Path integral formulation of noncommutative quantum mechanics

Abstract

We propose a phase-space path integral formulation of noncommutative quantum mechanics, and prove its equivalence to the operatorial formulation. As an illustration, the partition function of a noncommutative two-dimensional harmonic oscillator is calculated.Noncommutative quantum mechanics represents a natural extension of usual quantum mechanics, in which one allows nonvanishing commutators also between the coordinates, and between the momenta. Denoting the coordinates and the momenta collectively by {x i }, in (2+1)-dimensions one has the commutation relations

Noncommutative quantum mechanics and Bohm’s ontological interpretation

“…(20) at x i = x i (t). In order to find a good equation for the x i (t)'s, it is interesting to consider the Heisenberg formulation and the equations of motion for the observables (see, for example, [13]). For the variablesx i they are given by…”

The Klein-Gordon and the Dirac Oscillators in a Noncommutative Space

“…[6]. A simple insight on the role of noncommutativity in field theory can be obtained by studying the one particle sector, which prompted an interest in the study of noncommutative quantum mechanics (NCQM) [7,8,9,10,11,12,13,14]. In these studies some attention was paid to two-dimensional NCQM and its relation to the Landau problem.…”

Field Quantization on Noncommutative Phase Space

“…In literature the NC quantum mechanics and NC quantum field theory have been studied extensively, in literatures many interesting topics of NC quantum theories have been extensively investigated, from the Aharonov-Bohm effect to the quantum Hall effect [1][2][3][4][5][6][7]. Recently quantum theory of NC fields has been proposed as a possible field theory generalization of NC quantum mechanics [8][9][10][11]. In this paper, we devote our mind to discuss the energy of electromagnetic field on NC phase space.…”