Direct construction of scalar quantum fields by L{é}vy fields -- nontrivial exact Wightman fields in a wider field with a relaxed Gårding-Wightman Axioms-
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866913053982851072 |
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| author | Albeverio, Sergio Kawasaki, Suji Yahagi, Yumi Yoshida, Minoru W. |
| author_facet | Albeverio, Sergio Kawasaki, Suji Yahagi, Yumi Yoshida, Minoru W. |
| contents | 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 $<{\cal H}, U, ψ, D>$, within a relaxed framework of the Gårding-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{é}vy random fields}} on ${\mathbb R}^d$. The first constructed $<{\cal H}, U, ψ, D>$, here, satisfy all the requirements of the the Gårding-Wightman Axioms, except that the field operators $ψ(f)$ 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 Gårding-Wightman Axioms. Secondly, by taking the adequate subspaces of ${\cal H}$, non trivial exact Wightman quantum fields, which satisfy all the requirements of the Gårding-Wightman Axioms, are constructed actually.
keywords: Axiomatic quantum field theory, Gårding-Wightman axioms, Bochner-Minlos theorem, L{é}vy fields on ${\mathbb R}^d$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_20359 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Direct construction of scalar quantum fields by L{é}vy fields -- nontrivial exact Wightman fields in a wider field with a relaxed Gårding-Wightman Axioms- Albeverio, Sergio Kawasaki, Suji Yahagi, Yumi Yoshida, Minoru W. Mathematical Physics Functional Analysis Probability 31C25, 46E27, 46N30, 46N50, 47D07, 60H15, 60J46, 60J75, 81S20 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 $<{\cal H}, U, ψ, D>$, within a relaxed framework of the Gårding-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{é}vy random fields}} on ${\mathbb R}^d$. The first constructed $<{\cal H}, U, ψ, D>$, here, satisfy all the requirements of the the Gårding-Wightman Axioms, except that the field operators $ψ(f)$ 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 Gårding-Wightman Axioms. Secondly, by taking the adequate subspaces of ${\cal H}$, non trivial exact Wightman quantum fields, which satisfy all the requirements of the Gårding-Wightman Axioms, are constructed actually. keywords: Axiomatic quantum field theory, Gårding-Wightman axioms, Bochner-Minlos theorem, L{é}vy fields on ${\mathbb R}^d$. |
| title | Direct construction of scalar quantum fields by L{é}vy fields -- nontrivial exact Wightman fields in a wider field with a relaxed Gårding-Wightman Axioms- |
| topic | Mathematical Physics Functional Analysis Probability 31C25, 46E27, 46N30, 46N50, 47D07, 60H15, 60J46, 60J75, 81S20 |
| url | https://arxiv.org/abs/2604.20359 |