Relativistic quantum field theory with a fundamental length

Abstract

Since there are indications (from string theory and concrete models) that one must consider relativistic quantum field theories with a fundamental length the question of a suitable framework for such theories arises. It is immediately evident that quantum field theory in terms of tempered distributions and even in terms of Fourier hyperfunctions cannot meet the (physical) requirements. We argue that quantum field theory in terms of ultra-hyperfunctions is a suitable framework. For this we propose a set of axioms for the fields and for the sequence of vacuum expectation values of the fields, prove their equivalence, and we give a class of models (analytic, but not entire functions of free fields).

Solution of a linearized model of Heisenberg’s fundamental equation II

“…Then Theorem 6.3 of [1] says that the power series (3.1) defines an ultra-hyperfunction quantum field with fundamental length ℓ…”

“…In Section 2 we show that the n-point functionals constructed in this way satisfy the spinor version of the functional characterization of our condition of extended causality of [1]. In order to verify the remaining defining conditions of our relativistic field theory with a fundamental length we use Wick power series to define the theory.…”

“…contains a parameter l of the dimension of length and accordingly one might speculate that this parameter can play the rôle of the fundamental length of a quantum field theory with a fundamental length as introduced in [1]. Unfortunately, nobody knows to solve this equation.…”

“…In [5], it is shown that the Wightman functions of ψ(x) are not tempered distributions used in the usual Wightman axioms but tempered ultra-hyperfunctions which are used to formulate the quantum field theory with a fundamental length in [1].…”

“…In Section 3 the convergence of the Wick power series for ρ(x) =: e gφ(x) 2 : is recalled form [1]. In the next section the important differential equation…”