Closed star product on noncommutative ℝ 3 and scalar field dynamics

Abstract

Abstract:We consider the noncommutative space R 3 θ , a deformation of R 3 for which the star product is closed for the trace functional. We study one-loop IR and UV properties of the 2-point function for real and complex noncommutative scalar field theories with quartic interactions and Laplacian on R 3 as kinetic operator. We find that the 2-point functions for these noncommutative scalar field theories have no IR singularities in the external momenta, indicating the absence of UV/IR mixing. We also find that the 2-point functions are UV finite with the deformation parameter θ playing the role of a natural UV cut-off. The possible origin of the absence of UV/IR mixing in noncommutative scalar field theories on R 3 θ as well as on R 3 λ , another deformation of R 3 , is discussed.

Involutive representations of coordinate algebras and quantum spaces

“…From the above discussion, these latter must satisfy (2.17)-(2.18) which determine χ µ once ϕ µν fulfilling (2.16) is obtained from (2.15 43) in which g(∆) is still given by (2.41), (2.42). Hence, the poly-differential representation used in [44] can be extended to a * -representation which, by further assuming that l(∆) is a real functional, 6 is defined by the following expression…”

“…One well-known example heavily used in earlier studies of quantum properties of NCFT on Moyal spaces [31][32][33] (for a review on gauge theories on Moyal spaces see [34]; families of star products on the Moyal space R 4 θ have been constructed in [35]) is the Weyl quantization map, linked to the Wigner-Weyl transform, giving rise to the Moyal product. Other star-products related to κ-Minkowski spaces as well as deformations of R 3 with su(2) noncommutativity have also appeared and used to construct and study NCFT on these spaces [36][37][38][39][40][41][42][43][44][46][47][48] (for a general construction see [45]). The use of star-product formulation of NCFT is often convenient for fast construction of reasonable functional actions but may lead to technical difficulties whenever the star-product is represented by a complicated formula and/or is not closed for a trace functional.…”

“…Differential representations have been used in various contexts in the Physics literature, see e.g. [36][37][38][39][40][41][42][43][44][45][46][47][48] and references therein, some being * -representations, some being not. The use of a non * -representation may lead to troublesome features.…”

“…as algebras (see [27,43]), where U(su (2)) is the universal enveloping algebra of su (2), we focus now on the particular class of SO(3)-equivariant 3 representations π which therefore correspond to ϕ αµ of the form…”

“…Then, the analysis of [43] can be straightforwardly adapted to scalar noncommutative field theories with functional actions (in obvious notations) given by…”