons action. Recently Witten has been working with an attempt to understand in terms of quantum theory the so called Khovanov polynomial associated with a much more abstract link invariant whose interpretation and real understanding remains still open.

The attempts to understand Witten's thoughts lead to a series of questions unavoidably culminating to the frustrating "Why I do not have the brain of Witten making perhaps possible to answer these questions?". This one must just accept. In this article I summarize some thoughts inspired by the associations of the talk of Witten with quantum TGD and with the model of DNA as topological quantum computer. In my own childish manner I dare believe that these associations are interesting and dare also hope that some more brainy individual might take them seriously.

An idea inspired by TGD approach which also main streamer might find interesting is that the Jones invariant defined as vacuum expectation for a Wilson loop in 2+1-D space-time generalizes to a vacuum expectation for a collection of Wilson loops in 2+2-D space-time and could define an invariant for 2-D knots and for cobordisms of braids analogous to Jones polynomial. As a matter fact, it turns out that a generalization of gauge field known as gerbe is needed and that in TGD framework classical color gauge fields defined the gauge potentials of this field. Also topological string theory in 4-D space-time could define this kind of invariants. Of course, it might well be that this kind of ideas have been already discussed in literature.

Yangian Symmetry, Twistors, and TGD

Authors: Matti Pitkänen
Comments: 61 Pages.

There have been impressive steps in the understanding of N=4 maximally sypersymmetric YM theory possessing 4-D super-conformal symmetry. This theory is related by AdS/CFT duality to certain string theory in AdS₅× S⁵ background. Second stringy representation was discovered by Witten and is based on 6-D Calabi-Yau manifold defined by twistors. The unifying proposal is that so called Yangian symmetry is behind the mathematical miracles involved.

In the following I will discuss briefly the notion of Yangian symmetry and suggest its generalization in TGD framework by replacing conformal algebra with appropriate super-conformal algebras. Also a possible realization of twistor approach and the construction of scattering amplitudes in terms of Yangian invariants defined by Grassmannian integrals is considered in TGD framework and based on the idea that in zero energy ontology one can represent massive states as bound states of massless particles. There is also a proposal for a physical interpretation of the Cartan algebra of Yangian algebra allowing to understand at the fundamental level how the mass spectrum of n-particle bound states could be understood in terms of the n-local charges of the Yangian algebra.

Twistors were originally introduced by Penrose to characterize the solutions of Maxwell's equations. Kähler action is Maxwell action for the induced Kähler form of CP₂. The preferred extremals allow a very concrete interpretation in terms of modes of massless non-linear field. Both conformally compactified Minkowski space identifiable as so called causal diamond and CP₂ allow a description in terms of twistors. These observations inspire the proposal that a generalization of Witten's twistor string theory relying on the identification of twistor string world sheets with certain holomorphic surfaces assigned with Feynman diagrams could allow a formulation of quantum TGD in terms of 3-dimensional holomorphic surfaces of CP₃× CP₃ mapped to 6-surfaces dual CP₃× CP₃, which are sphere bundles so that they are projected in a natural manner to 4-D space-time surfaces. Very general physical and mathematical arguments lead to a highly unique proposal for the holomorphic differential equations defining the complex 3-surfaces conjectured to correspond to the preferred extremals of Kähler action.

A Possible Explanation for Shnoll Effect

Authors: Matti Pitkänen
Comments: 17 Pages.

Shnoll and collaborators have discovered strange repeating patterns of random fluctuations of physical observables such as the number n of nuclear decays in a given time interval. Periodically occurring peaks for the distribution of the number N(n) of measurements producing n events in a series of measurements as a function of n is observed instead of a single peak. The positions of the peaks are not random and the patterns depend on position and time varying periodically in time scales possibly assignable to Earth-Sun and Earth-Moon gravitational interaction.

These observations suggest a modification of the expected probability distributions but it is very difficult to imagine any physical mechanism in the standard physics framework. Rather, a universal deformation of predicted probability distributions would be in question requiring something analogous to the transition from classical physics to quantum physics.

The hint about the nature of the modification comes from the TGD inspired quantum measurement theory proposing a description of the notion of finite measurement resolution in terms of inclusions of so called hyper-finite factors of type II₁ (HFFs) and closely related quantum groups. Also p-adic physics -another key element of TGD- is expected to be involved. A modification of a given probability distribution P(nkenovert λ_i) for a positive integer valued variable n characterized by rational-valued parameters λ_i is obtained by replacing n and the integers characterizing λ_i with so called quantum integers depending on the quantum phase q_m=exp(i2π/m). Quantum integer n_q must be defined as the product of quantum counterparts p_q of the primes p appearing in the prime decomposition of n. One has p_q= sin(2π p/m)/sin(2π/m) for p≠ P and p_q=P for p=P. m must satisfy m≥ 3, m≠ p, and m≠ 2p.

The quantum counterparts of positive integers can be negative. Therefore quantum distribution is defined first as p-adic valued distribution and then mapped by so called canonical identification I to a real distribution by the map taking p-adic -1 to P and powers Pⁿ to P^-n and other quantum primes to themselves and requiring that the mean value of n is for distribution and its quantum variant. The map I satisfies I(∑ P_n)=∑ I(P_n). The resulting distribution has peaks located periodically with periods coming as powers of P. Also periodicities with peaks corresponding to n=n⁺n^-, n⁺_q>0 with fixed n^-_q<0, are predicted. These predictions are universal and easily testable. The prime P and integer m characterizing the quantum variant of distribution can be identified from data. The shapes of the distributions obtained are qualitatively consistent with the findings of Shnoll but detailed tests are required to see whether the number theoretic predictions are correct.

The periodic dependence of the distributions would be most naturally assignable to the gravitational interaction of Earth with Sun and Moon and therefore to the periodic variation of Earth-Sun and Earth-Moon distances. The TGD inspired proposal is that the p-dic prime P and integer m characterizing the quantum distribution are determined by a process analogous to a state function reduction and their most probably values depend on the deviation of the distance R through the formulas Δ p/p≈ k_pΔ R/R and Δ m/m≈ k_mΔ R/R. The p-adic primes assignable to elementary particles are very large unlike the primes which could characterize the empirical distributions. The hierarchy of Planck constants allows the gravitational Planck constant assignable to the space-time sheets mediating gravitational interactions to have gigantic values and this allows p-adicity with small values of the p-adic prime P.

