Authors: Carlos Castro
Modifications of the Weyl-Heisenberg algebra $ [ { \bf x}^i, {\bf p}^j ] = i \hbar g^{ij} ( {\bf p } ) $ are proposed where the classical limit $g_{ij} ( p ) $ corresponds to a metric in (curved) momentum spaces. In the simplest scenario, the $ 2D$ de Sitter metric of constant curvature in momentum space furnishes a hierarchy of modified uncertainty relations leading to a minimum value for the position uncertainty $ \Delta x $. The first uncertainty relation of this hierarchy has the same functional form as the $stringy$ modified uncertainty relation with a Planck scale minimum value for $ \Delta x = L_P $ at $ \Delta p = p_{Planck} $. We proceed with a discussion of the most general curved phase space scenario (cotangent bundle of spacetime) and provide the noncommuting phase space coordinates algebra in terms of the symmetric $ g_{ ( \mu \nu ) } $ and nonsymmetric $ g_{ [ \mu \nu ] } $ metric components of a Hermitian complex metric $ g_{ \mu \nu} = g_{ ( \mu \nu ) } + i g_{ [ \mu \nu ] } $, such $ g_{ \mu \nu} = (g_{ \nu \mu})^*$. Yang's noncommuting phase-space coordinates algebra, combined with the Schrodinger-Robertson inequalities involving angular momentum eigenstates, reveals how a quantized area operator in units of $ L_P^2$ emerges like it occurs in Loop Quantum Gravity (LQG). Some final comments are made about Fedosov deformation quantization, Noncommutative and Nonassociative gravity.
Comments: 10 Pages. Submitted to Physical Review D
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[v1] 2016-04-17 02:56:34
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