Involutive representations of coordinate algebras and quantum spaces

Abstract

We show that su(2) Lie algebras of coordinate operators related to quantum spaces with su(2) noncommutativity can be conveniently represented by SO(3)-covariant poly-differential involutive representations. We show that the quantized plane waves obtained from the quantization map action on the usual exponential functions are determined by polar decomposition of operators combined with constraint stemming from the Wigner theorem for SU(2). Selecting a subfamily of * -representations, we show that the resulting star-product is equivalent to the Kontsevich product for the Poisson manifold dual to the finite dimensional Lie algebra su(2). We discuss the results, indicating a way to extend the construction to any semi-simple non simply connected Lie group and present noncommutative scalar field theories which are free from perturbative UV/IR mixing.

Localization and reference frames in κ -Minkowski spacetime

“…Further structures, such as smoothness, are encoded in other operators such as the Dirac operator, or its generalizations (for a review see for example [17]). Usually one introduces a deformation of this algebra by defining a noncommutative deformed -product so that the -commutator [x µ , x ν ] = x µ x ν − x ν x µ reproduces (1.1), usually based on the composition of plane waves [5,18]. There exist many versions of -products which reproduce the commutatutation relation (1.1), see e.g.…”

Vacuum energy and the cosmological constant problem inκ-Poincaré invariant field theories

“…Once the NC structure of spacetime is assumed, NC field theories arises very naturally [15][16][17]. These NC field theories can have features like non-locality, UV/IR mixing and completely different UV behavior from their commutative counterparts [18][19][20][21][22][23][24][25][26].…”

Quantum spaces, central extensions of Lie groups and related quantum field theories

“…The purpose of this paper is to select salient features developed in our recent works [15,16] on quantum spaces with su(2) noncommutativity, hereafter denoted generically by R 3 θ (see below). It appears that these latter can be modelled conveniently by exploiting a family of SO(3)-equivariant differential * -representations as we will show in a while.…”