Groenewold-Moyal product, α*-cohomology, and classification of translation-invariant non-commutative structures

Abstract

The theory of α*-cohomology is studied thoroughly and it is shown that in each cohomology class there exists a unique 2-cocycle, the harmonic form, which generates a particular Groenewold-Moyal star product. This leads to an algebraic classification of translation-invariant non-commutative structures and shows that any general translation-invariant non-commutative quantum field theory is physically equivalent to a Groenewold-Moyal non-commutative quantum field theory.

Boundaries in the Moyal plane

“…product [22]. In this regard, our study of the NC disc could contribute to the comparison between the use of Moyal and Voros products in the formulation of field theories [23][24][25][26][27][28] in the presence of boundaries. 3 We also point out that whereas the definition of B 2 F appears as the natural extension of the procedures used to define S 2 F and T 2 F , our definition -which does not rely on the symmetries of the base manifold-could in principle be applied to quantum fields confined in an arbitrary subregion of the NC plane.…”

Perturbative effective action for the photon in noncommutative QED2 and exactness of the Schwinger mass

“…Moreover, since there is a unique Moyal product in two-dimensional spacetime, thus according to (b), the given formula (3.3) definitely provides the scattering matrix for any given noncommutative energy-momentum preserving version of QED 2 . In other words, according to [45], for a two-dimensional spacetime, which presents only one α * -cohomology class of noncommutative translation-invariant star products, the calculated scattering matrix (3.3) is the only energy-momentum preserving modification of that obtained in standard Euclidean QED 2 .…”

“…(b) Any given translation-invariant noncommutative is α * -cohomologous to a unique Moyal star product [45].…”

“…In Refs. [44,45], it has been shown that for any ⋆ ′ there exists an appropriate function β : R 2 → R which for any set of functions, say {f 1 , . .…”

“…Essentially, using the theory of α * -cohomology [42,43] to characterize the quantum behaviors of translation-invariant noncommutative quantum field theories, has shown that [44,45] (a) The whole quantum behavior of two translation-invariant noncommutative versions of a specific renormalizable quantum field theory with two α * -cohomologous star products, say ⋆ 1 and ⋆ 2 , coincide precisely. In fact, the equality of scattering matrices of theories L ⋆ 1 and L ⋆ 2 holds for any renormalizable theory L and for each ⋆ 1 ∼ ⋆ 2 [44].…”