A Possible Explanation for Shnoll Effect

Authors: Matti Pitkänen
Comments: 17 Pages.

Shnoll and collaborators have discovered strange repeating patterns of random fluctuations of physical observables such as the number n of nuclear decays in a given time interval. Periodically occurring peaks for the distribution of the number N(n) of measurements producing n events in a series of measurements as a function of n is observed instead of a single peak. The positions of the peaks are not random and the patterns depend on position and time varying periodically in time scales possibly assignable to Earth-Sun and Earth-Moon gravitational interaction.

These observations suggest a modification of the expected probability distributions but it is very difficult to imagine any physical mechanism in the standard physics framework. Rather, a universal deformation of predicted probability distributions would be in question requiring something analogous to the transition from classical physics to quantum physics.

The hint about the nature of the modification comes from the TGD inspired quantum measurement theory proposing a description of the notion of finite measurement resolution in terms of inclusions of so called hyper-finite factors of type II₁ (HFFs) and closely related quantum groups. Also p-adic physics -another key element of TGD- is expected to be involved. A modification of a given probability distribution P(nkenovert λ_i) for a positive integer valued variable n characterized by rational-valued parameters λ_i is obtained by replacing n and the integers characterizing λ_i with so called quantum integers depending on the quantum phase q_m=exp(i2π/m). Quantum integer n_q must be defined as the product of quantum counterparts p_q of the primes p appearing in the prime decomposition of n. One has p_q= sin(2π p/m)/sin(2π/m) for p≠ P and p_q=P for p=P. m must satisfy m≥ 3, m≠ p, and m≠ 2p.

The quantum counterparts of positive integers can be negative. Therefore quantum distribution is defined first as p-adic valued distribution and then mapped by so called canonical identification I to a real distribution by the map taking p-adic -1 to P and powers Pⁿ to P^-n and other quantum primes to themselves and requiring that the mean value of n is for distribution and its quantum variant. The map I satisfies I(∑ P_n)=∑ I(P_n). The resulting distribution has peaks located periodically with periods coming as powers of P. Also periodicities with peaks corresponding to n=n⁺n^-, n⁺_q>0 with fixed n^-_q<0, are predicted. These predictions are universal and easily testable. The prime P and integer m characterizing the quantum variant of distribution can be identified from data. The shapes of the distributions obtained are qualitatively consistent with the findings of Shnoll but detailed tests are required to see whether the number theoretic predictions are correct.

The periodic dependence of the distributions would be most naturally assignable to the gravitational interaction of Earth with Sun and Moon and therefore to the periodic variation of Earth-Sun and Earth-Moon distances. The TGD inspired proposal is that the p-dic prime P and integer m characterizing the quantum distribution are determined by a process analogous to a state function reduction and their most probably values depend on the deviation of the distance R through the formulas Δ p/p≈ k_pΔ R/R and Δ m/m≈ k_mΔ R/R. The p-adic primes assignable to elementary particles are very large unlike the primes which could characterize the empirical distributions. The hierarchy of Planck constants allows the gravitational Planck constant assignable to the space-time sheets mediating gravitational interactions to have gigantic values and this allows p-adicity with small values of the p-adic prime P.

A New Dark Energy Force Theoretically Calculates Faster-than-light-neutrinos.

Authors: Dan Visser
Comments: 13 Pages.

Version2 is written after my paper “Duonistic Neutrinos Violate Relativity”. A theoretical calculation with a new dark energy force formula discloses the correctness of the experimental faster-than-light-neutrinos in the CERN-San Grasso experiment. The formulation in this paper theoretically confirms that Einstein’s Relativity could be violated. This introduces the obligation to accept a new cosmological model, called the Double Torus hypothesis . The theoretical calculation in this paper is based on a new momentum of dark energy force. This paper theoretically calculates 62.8 nanosecond for the experimental detected early-arrival of muon-neutrinos related to how light-in-vacuum would have arrived. This is a marvelous close match compared to the ((60.7 ± 6.9 (stat.) ± 7.4 (sys.)) nanosecond found during the ‘neutrino-flight path’ from CERN to San Grasso in the OPERA-project. However, this version-2 paper also makes clear neutrinos can only go faster-than-light in neutrino-pairs: I call these pairs Duonistic Neutrinos! Just as in the paper “Duonistic Neutrinos Violate Relativity”. This paper presents the set of equations to prove that.
Category: Mathematical Physics

Black Hole vs. Variable Rest Mass Neutron Star

Authors: DT Froedge
Comments: 13 Pages. Clarifying some points of previous version

In a previous paper we have discussed the conjecture of a variable particle rest mass as a function of gravitational potential [1]. This paper discuses the implications, in regard to a large neutron star, and contrast the difference between the predicted phenomena, and Black Hole theory as put fourth by standard GR. As most know, Einstein was not convinced of the existence of Black Holes, but modern solutions of the GR field equations appear to agree with the experimental evidence. There are some problems however, as are well known, the explanations for the diffuse, and the persistent source gamma ray emissions from the galactic center, do not have an adequate explanation, and the energy engines driving Quasars and AGNs are not sufficiently explained. This paper will explore the differences between VRM neutron stars, and Black Holes, for the purpose of identifying detectable, and measurable phenomena. The validity of this theory will be established on the finding of massive neutron stars or massive pulsars. >3 suns.
Category: Mathematical Physics

Exact Solution of Viscous-Plastic Flow Equations for Glacier Dynamics in 2-Dimensional Case.

Authors: Sergey V. Ershkov
Comments: 9 pages

Here is presented a new exact solution of Ice dynamics in Glaciers in terms of viscousplastic theory of movements, for 2-dimensional case: x (t) = y (t). In general case, 2-D solution of Ice dynamics could be classified as Riccati's type. Due to a very special character of Riccati's type equation, it's general solution is proved to have a proper gap of components of such a solution.
Category: Mathematical Physics

Exact Solution of Viscous-Plastic Flow Equations for Glacier Dynamics in 2-Dimensional Case.

