Test Functions Space in Noncommutative Quantum Field Theory

It is proven that the $\star$-product of field operators implies that the space of test functions in the Wightman approach to noncommutative quantum field theory is one of the Gel'fand-Shilov spaces $S^{\beta}$ with $\beta < 1/2$. This class of test functions smears the noncommutative Wightman functions, which are in this case generalized distributions, sometimes called hyperfunctions. The existence and determination of the class of the test function spaces in NC QFT is important for any rigorous treatment in the Wightman approach.Comment: 10 pages, clarification of formula (1.6), typos fixed, minor language correction

Towards an axiomatic formulation of noncommutative quantum field theory. II

Classical results of the axiomatic quantum field theory – irreducibility of the set of field operators, Reeh and Schlieder's theorems and generalized Haag's theorem are proven in SO(1,1) invariant quantum field theory, of which an important example is noncommutative quantum field theory. In SO(1,3) invariant theory new consequences of generalized Haag's theorem are obtained. It has been proven that the equality of four-point Wightman functions in two theories leads to the equality of elastic scattering amplitudes and thus the total cross-sections in these theories.Peer reviewe

Seiberg-Witten map with Lorentz-invariance and gauge-covariant star product

We develop the Seiberg-Witten map using the gauge-covariant star product with the noncommutativity tensor theta(mu v)(x). The latter guarantees the Lorentz invariance of the theory. The usual form of this map and its other recent generalizations do not consider such a covariant star product. We construct the Seiberg-Witten map for the gauge parameter, the gauge field and the strength tensor to the first order in the noncommutativity parameter theta(mu v)(x). Prescription for the generalization of the map to higher orders is also given. Interestingly, the associativity of the covariant star product both in the first and second orders requires the same constraints, namely, on the theta(mu v)(x) and on the space-time connection. This fact suggests that the same constraints could be enough to ensure the associativity in all orders. The resulting Seiberg-Witten map applies both to the internal and space-time gauge theories. Comparisons with the Seiberg-Witten map based on other (non-covariant) star products are given and some characteristic properties are also presented. As an application, we consider the GL(2, C) noncommutative gauge theory of gravitation, in which it is shown that the connection determines a space-time with symplectic structure (as proposed by Zumino et al [33]). This example shows that the constraints required for the associativity of the gauge-covariant star product can be satisfied. The presented GL(2, C) noncommutative gauge theory of gravitation is also compared to the one (given by Chamseddine [44]) with non-covariant star product. (C) 2022 The Author(s). Published by Elsevier B.V.Peer reviewe