Noncommutative quantum mechanics as a gauge theory

Abstract

The classical counterpart of noncommutative quantum mechanics is a constrained system containing only second-class constraints. The embedding procedure formulated by Batalin, Fradkin and Tyutin (BFT) enables one to transform this system into an Abelian gauge theory exhibiting only first class constraints. The appropriateness of the BFT embedding, as implemented in this work, is verified by showing that there exists a one to one mapping linking the second-class model with the gauge invariant sector of the gauge theory. As is known, the functional quantization of a gauge theory calls for the elimination of its gauge freedom. Then, we have at our disposal an infinite set of alternative descriptions for noncommutative quantum mechanics, one for each gauge. We study the relevant features of this infinite set of correspondences. The functional quantization of the gauge theory is explicitly performed for two gauges and the results compared with that corresponding to the second-class system. Within the operator framework the gauge theory is quantized by using Dirac's method.

Twist deformation of rotationally invariant quantum mechanics

“…[17], where the noncommutative hydrogen atom was discussed, and [18]. In [19] (see also [20] and [21]) the classical counterpart of the non-commutative quantum mechanics is shown to be a constrained system. In [22] investigations of a dynamical (i.e.…”

From Noncommutative Sphere to Nonrelativistic Spin

“…Equations (10) can be used to exclude allπ i . Then one deal with the remaining variables J i obeying SO(3) algebra (7) and subject the constraint (11). Finite dimensional irreducible representations of the angular momentum algebra are numbered by the spin s. The condition (11) fixes the spin s = 1 2 .…”

“…It seems to be reasonable approach, since at least the pioneer NC models [7,8,6] all can be properly treated by this way [9,8]. Moreover, following this way, with any classical mechanical system one associates its NC version [10,11,12,13,14].…”

“…Since NC geometry turns out to be useful tool for reformulation and investigation of some problems in classical and quantum mechanics [10,11,12,12,14,15,16] as well as in QFT [17], there are attempts to formulate it on a more fundamental level (see [18] for the recent review). One of the barriers here is relativistic (Galilean) invariance.…”

Noncommutativity due to spin

“…It is known that canonical NCQM [4] can be obtained as a result of quantization of classical models, see e.g., [10]- [12]. The corresponding action functional appears as an effective action in path integral representation of NCQM [13]- [15] and can be used for study of global and local symmetries of the system [16], etc. The first question is if there exists a classical model, which after quantization lead to the spin noncommutativity.…”