Authors: Sergey V. Ershkov
Comments: 7 pages

Here is presented a new exact solution of Ice dynamics in Glaciers in terms of viscousplastic theory of movements, for 2-dimensional case: x (t) = y (t). In general case, 2-D solution of Ice dynamics could be classified as Riccati's type. Due to a very special character of Riccati's type equation, it's general solution is proved to have a proper gap of components of such a solution.
Category: Mathematical Physics

Hard Theoretical Evidence for the Dark Energy Force Formula in a Double Torus Universe.

Authors: Dan Visser
Comments: 5 Pages.

This paper shows how my ‘dark energy force formula’ emerges five more space- and two more time-dimensions in nature. The ‘formula’ is earlier described in vixra-papers, announcing the universe is a Double Torus of dark energy and dark matter. The ‘formula’ represents a completely different force than the cosmological constant of Einstein, which is used to explain the accelerated expansion in Big Bang cosmology. With this in mind, two independent experimental investigations have given additional proof for my ‘dark energy force formula’, as follows: 1) The ratio of five extra space-dimensions and two extra time dimensions represent the up- and down electron-spin, behaving in a ‘chessboard-structure’ and associated with grapheme-experiments. 2) A computer-simulation shows a Double Torus, artificially emerging from two colliding black-holes. These results match with my hypothesis that the universe indeed is a Double Torus of dark energy and dark matter, because my (new) ’dark energy force formula‘ matches these two investigations. This (new) force could therefore have a link to topological insulators to guide light without scattering in quantum-computers.
Category: Mathematical Physics

A Note on the Action at a Distance

Authors: José Francisco García Juliá
Comments: 5 Pages.

We consider that the action at a distance is carried out by virtual carriers of the force, which are created and annihilated in the vacuum by the field in a time less than that of the Heisenberg's uncertainty.
Category: Mathematical Physics

A Note on the Action at a Distance

Authors: José Francisco García Juliá
Comments: 5 Pages.

We consider that the action at a distance is carried out by virtual carriers of the force, which are created and annihilated in the vacuum by the field in a time less than that of the Heisenberg's uncertainty.
Category: Mathematical Physics

A Note on the Action at a Distance

Authors: José Francisco García Juliá
Comments: 3 Pages.

We consider that the action at a distance is carried out by virtual carriers of the force, which are created and annihilated in the vacuum by the field in a time less than that of the Heisenberg's uncertainty.
Category: Mathematical Physics

Quintessence-Momentum as Link Between Mass and Charge

Authors: Malcolm Macleod
Comments: 3 pages. v1 in Russian, v4 in English

The natural constants, G; h; e, μ₀ and m_e are presented as geometrical shapes in terms of Planck momentum, α (Sommerfeld fine structure constant) and c. A square root solution of Planck momentum denoted Quintessence-momentum Q links the mass and charge constants. The electron formula describes a dimensionless magnetic monopole. The Rydberg constant R1, the most accurate of the natural constants, is used for crossreference, the solutions are consistent with CODATA 2010 precision.
Category: Mathematical Physics

Quintessence-Momentum as Link Between Mass and Charge

Authors: Malcolm Macleod
Comments: 3 pages. v1 in Russian, v3 in English

A mathematical description of the natural constants, G, h, e, μ₀ m_e R_∞, is presented in terms of momentum Q, alpha (Sommerfeld fine structure constant) and c. This momentum is referred to as Quintessence-momentum and is the square root of Planck momentum. The formulas describe geometrical forms, the units are consistent with corresponding SI units and the numerical values, including the Rydberg constant and the vacuum permeability, are consistent with CODATA 2006.
Category: Mathematical Physics

Quintessence-Momentum as Link Between Mass and Charge

Authors: Malcolm Macleod
Comments: 3 pages. v1 in Russian, v2 in English

This paper suggests a 'quantity of momentum', a square root of Planck momentum, here referred to as Quintessence-momentum, as a natural unit that is common to both mass and charge. In terms of this Quintessence momentum Q, alpha (Sommerfeld fine structure constant) and c; geometrical formulas for the natural physical constants and the electron mass are proposed. Results are consistent with CODATA 2006.
Category: Mathematical Physics

Scale Dimension as the Fifth Dimension of Spacetime

Authors: Sergey G. Fedosin
Comments: 5 Pages. Turkish Journal of Physics, 2012, Vol. 36, No 3, P. 461 – 464.

The scale dimension discovered in the theory of infinite nesting of matter is studied from the perspective of physical implementation of well-studied four-and n-dimensional geometric objects. Adding the scale dimension to Minkowski four-dimensional space means the necessity to use the five-dimensional spacetime.
Category: Mathematical Physics

Scale Dimension as the Fifth Dimension of Spacetime

Authors: Sergey G. Fedosin
Comments: 4 pages. v1 in Russian, v2 in English

The scale dimension which is discovered in the theory of infinite nesting of matter is studied from the perspective of the physical implementation of well-studied four-and n-dimensional geometric objects. Adding of the scale dimension to Minkowski space means the need to use the five-dimensional spacetime.
Category: Mathematical Physics

Fine Structure Constant α ~ 1/137.036 and Blackbody Radiation Constant α_R ~ 1/157.555

Authors: Ke Xiao
Comments: 5 pages

The fine structure constant α = e²/hc ~ 1/137.036 and the blackbody radiation constant α_R = e²(a_R/k⁴_B)^1/3 ~ 1/157.555 are linked by prime numbers. The blackbody radiation constant is a new method to measure the fine structure constant. It also links the fine structure constant to the Boltzmann constant.
Category: Mathematical Physics

32 Point Groups of Three Dimensional Crystal Cells Described by 5 Bits

Authors: Giuliano Bettini
Comments: 9 pages, v3 in Italian, v2 in English, corrections to the tables, and a new table added.

There are 32 possible combinations of symmetry operations that define the external symmetry of crystals. These 32 possible combinations result in the 32 crystal classes. But for a radar engineer it is inevitable to associate "32" to "5 bits". I submit a tentative classification of the 32 crystal classes with a 5 bit classification, obviously with a (tentative) physical meaning of each bit. Each bit means a physical property.
Category: Mathematical Physics

32 Point Groups of Three Dimensional Crystal Cells Described by 5 Bits

Authors: Giuliano Bettini
Comments: 9 pages, v1 in Italian, v2 in English, corrections to the tables, and a new table added.

