Wigner-Weyl-Moyal Formalism on Algebraic Structures

Details Wigner-Weyl-Moyal Formalism on Algebraic Structures Wigner-Weyl-Moyal Formalism on Algebraic Structures

1996-08-28

arxiv.org/abs/quant-ph/9608042v1

Int.J.Theor.Phys. 37 (1998) 697-758

Physics

Quantum Physics

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We first introduce the Wigner-Weyl-Moyal formalism for a theory whose phase-space is an arbitrary Lie algebra. We also generalize to quantum Lie algebras and to supersymmetric theories. It turns out that the non-commutativity leads to a deformation of the classical phase-space: instead of being a vector space it becomes a manifold, the topology of which is given by the commutator relations. It is shown in fact that the classical phase-space, for a semi-simple Lie algebra, becomes a homogenous symplectic manifold. The symplectic product is also deformed. We finally make some comments on how to generalize to $C^*$-algebras and other operator algebras too.

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