Noncommutative ℝ d $$ {\mathrm{\mathbb{R}}}^d $$ via closed star product

Abstract

Abstract:We consider linear star products on R d of Lie algebra type. First we derive the closed formula for the polydifferential representation of the corresponding Lie algebra generators. Using this representation we define the Weyl star product on the dual of the Lie algebra. Then we construct a gauge operator relating the Weyl star product with the one which is closed with respect to some trace functional, Tr (f g) = Tr (f · g). We introduce the derivative operator on the algebra of the closed star product and show that the corresponding Leibniz rule holds true up to a total derivative. As a particular example we study the space R 3 θ with su(2) type noncommutativity and show that in this case the closed star product is the one obtained from the Duflo quantization map. As a result a Laplacian can be defined such that its commutative limit reproduces the ordinary commutative one. The deformed Leibniz rule is applied to scalar field theory to derive conservation laws and the corresponding noncommutative currents.

Closed star product on noncommutative ℝ 3 and scalar field dynamics

“…This interesting space has been introduced recently in [62] where it has been shown that the closed star product ⋆ D is linked to the Duflo quantization map [63][64][65]. 4 The main motivation of [62] was to provide a first attempt to clarify the possible origin of the observed mild perturbative behavior of the NCFT on R 3 λ as stemming either from the particular form of the kinetic operator used in [54,56] or being rooted to another yet unidentified property of noncommutative spaces with su(2) noncommutativity to which R 3 λ and R 3 θ belong. The use of a closed star product enables us to deal with scalar NCFT on R 3 θ in which the kinetic operator is the usual Laplacian on R 3 .…”

“…The use of a closed star product enables us to deal with scalar NCFT on R 3 θ in which the kinetic operator is the usual Laplacian on R 3 . Such NCFT were introduced in [62] and some corresponding classical properties were examined. Here, we study one-loop IR and UV properties of the 2-point function for such real and complex noncommutative scalar field theories.…”

“…Let π :X µ → π(X µ ) = π(x σ , ∂ σ ) acting on functions of M(R 3 ) denote this representation, still a Lie algebra morphism. One finds [62] …”

“…Now, from (2.4) and f ⋆ W g = W −1 (W (f )W (g)), one can write 17) while the explicit expression of T computed for the su(2) case in [62] is given by…”

“…For a discussion of this point, see [62] where it is also pointed out that ⋆ D is actually the Kontsevich product [70].…”