There are 32 possible combinations of symmetry operations that define the external symmetry of crystals. These 32 possible combinations result in the 32 crystal classes. But for a radar engineer it is inevitable to associate "32" to "5 bits". I submit a tentative classification of the 32 crystal classes with a 5 bit classification, obviously with a (tentative) physical meaning of each bit. Each bit means a physical property.
Category: Mathematical Physics

Further on Non-Cartesian Systems

Authors: Elemér E Rosinger
Comments: 9 pages

A class of non-Cartesian physical systems, [7], are those whose composite state spaces are given by significantly extended tensor products. A more detailed presentation of the way such extended tensor products are constructed is offered, based on a step by step comparison with the construction of usual tensor products. This presentation clarifies the extent to which the extended tensor products are indeed more general than the usual ones.
Category: Mathematical Physics

Deeper Properties Through Dark and Visible-Matter in a New Cosmological Twin-Tori Model (TTM).

Authors: Dan Visser
Comments: 5 pages

A new cosmological model, named the Twin-Tori Model (TTM)[1], postulates a dark energy force F_de , which empowers the dynamic of a lower order universe, well known as the big bang. In this paper I introduce the 1st derivative F'_de of this dark energy force to reveal deeper properties of the TTM, such as: why quantummechanics exists in the big bang, why dark matter and visible matter are equally responsible for gravity in galaxies for 1/4 of the density of dark matter at a specific length, why the big bang universe is recalculated by subquantumlevel-information below the Plancklength, and why the impression of space-expansion is due to the higher order cosmological model TTM.
Category: Mathematical Physics

Thought-Experiment Provides a Formula for (New) Dark Energy Force.

Authors: Dan Visser
Comments: 12 Pages.

A (new) ‘dark energy force-formula’ was introduced on April 10 2004 on Dan Visser’s website. His (new) formula was picked up by a PhD-mathematician and - Physics, Christopher Forbes (UK) in the summer of 2009, leading to email-contact among them, and resulting in a publication of a general mathematical expression, whereof Dan Visser’s (new) ‘dark energy force-formula’ came out as a solution. The derivation of this (new) force was then published in the Vixra-archive on October 7 2010 (in retrospective). The ‘thought-experiment’ is described as a mathematical exercise, which in the end is expressed as Dan Visser's (new) ‘dark energy force-formula’. Since then only textual changes were made for the benefit of a better explanation of the hypothesis, without altering the original mathematical content. The main issue in the ‘thought-experiment’ is a ‘non-relativistic scaling principle’, characterized as ‘scaling-away’ two black holes from each other (small and large), as well as ‘melting them together’. Both ‘movements’ presents a ‘change of dark information’. This was further analyzed, synthesized, combined and translated mathematically by Dan Visser, into a force of ‘extra time’ hitting an observer’s eye. The result is the (new) ‘dark energy force’, a ‘non-relativistic force’.
Category: Mathematical Physics

Summary of the Zeta Regularization Method Applied to the Calculation of Divergent Series and Divergent Integrals

Authors: Jose Javier Garcia Moreta
Comments: 20 Pages.

•ABSTRACT: We study a generalization of the zeta regularization method applied to the case of the regularization of divergent integrals for positive ‘s’ , using the Euler Maclaurin summation formula, we manage to express a divergent integral in term of a linear combination of divergent series , these series can be regularized using the Riemann Zeta function s >0 , in the case of the pole at s=1 we use a property of the Functional determinant to obtain the regularization , with the aid of the Laurent series in one and several variables we can extend zeta regularization to the cases of integrals , we believe this method can be of interest in the regularization of the divergent UV integrals in Quantum Field theory since our method would not have the problems of the Analytic regularization or dimensional regularization •Keywords: = Riemann Zeta function, Functional determinant, Zeta regularization, divergent series .
Category: Mathematical Physics

A Multiple Particle System Equation Underlying the Klein-Gordon-Dirac-Schrödinger Equations

Authors: DT Froedge
Comments: 36 Pages. V032112 ongoing

The purpose of this paper is to illustrate a fundamental, multiple particle, system equation for which the Klein-Gordon-Dirac-Schrödinger equations are, and single particle special cases. The basic concept is that there is a broader picture, based on a more general equation that includes the entire system of particles. The first part will be to postulate an equation, and then, by then by defining an action field based on the endpoint action of the particles in the system, develop a solution which properly illustrates internal dynamics as well as particle interactions. The complete function has both real, and imaginary, as well as timelike and spacelike parts, each of which are separable into independent expressions that define particle properties. In the same manner that eigenvalues of the Schrödinger equation represents energy levels of an atomic system, particle masses are eigenvalues in an interacting universe of particles. The Dirac massive and massless equation and solution will be shown as factorable independent parts of the Systemfunction. A clear relation between the classical and quantum properties of particles is made, increasing the scope of QM.
Category: Mathematical Physics

A Multiple Particle System Equation Underlying the Klein-Gordon-Dirac-Schrödinger Equations

Authors: DT Froedge
Comments: 39 Pages. For presentation April 12 APS Atlanta Verson V012612 (ongoing)

The purpose of this paper is to illustrate a fundamental, multiple particle, system equation for which the Klein-Gordon-Dirac-Schrödinger equations are approximations, and single particle special cases. The basic concept is that there is a broader picture, based on a more general equation that includes the entire system of particles. The first part will be to postulate an equation, and then, by then by defining an action field based on the endpoint action of the particles in the system, develop a solution which properly illustrates internal dynamics as well as particle interactions. The complete function has both real, and imaginary, as well as timelike and spacelike parts, each of which are separable into independent expressions that define particle properties. In the same manner that eigenvalues of the Schrödinger equation represents energy levels of an atomic system, particle masses are eigenvalues in an interacting universe of particles. The Dirac massive and massless equation and solution will be shown as factorable independent parts of the Systemfunction. A clear relation between the classical and quantum properties of particles is made, increasing the scope of QM.
Category: Mathematical Physics

A Multiple Particle System Equation Underlying the Klein-Gordon-Dirac-Schrödinger Equations

Authors: DT Froedge
Comments: 39 Pages.

