Noncommutativity in Configuration Space Induced by A Conjugate Magnetic Field in Phase Space

Abstract

As is well known, an external magnetic field in configuration space coupled to a quantum dynamics induces noncommutativity in its velocity momentum space. By phase space duality, an external vector potential in the conjugate momentum sector of the system induces noncommutativity in its configuration space. Such a rationale for noncommutativity is explored herein for an arbitrary configuration space of Euclidean geometry. Ordinary quantum mechanics with a commutative configuration space is revisited first. Through the introduction of an arbitrary positive definite $*$-product, a one-to-one correspondence between the Hilbert space of abstract quantum states and that of the enveloping algebra of the position quantum operators is identified. A parallel discussion is then presented when configuration space is noncommutative, and thoroughly analysed when the conjugate magnetic field is momentum independent and nondegenerate. Once again the space of quantum states may be identified with the enveloping algebra of the noncommutative position quantum operators. Furthermore when the positive definite $*$-product is chosen in accordance with the value of the conjugate magnetic field which determines the commutator algebra of the coordinate operators, these operators span a Fock algebra of which the canonical coherent states are the localised noncommutative quantum analogues of the sharp and structureless local points of the associated commutative configuration space geometry. These results generalise and justify a posteriori within the context of ordinary canonical quantisation the heuristic approach to quantum mechanics in the noncommutative Euclidean plane as constructed and developed by F. G. Scholtz and his collaborators.