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-

TL;DR

This paper constructs scalar quantum fields within a relaxed G{ a}rding-Wightman framework using Levy random fields, leading to nontrivial exact Wightman fields satisfying all axioms.

Contribution

It introduces a novel construction of quantum fields via Levy fields under relaxed axioms, advancing the understanding of exact relativistic quantum field models.

Findings

Constructed Hermitian scalar quantum fields satisfying most G{ a}rding-Wightman axioms.

Achieved nontrivial exact Wightman fields fulfilling all axioms.

Demonstrated the use of Levy random fields in quantum field construction.

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 ${\mathbb R}^d$.