The purpose of this paper is to illustrate a fundamental, multiple particle, system equation for which the Klein-Gordon-Dirac-Schrödinger equations are single particle special cases. The basic concept is that there is a broader picture, based on a more general equation that includes the entire system of particles. The first part will be to postulate an equation, and then, by modifying the methods of Path Integrals, develop a solution which describes the internal dynamics as well as particle interactions of quantum particles. The complete function has both real and imaginary, as well as timelike and spacelike parts, each of which are separable into independent expressions that define particle properties. In the same manner that eigenvalues of the Schrödinger equation represents energy levels of an atomic system, particle are eigenvalues in an interacting universe of particles. The Dirac massive and massless equation and solution will be shown as factorable independent components. A clear distinction between the classical and quantum properties of particles is made, increasing the scope of QM.
Category: Mathematical Physics

A Multiple Particle System Equation Underlying the Klein-Gordon-Dirac-Schrödinger Equations

Authors: D.T. Froedge
Comments: 30 pages.

The purpose of this paper is to illustrate a fundamental, multiple particle, system equation for which the Klein-Gordon-Dirac-Schrödinger equations are single particle special cases. In the same manner that eigenvalues of the Schrödinger equation represents energy levels of an interacting atomic system, eigenvalues represent particle energies in an interacting system of particles. An equation and a solution is proposed that treats all of the particles in the universe as a single system. The proposed solution is a descriptor of a symmetric, light speed expanding group of interacting particles having familiar constituents.
Category: Mathematical Physics

A Multiple Particle System Equation Underlying the Klein-Gordon-Dirac-Schrödinger Equations

Authors: D.T. Froedge
Comments: 17 pages 38 equations 98kb

The purpose of this paper is to illustrate a fundamental, multiple particle, system equation for which the Klein-Gordon-Dirac-Schrödinger equations are single particle special cases. In the same manner that eigenvalues of the Schrödinger equation represents energy levels of an interacting atomic system, eigenvalues represent particle energies in an interacting system of particles. An equation is proposed that has vector solutions defined in Dirac, or Clifford algebra, that treats all of the particles in the universe as a single system. The proposed solution is a descriptor of a symmetric, light speed expanding group of interacting particles having real, as well as the familiar QM constituents.
Category: Mathematical Physics

A Hamiltonian Whose Energies Are the Zeros of the Riemann XI-Function

Authors: Joes Javier Garcia Moreta
Comments: 11 Pages.

• ABSTRACT: We give an spectral interpretation of the non-trivial zeros of the Riemann Xi-function. The main idea of the paper is to find a Hamiltonian (Hermitian) operator in the form with an even potential whose energies are precisely , the zeros of the Riemann Xi-function . In order to obtain this Hamiltonian we use the WKB method and the Bohr-sommerfeld quantization condition for the energies. We also prove the fact that the Riemann Xi-function is proportional to the Functional determinant in the sense of a zeta-regularized determinant In this paper and for simplicity we use units so • Keywords: = Riemann Hypothesis, Functional determinant, WKB semiclassical Approximation , Trace formula ,Bolte’s law.
Category: Mathematical Physics

The Geometry of CP₂ and its Relationship to Standard Model

Authors: Matti Pitkänen
Comments: 13 Pages.

This appendix contains basic facts about CP₂ as a symmetric space and Kähler manifold. The coding of the standard model symmetries to the geometry of CP₂, the physical interpretation of the induced spinor connection in terms of electro-weak gauge potentials, and basic facts about induced gauge fields are discussed
Category: Mathematical Physics

Could the Dynamics of Kähler Action Predict the Hierarchy of Planck Constants?

Authors: Matti Pitkänen
Comments: 5 Pages.

The original justification for the hierarchy of Planck constants came from the indications that Planck constant could have large values in both astrophysical systems involving dark matter and also in biology. The realization of the hierarchy in terms of the singular coverings and possibly also factor spaces of CD and CP₂ emerged from consistency conditions. It however seems that TGD actually predicts this hierarchy of covering spaces. The extreme non-linearity of the field equations defined by Kähler action means that the correspondence between canonical momentum densities and time derivatives of the imbedding space coordinates is 1-to-many. This leads naturally to the introduction of the covering space of CD x CP₂, where CD denotes causal diamond defined as intersection of future and past directed light-cones.
Category: Mathematical Physics

Weak Form of Electric-Magnetic Duality and Its Implications

Authors: Matti Pitkänen
Comments: 25 Pages.

The notion of electric magnetic duality emerged already two decades ago in the attempts to formulate the Kähler geometry of the "world of classical worlds". Quite recently a considerable step of progress took place in the understanding of this notion. This concept leads to the identification of the physical particles as string like objects defined by magnetic charged wormhole throats connected by magnetic ux tubes. The second end of the string contains particle having electroweak isospin neutralizing that of elementary fermion and the size scale of the string is electro-weak scale would be in question. Hence the screening of electro-weak force takes place via weak confinement. This picture generalizes to magnetic color confinement. Electric-magnetic duality leads also to a detailed understanding of how TGD reduces to almost topological quantum field theory. A surprising outcome is the necessity to replace CP₂ Kähler form in Kähler action with its sum with S² Kähler form.
Category: Mathematical Physics

How to Define Generalized Feynman Diagrams?

Authors: Matti Pitkänen
Comments: 16 Pages.

Generalized Feynman diagrams have become the central notion of quantum TGD and one might even say that space-time surfaces can be identified as generalized Feynman diagrams. The challenge is to assign a precise mathematical content for this notion, show their mathematical existence, and develop a machinery for calculating them. Zero energy ontology has led to a dramatic progress in the understanding of generalized Feynman diagrams at the level of fermionic degrees of freedom. In particular, manifest finiteness in these degrees of freedom follows trivially from the basic identifications as does also unitarity and non-trivial coupling constant evolution. There are however several formidable looking challenges left.

1. One should perform the functional integral over WCW degrees of freedom for fixed values of on mass shell momenta appearing in the internal lines. After this one must perform integral or summation over loop momenta.
2. One must define the functional integral also in the p-adic context. p-Adic Fourier analysis relying on algebraic continuation raises hopes in this respect. p-Adicity suggests strongly that the loop momenta are discretized and ZEO predicts this kind of discretization naturally.

Physics as Generalized Number Theory: Infinite Primes

Authors: Matti Pitkänen
Comments: 38 Pages.

The focus of this book is the number theoretical vision about physics. This vision involves three loosely related parts.

