Emergence of a geometric phase shift in planar noncommutative quantum mechanics

Fingerprints of the quantum space-time in time-dependent quantum mechanics: An emergent geometric phase

“…A representation of NC coordinate sub-algebra in ( 26) is furnished by the Hilbert space [11,31,32,33],…”

“…fulfilling [T R θ , X R θ ] = −iθ , appropriate for the generators of opposite algebra Âo θ . Finally coming to the coherent state representation of momenta operators in (32) we observe that since momenta operators act adjointly, their actions can be written equivalently as the corresponding differences between left and right actions yielding,…”

“…where the coordinate space variables are replaced with T and X given by (32). Using this Hamiltonian in the Schrödinger constraint equation (48), and taking its overlap with the coherent state basis (38), we get by using (32),…”

“…This phase directly depends on the non-commutative parameter θ in our expression, computed upto first order in θ and vanishes in commutative limit θ → 0 . It should be noted in this context that our computations, carried out in Heisenberg picture, as in [32], is rather a bit of an unconventional approach to calculate the geometric phase. Its primary advantage is that, it provides a natural framework for semi-classical correspondence.…”

Particle in uniform field in noncommutative space with preserved time reversal and rotational symmetries

“…Many different deformed algebras leading to the minimal length were proposed. One can distinguish three types of the algebras: noncommutative algebras of canonical type (see, for instance, [3][4][5][6]), noncommutative algebras of Lie type (see, for instance, [7][8][9]), nonlinear deformed algebras (see, for instance, [10][11][12]). Noncommutative algebras of canonical type are the most simple algebras which describe space quantization on the Planck scale.…”