Gaussian β-Markov Path Spaces without a Spectral Gap and the Araki--Woods Representations
Abstract
Dereziński and Gérard constructed the Gaussian β-Markov path space of a free Bose field in a form equivalent to the Araki--Woods representation, which suits nonequilibrium statistical mechanics with a quantum-field environment, but only for one-particle Hamiltonians with a spectral gap. We extend this construction, together with the equivalence to the Araki--Woods representation, to one-particle Hamiltonians without a spectral gap. For one-particle Hamiltonians joined by isometric embeddings that intertwine their spectral measures and test spaces, as for nested momentum lattices, and for compatible regularizers, the path spaces form a projective system whose limit carries the time translations and the reflection, which gives a quasi-local construction. The extension is intended for constructive quantum field theory at positive temperature and equilibrium statistical mechanics, for instance in the problem of the absence of Bose--Einstein condensation of quasiparticles whose number is not conserved.
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BibTeX
@misc{sekine2026gaussian,
title = {{Gaussian $β$-Markov Path Spaces without a Spectral Gap and the Araki--Woods Representations}},
author = {Yoshitsugu Sekine},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.34783},
eprint = {2610.34783},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.34783},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Gaussian β-Markov Path Spaces without a Spectral Gap and the Araki--Woods Representations
AU - Yoshitsugu Sekine
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.34783
UR - https://arxiv.org/abs/2610.34783
AB - Dereziński and Gérard constructed the Gaussian β-Markov path space of a free Bose field in a form equivalent to the Araki--Woods representation, which suits nonequilibrium statistical mechanics with a quantum-field environment, but only for one-particle Hamiltonians with a spectral gap. We extend this construction, together with the equivalence to the Araki--Woods representation, to one-particle Hamiltonians without a spectral gap. For one-particle Hamiltonians joined by isometric embeddings that intertwine their spectral measures and test spaces, as for nested momentum lattices, and for compatible regularizers, the path spaces form a projective system whose limit carries the time translations and the reflection, which gives a quasi-local construction. The extension is intended for constructive quantum field theory at positive temperature and equilibrium statistical mechanics, for instance in the problem of the absence of Bose--Einstein condensation of quasiparticles whose number is not conserved.
ER -