1. The fusion of real physic and various p-adic physics to a single coherent whole by generalizing the number concept by fusing real numbers and various p-adic number fields along common rationals. Extensions of p-adic number fields can be introduced by gluing them along common algebraic numbers to reals. Algebraic continuation of the physics from rationals and their their extensions to various number fields (generalization of completion process for rationals) is the key idea, and the challenge is to understand whether how one could achieve this dream. A profound implication is that purely local p-adic physics would code for the p-adic fractality of long length length scale real physics and vice versa, and one could understand the origins of p-adic length scale hypothesis.
2. Second part of the vision involves hyper counterparts of the classical number fields defined as subspaces of their complexifications with Minkowskian signature of metric. Allowed space-time surfaces would correspond to what might be called hyper-quaternionic sub-manifolds of a hyper-octonionic space and mappable to M⁴× CP₂ in natural manner. One could assign to each point of space-time surface a hyper-quaternionic 4-plane which is the plane defined by the modified gamma matrices but not tangent plane in general. Hence the basic variational principle of TGD would have deep number theoretic content.
3. The third part of the vision involves infinite primes identifiable in terms of an infinite hierarchy of second quantized arithmetic quantum fields theories on one hand, and as having representations as space-time surfaces analogous to zero loci of polynomials on the other hand. Single space-time point would have an infinitely complex structure since real unity can be represented as a ratio of infinite numbers in infinitely many manners each having its own number theoretic anatomy. Single space-time point would be in principle able to represent in its structure the quantum state of the entire universe. This number theoretic variant of Brahman=Atman identity would make Universe an algebraic hologram.

   Number theoretical vision suggests that infinite hyper-octonionic or -quaternionic primes could could correspond directly to the quantum numbers of elementary particles and a detailed proposal for this correspondence is made. Furthermore, the generalized eigenvalue spectrum of the Chern-Simons Dirac operator could be expressed in terms of hyper-complex primes in turn defining basic building bricks of infinite hyper-complex primes from which hyper-octonionic primes are obtained by dicrete SU(3) rotations performed for finite hyper-complex primes.

Besides this holy trinity I will discuss loosely related topics. Included are possible applications of category theory in TGD framework; TGD inspired considerations related to Riemann hypothesis; topological quantum computation in TGD Universe; and TGD inspired approach to Langlands program.

Category: Mathematical Physics

Physics as Generalized Number Theory: Infinite Primes

Authors: Matti Pitkänen
Comments: 695 Pages.

The focus of this book is the number theoretical vision about physics. This vision involves three loosely related parts.

1. The fusion of real physic and various p-adic physics to a single coherent whole by generalizing the number concept by fusing real numbers and various p-adic number fields along common rationals. Extensions of p-adic number fields can be introduced by gluing them along common algebraic numbers to reals. Algebraic continuation of the physics from rationals and their their extensions to various number fields (generalization of completion process for rationals) is the key idea, and the challenge is to understand whether how one could achieve this dream. A profound implication is that purely local p-adic physics would code for the p-adic fractality of long length length scale real physics and vice versa, and one could understand the origins of p-adic length scale hypothesis.
2. Second part of the vision involves hyper counterparts of the classical number fields defined as subspaces of their complexifications with Minkowskian signature of metric. Allowed space-time surfaces would correspond to what might be called hyper-quaternionic sub-manifolds of a hyper-octonionic space and mappable to M⁴× CP₂ in natural manner. One could assign to each point of space-time surface a hyper-quaternionic 4-plane which is the plane defined by the modified gamma matrices but not tangent plane in general. Hence the basic variational principle of TGD would have deep number theoretic content.
3. The third part of the vision involves infinite primes identifiable in terms of an infinite hierarchy of second quantized arithmetic quantum fields theories on one hand, and as having representations as space-time surfaces analogous to zero loci of polynomials on the other hand. Single space-time point would have an infinitely complex structure since real unity can be represented as a ratio of infinite numbers in infinitely many manners each having its own number theoretic anatomy. Single space-time point would be in principle able to represent in its structure the quantum state of the entire universe. This number theoretic variant of Brahman=Atman identity would make Universe an algebraic hologram.

   Number theoretical vision suggests that infinite hyper-octonionic or -quaternionic primes could could correspond directly to the quantum numbers of elementary particles and a detailed proposal for this correspondence is made. Furthermore, the generalized eigenvalue spectrum of the Chern-Simons Dirac operator could be expressed in terms of hyper-complex primes in turn defining basic building bricks of infinite hyper-complex primes from which hyper-octonionic primes are obtained by dicrete SU(3) rotations performed for finite hyper-complex primes.

Besides this holy trinity I will discuss loosely related topics. Included are possible applications of category theory in TGD framework; TGD inspired considerations related to Riemann hypothesis; topological quantum computation in TGD Universe; and TGD inspired approach to Langlands program.

Category: Mathematical Physics

Physics as Generalized Number Theory: Classical Number Fields

Authors: Matti Pitkänen
Comments: 28 Pages.

Physics as a generalized number theory program involves three threads: various p-adic physics and their fusion together with real number based physics to a larger structure, the attempt to understand basic physics in terms of classical number fields discussed in this article, and infinite primes whose construction is formally analogous to a repeated second quantization of an arithmetic quantum field theory. In this article the connection between standard model symmetries and classical number fields is discussed. The basis vision is that the geometry of the infinite-dimensional WCW ("world of classical worlds") is unique from its mere existence. This leads to its identification as union of symmetric spaces whose Kähler geometries are fixed by generalized conformal symmetries. This fixes space-time dimension and the decomposition M⁴ x S and the idea is that the symmetries of the Kähler manifold S make it somehow unique. The motivating observations are that the dimensions of classical number fields are the dimensions of partonic 2-surfaces, space-time surfaces, and imbedding space and M⁸ can be identified as hyper-octonions- a sub-space of complexified octonions obtained by adding a commuting imaginary unit. This stimulates some questions. Could one understand S = CP₂ number theoretically in the sense that M⁸ and H = M⁴ x CP₂ be in some deep sense equivalent ("number theoretical compactification" or M⁸ - H duality)? Could associativity define the fundamental dynamical principle so that space-time surfaces could be regarded as associative or co-associative (defined properly) sub-manifolds of M⁸ or equivalently of H. One can indeed define the associativite (co-associative) 4-surfaces using octonionic representation of gamma matrices of 8-D spaces as surfaces for which the modified gamma matrices span an associate (co-associative) sub-space at each point of space-time surface. Also M⁸ - H duality holds true if one assumes that this associative sub-space at each point contains preferred plane of M⁸ identifiable as a preferred commutative or co-commutative plane (this condition generalizes to an integral distribution of commutative planes in M⁸). These planes are parametrized by CP₂ and this leads to M⁸ - H duality. WCW itself can be identified as the space of 4-D local sub-algebras of the local Clifford algebra of M⁸ or H which are associative or co-associative. An open conjecture is that this characterization of the space-time surfaces is equivalent with the preferred extremal property of Kähler action with preferred extremal identified as a critical extremal allowing infinite-dimensional algebra of vanishing second variations.
Category: Mathematical Physics

Physics as Generalized Number Theory: P-Adic Physics and Number Theoretic Universality

Authors: Matti Pitkänen
Comments: 51 Pages.

