We show that a considerable part of the theory of (ultra)distributions and hyperfunctions can be extended to more singular generalized functions, starting from an angular localizability notion introduced previously. Such an extension is needed to treat gauge quantum field theories with indefinite metric in a generic covariant gauge. Prime attention is paid to the generalized functions defined on the Gelfand-Shilov spaces S 0 α which gives the widest framework for construction of gauge-like models. We associate a similar test function space with every open and every closed cone, show that these spaces are nuclear and obtain the required formulas for their tensor products. The main results include the generalization of the Paley-Wiener-Schwartz theorem to the case of arbitrary singularity and the derivation of the relevant theorem on holomorphic approximation.1

We prove that the distributions defined on the Gelfand-Shilov spaces S β α with β < 1, and hence more singular than hyperfunctions, retain the angular localizability property. Specifically, they have uniquely determined support cones. This result enables one to develop a distribution-theoretic techniques suitable for the consistent treatment of quantum fields with arbitrarily singular ultraviolet and infrared behavior. The proof covering the most general and difficult case β = 0 is based on the use of the theory of plurisubharmonic functions and Hörmander's L

A new, more general derivation of the spin-statistics and PCT theorems is presented. It uses the notion of the analytic wave front set of (ultra)distributions and, in contrast to the usual approach, covers nonlocal quantum fields. The fields are defined as generalized functions with test functions of compact support in momentum space. The vacuum expectation values are thereby admitted to be arbitrarily singular in their space-time dependence. The local commutativity condition is replaced by an asymptotic commutativity condition, which develops generalizations of the microcausality axiom previously proposed.

In this paper, we introduce the condition of θ-locality which can be used as a substitute for microcausality in quantum field theory on noncommutative spacetime. This condition is closely related to the asymptotic commutativity which was previously used in nonlocal QFT. Heuristically, it means that the commutators of observables behave at large spacelike separation like exp(−|x − y| 2 /θ), where θ is the noncommutativity parameter. The rigorous formulation given in the paper implies averaging fields with suitable test functions. We define a test function space which most closely corresponds to the Moyal ⋆-product and prove that this space is a topological algebra under the star product. As an example, we consider the simplest normal ordered monomial : φ ⋆ φ : and show that it obeys the θ-locality condition.

We revisit the question of microcausality violations in quantum field theory on noncommutative spacetime, taking $O(x)=:\phi\star\phi:(x)$ as a sample observable. Using methods of the theory of distributions, we precisely describe the support properties of the commutator [O(x),O(y)] and prove that, in the case of space-space noncommutativity, it does not vanish at spacelike separation in the noncommuting directions. However, the matrix elements of this commutator exhibit a rapid falloff along an arbitrary spacelike direction irrespective of the type of noncommutativity. We also consider the star commutator for this observable and show that it fails to vanish even at spacelike separation in the commuting directions and completely violates causality. We conclude with a brief discussion about the modified Wightman functions which are vacuum expectation values of the star products of fields at different spacetime points.Comment: LaTeX, 22 pages; v2: minor updates to agree with published version, added referenc