The classical and commutative limits of noncommutative quantum mechanics: a superstar * Wigner-Moyal equation
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 theta-dependent, momentum-dependent 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 <FONT FACE=Symbol>q ¹</FONT> 0 there is no classical limit. Classical limit exists only if <FONT FACE=Symbol>q ®</FONT> 0 at least as fast as h ->0, but this limit does not yield Newtonian mechanics, unless the limit of theta/h vanishes as <FONT FACE=Symbol>q ®</FONT> 0. For another angle towards this issue we formulate the NC version of the continuity equation both from an explicit expansion in orders of theta and from a Noether's theorem conserved current argument. We also examine the Ehrenfest theorem in the NCQM context.
| Main Authors: | , |
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| Format: | Digital revista |
| Language: | English |
| Published: |
Sociedade Brasileira de Física
2005
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| Online Access: | http://old.scielo.br/scielo.php?script=sci_arttext&pid=S0103-97332005000200019 |
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