Trace formulas, a Golden-Thompson inequality and classical limit in boson Fock space
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 Let \(\mu_\beta\) denote the probability measure on \(E\) obtained from another Gaussian process, where \((f,f)\) has been replaced by \((f,\text{coth}(\beta A/2)f)\).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.
scientific article; zbMATH DE number 871726