The representations of the Heisenberg algebra in Krein spaces, more generally in weakly complete inner product spaces, are classified under general regularity and irreducibility conditions. Besides the Fock representation, two other types appear; one with negative, the other with a two-sided discrete spectrum of the number operator.

It is proven that the ⋆-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 β with β < 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.

We derive the analytical properties of the elastic forward scattering amplitude of two scalar particles from the axioms of the noncommutative quantum field theory. For the case of only space-space noncommutativity, i.e. θ 0i = 0, we prove the dispersion relation which is similar to the one in commutative quantum field theory. The proof in this case is based on the existence of the analog of the usual microcausality condition and uses the LehmannSymanzik-Zimmermann (LSZ) or equivalently the Bogoliubov-Medvedev-Polivanov (BMP) reduction formalisms. The existence of the latter formalisms is also shown. We remark on the general noncommutative case, θ 0i = 0, as well as on the nonforward scattering amplitude and mention their peculiarities.

The axiomatic approach based on Wightman functions is developed in noncommutative field theory. We prove that the main results of the axiomatic approach remain valid if the noncommutativity affects only the spatial variables.

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.