Physics as a generalized number theory program involves three threads: various p-adic physics and their fusion together with real number based physics to a larger structure, the attempt to understand basic physics in terms of classical number fields (in particular, identifying associativity condition as the basic dynamical principle), and infinite primes whose construction is formally analogous to a repeated second quantization of an arithmetic quantum field theory. In this article p-adic physics and the technical problems relates to the fusion of p-adic physics and real physics to a larger structure are discussed. The basic technical problems relate to the notion of definite integral both at space-time level, imbedding space level and the level of WCW (the "world of classical worlds"). The expressibility of WCW as a union of symmetric spacesleads to a proposal that harmonic analysis of symmetric spaces can be used to define various integrals as sums over Fourier components. This leads to the proposal the p-adic variant of symmetric space is obtained by a algebraic continuation through a common intersection of these spaces, which basically reduces to an algebraic variant of coset space involving algebraic extension of rationals by roots of unity. This brings in the notion of angle measurement resolution coming as Δφ = 2π/pⁿ for given p-adic prime p. Also a proposal how one can complete the discrete version of symmetric space to a continuous p-adic versions emerges and means that each point is effectively replaced with the p-adic variant of the symmetric space identifiable as a p-adic counterpart of the real discretization volume so that a fractal p-adic variant of symmetric space results. If the Kähler geometry of WCW is expressible in terms of rational or algebraic functions, it can in principle be continued the p-adic context. One can however consider the possibility that that the integrals over partonic 2-surfaces defining ux Hamiltonians exist p-adically as Riemann sums. This requires that the geometries of the partonic 2-surfaces effectively reduce to finite sub-manifold geometries in the discretized version of δM₊⁴. If Kähler action is required to exist p-adically same kind of condition applies to the space-time surfaces themselves. These strong conditions might make sense in the intersection of the real and p-adic worlds assumed to characterized living matter.
Category: Mathematical Physics

Construction of Configuration Space Spinor Structure

Authors: Matti Pitkänen
Comments: 95 Pages.

There are three separate approaches to the challenge of constructing WCW Kähler geometry and spinor structure. The first approach relies on a direct guess of Kähler function. Second approach relies on the construction of Kähler form and metric utilizing the huge symmetries of the geometry needed to guarantee the mathematical existence of Riemann connection. The third approach discussed in this article relies on the construction of spinor structure based on the hypothesis that complexified WCW gamma matrices are representable as linear combinations of fermionic oscillator operator for the second quantized free spinor fields at space-time surface and on the geometrization of super-conformal symmetries in terms of spinor structure. This implies a geometrization of fermionic statistics. The basic philosophy is that at fundamental level the construction of WCW geometry reduces to the second quantization of the induced spinor fields using Dirac action. This assumption is parallel with the bosonic emergence stating that all gauge bosons are pairs of fermion and antifermion at opposite throats of wormhole contact. Vacuum function is identified as Dirac determinant and the conjecture is that it reduces to the exponent of Kähler function. In order to achieve internal consistency induced gamma matrices appearing in Dirac operator must be replaced by the modified gamma matrices defined uniquely by Kähler action and one must also assume that extremals of Kähler action are in question so that the classical space-time dynamics reduces to a consistency condition. This implies also super-symmetries and the fermionic oscillator algebra at partonic 2-surfaces has intepretation as N = 1 generalization of space-time supersymmetry algebra different however from standard SUSY algebra in that Majorana spinors are not needed. This algebra serves as a building brick of various super-conformal algebras involved. The requirement that there exist deformations giving rise to conserved Noether charges requires that the preferred extremals are critical in the sense that the second variation of the Kähler action vanishes for these deformations. Thus Bohr orbit property could correspond to criticality or at least involve it. Quantum classical correspondence demands that quantum numbers are coded to the properties of the preferred extremals given by the Dirac determinant and this requires a linear coupling to the conserved quantum charges in Cartan algebra. Effective 2-dimensionality allows a measurement interaction term only in 3-D Chern-Simons Dirac action assignable to the wormhole throats and the ends of the space-time surfaces at the boundaries of CD. This allows also to have physical propagators reducing to Dirac propagator not possible without the measurement interaction term. An essential point is that the measurement interaction corresponds formally to a gauge transformation for the induced Kähler gauge potential. If one accepts the weak form of electric-magnetic duality Kähler function reduces to a generalized Chern-Simons term and the effect of measurement interaction term to Kähler function reduces effectively to the same gauge transformation. The basic vision is that WCW gamma matrices are expressible as super-symplectic charges at the boundaries of CD. The basic building brick of WCW is the product of infinite-D symmetric spaces assignable to the ends of the propagator line of the generalized Feynman diagram. WCW Kähler metric has in this case "kinetic" parts associated with the ends and "interaction" part between the ends. General expressions for the super-counterparts of WCW ux Hamiltoniansand for the matrix elements of WCW metric in terms of their anticommutators are proposed on basis of this picture.
Category: Mathematical Physics

Construction of Configuration Space Geometry from Symmetry Principles

Authors: Matti Pitkänen
Comments: 26 Pages.

