A Survey of Star Product Geometry

Abstract

The subniitttjd iii,inuscript lids been creat~*d by the University of Chicago as Operator of Argonne National Laboratory ("Argonnu"! under Contract No. W-31-109-ENG.

Noncommutative AdS2/CFT1 duality: The case of massive and interacting scalar fields

“…Then one can pass from the operator algebra generated byX μ to the algebra of their symbols X μ ðx; yÞ, with the operator product replaced by the Moyal-Weyl star product, ⋆. (for a nice review, see [24]) The Moyal-Weyl star product of any two symbols F and G is defined in the standard way by…”

Generalized coherent state approach to star products and applications to the fuzzy sphere

“…One of the attractive features of the fuzzy sphere as a noncommutative space is that it is covariant with respect to the same symmetry as the standard sphere, namely SO (3). (This is in contrast to the case of the quantum sphere.…”

“…The most well studied star product is often referred to as the Moyal star product [1], [2]. (For a nice review see [3].) It allows for a quantum mechanical description on phase space.…”

“…Associativity is once again assured, and the results can be applied to general two dimensional Poisson structures. From our approach one easily recovers the Moyal star product (or more precisely, an equivalent construction due to Voros [16], [3]), and also a star product for the fuzzy sphere. The generalized coherent states have the usual property that they form an over complete basis for an infinite dimensional Hilbert space.…”

Covariant Noncommutative Field Theory

“…where θ µν is a constant tensor, is done by means the Weyl-Wigner-Moyal correspondence [5], which establishes an equivalence between the algebra of operators A of these noncommutative variables (with the usual product of operators) and the algebra of smooth functions A θ (with the noncommutative Moyal product). This correspondence expresses a Lie algebra isomorphism…”