Carlos Castro Perelman (2016)

Modifications of the Weyl-Heisenberg algebra [ x^i, p^j ] = i \hbar g^{ij} ( 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.