Trace formulas, a Golden-Thompson inequality and classical limit in boson Fock space
type: Item
instance of: scholarly article
title: Trace formulas, a Golden-Thompson inequality and classical limit in boson Fock space
zbMATH Open document ID: 0894.47059
author: Asao Arai
published in: Journal of Functional Analysis
publication date: 1996-06-18T00:00:00Z
review text: A functional integral representation is given for various traces of operators in connection with semigroups on bosonic Fock spaces obtained from a second quantization procedure (using the imaginary time formalism). It thus extends previous results obtained by B. Simon which once proved to be a useful tool for constructive quantum field theory. However, unlike in the standard situation, where \(A\) is the relativistic energy, Arai chooses a more general unperturbed Hamiltonian \(H_0= d\Gamma(A)\), where \(A\) may be any strictly positive selfadjoint operator on some real separable Hilbert space \({\mathcal H}\) (the one-particle space of physics) satisfying \(\| Af\|\geq c\| f\|\). Later, in Section IV, Arai imposes a more stringent condition that unfortunately excludes the standard situation. Let \(E\) be the underlying probability space with measure \(\mu\) for the Gaussian random process indexed by \({\mathcal H}\) so that the Fock space may be identified with \(L^2(E, \mu)\). Arai considers a perturbation of \(H_0\) by a multiplication operator \(V\) given by some real-valued measurable function on \(E\). Let \(\mu_\beta\) denote the probability measure on \(E\) obtained from another Gaussian process, where \((f,f)\) has been replaced by \((f,\text{coth}(\beta A/2)f)\). Then the main result is a Golden-Thompson like inequality: \[ \exp(-\beta \int V d\mu_\beta)\leq \text{tr }e^{-\beta(H_0+ v)}/\text{tr }e^{-\beta H_0}\leq \int e^{-\beta V}d\mu_\beta. \] The results are applied to a field theory model of a self{}interacting scalar field and the classical limit \((\hbar\to 0)\) is constructed.
Mathematics Subject Classification ID: 47N50
Mathematics Subject Classification ID: 81T08
Mathematics Subject Classification ID: 81T10
Mathematics Subject Classification ID: 81Q10
Mathematics Subject Classification ID: 81Q15
zbMATH DE Number: 871726
zbMATH Keywords: trace formulas
zbMATH Keywords: Golden-Thompson inequality
zbMATH Keywords: functional integral representation
zbMATH Keywords: traces of operators
zbMATH Keywords: semigroups on bosonic Fock spaces
zbMATH Keywords: second quantization
zbMATH Keywords: imaginary time formalism
zbMATH Keywords: perturbation
zbMATH Keywords: multiplication operator
zbMATH Keywords: Gaussian process
reviewed by: Gert Roepstorff
MaRDI profile type: Publication
full work available at URL: jfan.1996.0036
OpenAlex ID: W2077724342
DOI: 10.1006/JFAN.1996.0036
description: scientific article; zbMATH DE number 871726