Direct construction of scalar quantum fields by L{\'e}vy fields -- nontrivial exact Wightman fields in a wider field with a relaxed G{\aa}rding-Wightman Axioms-

Abstract

This paper introduces partial results, in the current situation, of ongoing considerations corresponding to the above title. A construction on exact relativistic quantum field model with the space time dimension $d \in {\mathbb N}$, including the case where $d \geq 4$, is going to be discussed. Firstly, Hermitian scalar quantum fields $<{\cal H}, U, \psi, D>$, within a relaxed framework of the G{\aa}rding-Wightman Axioms, is constructed by making use of the stochastic calculus arguments with respect to the {\it{stationary additive random fields }} on ${\mathbb R}^d$, i.e., the {\it{L{\'e}vy random fields}} on ${\mathbb R}^d$. The first constructed $<{\cal H}, U, \psi, D>$, here, satisfy all the requirements of the the G{\aa}rding-Wightman Axioms, except that the field operators $\psi (f)$ with $f \in {\cal S}({\mathbb R}^d \to {\mathbb R})$ are symmetric operators on the physical Hilbert space ${\cal H}$, which situation is denoted here as {\it{a relaxed framework}} of the G{\aa}rding-Wightman Axioms. Secondly, by taking the adequate subspaces of ${\cal H}$, non trivial exact Wightman quantum fields, which satisfy all the requirements of the G{\aa}rding-Wightman Axioms, are constructed actually. keywords: Axiomatic quantum field theory, G{\aa}rding-Wightman axioms, Bochner-Minlos theorem, L{\'e}vy fields on ${\mathbb R}^d$.