Toward an axiomatic formulation of noncommutative quantum field theory

Abstract

We propose new Wightman functions as vacuum expectation values of products of field operators in the noncommutative space-time. These Wightman functions involve the ⋆-product. In the case of only space-space noncommutativity (θ0i = 0), we prove the CPT theorem using the noncommutative form of the Wightman functions. As a byproduct, one arrives at the general conclusion of the following theorem that the violation of CPT invariance implies the violation of not only Lorentz invariance, but also its subgroup of symmetry SO(1, 1)×SO(2). We also show that the spin-statistics theorem, for the simplest case of a scalar field, holds within this formalism.

Algebraic approach to quantum field theory on a class of noncommutative curved spacetimes

“…For this purpose the NC gravity solutions recently obtained in [24][25][26] will be helpful. It would also be fruitful to study the differences to other approaches to NC QFT [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17] within these models.…”

“…Even though some work on interacting QFTs on canonically deformed Euclidean or Minkowski space has been done using modified Feynman rules or other perturbative methods, the structure of free NC QFTs has still not been elucidated completely. For a collection of different approaches to QFT on Moyal-Weyl-or κ-deformed Minkowski spacetime see [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17] and references therein. Furthermore, there are attempts to generalize QFT to projective modules [18] and to spectral geometries [19].…”

LSZ reduction formula in noncommutative quantum field theory and its consequences

“…4 it was proposed that the expression for the noncommutative Wightman functions has the following form:…”

“…Relation (3) admits further generalization: for the symbols (fields) taken at different points one can define twisted tensor product 3,4 ϕ(…”

On Finite Noncommutativity in Quantum Field Theory

“…24,25 However, the finite noncommutativity we wish to obtain is of a different kind. We wish to have the two conditions:…”