There are three separate approaches to the challenge of constructing WCW Kähler geometry and spinor structure. The first one relies on a direct guess of Kähler function. Second approach relies on the construction of Kähler form and metric utilizing the huge symmetries of the geometry needed to guarantee the mathematical existence of Riemann connection. The third approach relies on the construction of spinor structure assuming that complexified WCW gamma matrices are representable as linear combinations of fermionic oscillator operator for the second quantized free spinor fields at space-time surface and on the geometrization of super-conformal symmetries in terms of spinor structure. In this article the construction of Kähler form and metric based on symmetries is discussed. The basic vision is that WCW can be regarded as the space of generalized Feynman diagrams with lines thickned to light-like 3-surfaces and vertices identified as partonic 2-surfaces. In zero energy ontology the strong form of General Coordinate Invariance (GCI) implies effective 2-dimensionality and the basic objects are pairs partonic 2-surfaces X² at opposite light-like boundaries of causal diamonds (CDs). The hypothesis is that WCW can be regarded as a union of infinite-dimensional symmetric spaces G/H labeled by zero modes having an interpretation as classical, non-quantum uctuating variables. A crucial role is played by the metric 2-dimensionality of the light-cone boundary δM₊⁴ + and of light-like 3-surfaces implying a generalization of conformal invariance. The group G acting as isometries of WCW is tentatively identified as the symplectic group of δM₊⁴ x CP₂ localized with respect to X². H is identified as Kac-Moody type group associated with isometries of H = M₊⁴ x CP₂ acting on light-like 3-surfaces and thus on X². An explicit construction for the Hamiltonians of WCW isometry algebra as so called ux Hamiltonians is proposed and also the elements of Kähler form can be constructed in terms of these. Explicit expressions for WCW ux Hamiltonians as functionals of complex coordinates of the Cartesisian product of the infinite-dimensional symmetric spaces having as points the partonic 2-surfaces defining the ends of the the light 3-surface (line of generalized Feynman diagram) are proposed.
Category: Mathematical Physics

Identification of the Configuration Space Kähler Function

Authors: Matti Pitkänen
Comments: 29 Pages.

There are two basic approaches to quantum TGD. The first approach, which is discussed in this article, is a generalization of Einstein's geometrization program of physics to an infinitedimensional context. Second approach is based on the identification of physics as a generalized number theory. The first approach relies on the vision of quantum physics as infinite-dimensional Kähler geometry for the "world of classical worlds" (WCW) identified as the space of 3-surfaces in in certain 8-dimensional space. There are three separate approaches to the challenge of constructing WCW Kähler geometry and spinor structure. The first approach relies on direct guess of Kähler function. Second approach relies on the construction of Kähler form and metric utilizing the huge symmetries of the geometry needed to guarantee the mathematical existence of Riemann connection. The third approach relies on the construction of spinor structure based on the hypothesis that complexified WCW gamma matrices are representable as linear combinations of fermionic oscillator operator for second quantized free spinor fields at space-time surface and on the geometrization of super-conformal symmetries in terms of WCW spinor structure. In this article the proposal for Kähler function based on the requirement of 4-dimensional General Coordinate Invariance implying that its definition must assign to a given 3-surface a unique space-time surface. Quantum classical correspondence requires that this surface is a preferred extremal of some some general coordinate invariant action, and so called Kähler action is a unique candidate in this respect. The preferred extremal has intepretation as an analog of Bohr orbit so that classical physics becomes and exact part of WCW geometry and therefore also quantum physics. The basic challenge is the explicit identification of WCW Kähler function K. Two assumptions lead to the identification of K as a sum of Chern-Simons type terms associated with the ends of causal diamond and with the light-like wormhole throats at which the signature of the induced metric changes. The first assumption is the weak form of electric magnetic duality. Second assumption is that the Kähler current for preferred extremals satisfies the condition jK ^ djK = 0 implying that the ow parameter of the ow lines of jK defines a global space-time coordinate. This would mean that the vision about reduction to almost topological QFT would be realized. Second challenge is the understanding of the space-time correlates of quantum criticality. Electric-magnetic duality helps considerably here. The realization that the hierarchy of Planck constant realized in terms of coverings of the imbedding space follows from basic quantum TGD leads to a further understanding. The extreme non-linearity of canonical momentum densities as functions of time derivatives of the imbedding space coordinates implies that the correspondence between these two variables is not 1-1 so that it is natural to introduce coverings of CD x CP₂. This leads also to a precise geometric characterization of the criticality of the preferred extremals.
Category: Mathematical Physics

Physics as Infinite-dimensional Geometry and Generalized Number Theory: Basic Visions

Authors: Matti Pitkänen
Comments: 33 Pages.

There are two basic approaches to the construction of quantum TGD. The first approach relies on the vision of quantum physics as infinite-dimensional Kähler geometry for the "world of classical worlds" identified as the space of 3-surfaces in in certain 8-dimensional space. Essentially a generalization of the Einstein's geometrization of physics program is in question. The second vision is the identification of physics as a generalized number theory. This program involves three threads: various p-adic physics and their fusion together with real number based physics to a larger structure, the attempt to understand basic physics in terms of classical number fields (in particular, identifying associativity condition as the basic dynamical principle), and infinite primes whose construction is formally analogous to a repeated second quantization of an arithmetic quantum field theory. In this article brief summaries of physics as infinite-dimensional geometry and generalized number theory are given to be followed by more detailed articles.
Category: Mathematical Physics

Zeta Regularization Applied to the Problem of Riemann Hypothesis and the Calculation of Divergent Integrals

Authors: Jose Javier Garcia Moreta
Comments: 18 Pages.

ABSTRACT: In this paper we review some results of our previous papers involving Riemann Hypothesis in the sense of Operator theory (Hilbert-Polya approach) and the application of the negative values of the Zeta function  (1 s) to the divergent integrals 1 0 s x dx    and to the problem of defining a consistent product of distributions of the form ( ) ( ) m n D  x D  x , in this paper we present new results of how the sums over the non-trivial zeros of the zeta function h( )    can be related to the Mangoldt function 0 (x) assuming Riemann Hypothesis.Throughout the paper we will use the notation ( ) ( ) R s  s meaning that we use the zeta regularization for the divergent series 0 s n n    s>0 or s=0
Category: Mathematical Physics

3x3 Unitary to Magic Matrix Transformations

Authors: Philip Gibbs
Comments: 5 pages

We prove that any 3x3 unitary matrix can be transformed to a magic matrix by multiplying its rows and columns by phase factors. A magic matrix is defined as one for which the sum of the elements in any row or column add to the same value. This result is relevant to recent observations on particle mixing matrices.
Category: Mathematical Physics