The classical and commutative limits of noncommutative quantum mechanics: a superstar * Wigner-Moyal equation

Abstract

We are interested in the similarities and differences between the quantum-classical (Q-C) and the noncommutative-commutative (NC-Com) correspondences. As one useful platform to address this issue we derive the superstar Wigner-Moyal equation for noncommutative quantum mechanics (NCQM). A superstar -product combines the usual phase space * star and the noncommutative star-product. Having dealt with subtleties of ordering present in this problem we show that the Weyl correspondence of the NC Hamiltonian has the same form as the original Hamiltonian, but with a non-commutativity parameter θ-dependent, momentumdependent shift in the coordinates. Using it to examine the classical and the commutative limits, we find that there exist qualitative differences between these two limits. Specifically, if θ = 0 there is no classical limit. Classical limit exists only if θ → 0 at least as fast as → 0, but this limit does not yield Newtonian mechanics, unless the limit of θ/ vanishes as θ → 0. For another angle towards this issue we formulate the NC version of the continuity equation both from an explicit expansion in orders of θ and from a Noether's theorem conserved current argument. We also examine the Ehrenfest theorem in the NCQM context. AimIn this program of investigation we ask the question whether there is any structural similarity or conceptual connection between the quantum-classical (Q-C) and the noncommutative-commutative (NC-Com) correspondences. We want to see if our understanding of the quantum-classical correspondence acquired in the last decade can aid us in any way to understand the physical attributes and meanings of a noncommutative space from the vantage point of the ordinary commutative space. We find that the case of quantum to classical transition in the context of noncommutative geometry is quite different from that in the ordinary (commutative) space. Specifically, if θ = 0 there is no classical limit. Classical limit exists only if θ → 0 at least as fast as → 0, but this limit does not yield Newtonian mechanics, unless the limit of θ/ vanishes as θ → 0. We make explicit this relationship by deriving a superstar Wigner-Moyal equation for noncommutative quantum mechanics (NCQM) and identifying the difference between the classical and the commutative limits. A superstar -product combines the usual phase space * star and the noncommutative -product [1].In this paper we focus on the nature of the commutative and classical limits of noncommutative quantum physics. We point out some subtleties which arise due to the ordering problem. When these issues are properly addressed we show that the classical correspondent to the NC Hamiltonian is indeed one with a θ-dependent, momentum-dependent shift in the coordinates. For another angle towards this issue we formulate the NC version of the continuity equation both from an explicit expansion in orders of θ and from a Noether's theorem conserved current argument. We also examine the Ehrenfest theorem in the NCQM context. I. CRITERIA FOR CLASSICALITYWe open this di...

Remarks on the formulation of quantum mechanics on noncommutative phase spaces

“…That is, nonrelativistic classical mechanics with * θ deformation in the spatial directions is not a limiting case of a quantum theory. This result was obtained in [22]. However, if θ ij is of the form θ ij , then the limit exists and we have a classical theory with nontrivial Poisson bracket between spatial coordinates.…”

Non-singular Brans–Dicke collapse in deformed phase space

“…The last equation shows that there is "minimum-distance principle" in the Moyal minisuperspace. Therefore, as it has been shown in [78], there is no classical limit at θ = 0. Explicitly, the classical limit exists only if θ → 0 at least as fast as → 0, but this limit does not yield a classical commutative setting, unless the limit of θ/ vanishes as θ → 0 [78].…”

Non Commutative Quantum Mechanics in Time - Dependent Backgrounds

“…It is clear that we have to keep both parameters non zero, λ = 0 and θ = 0. Instead of an algebra of commutators, some theoretical physicists (22), (23) consider its classical analogon involving Poisson brackets of functions on real variables. But the standard limit from quantum to classical mechanics that ish → 0 has as a result to vanish the third of the commutators (4), while the last one tends to infinity.…”