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数理解析研究所講究録

TRACE FORMULAS IN BOSON FOCK SPACES AND APPLICATIONS(Analysis of Operators on Gaussian Space and Quantum Probability Theory)

Asao Arai

Published Sep 1, 1995• 0 citations
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Abstract

Trace formulas for the heat semi-groups of second quantization operators and their perturbations in the abstract Boson Fock space are given in terms of path (functional) integral representations.As applications, an inequality of Golden-Thompson type and a classical limit are derived for the trace of the heat semi-group of a perturbed second quantization operator.The abstract results are applied to a model of $P(\emptyset)$ -type in quantum field theory.

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Trace formulas for the heat semi-groups of second quantization operators and their perturbations in the abstract Boson Fock space are given in terms of path (functional) integral representations.As applications, an inequality of Golden-Thompson type and a classical limit are derived for the trace of the heat semi-group of a perturbed second quantization operator.The abstract results are applied to a model of P(∅)P(\emptyset)P(∅) -type in quantum field theory.
For a general class of boson–fermion Hamiltonians H acting in the tensor product Hilbert space L2(ℝn) ⊗ ∧(ℂr) of L2(ℝn) and the fermion Fock space ∧(ℂr) over ℂr(n, r ∈ ℕ), we establish, in terms of an n-dimensional conditional oscillator measure, a functional integral representation for the trace Tr (F ⊗ zN f e-tH)(F ∈ L∞(ℝn), z ∈ ℂ∖{0}, t > 0), where N f is the fermion number operator on ∧(ℂr). We prove a Golden–Thompson type inequality for | Tr (F ⊗ zN f e-tH)|. Also we discuss applications to a model in supersymmetric quantum mechanics and present an improved version of the Golden–Thompson inequality in supersymmetric quantum mechanics given by Klimek and Lesniewski ([Lett. Math. Phys.21 (1991) 237–244]). An upper bound for the number of the supersymmetric states is given as well as a sufficient condition for the spontaneous supersymmetry breaking. Moreover, we derive a functional integral representation for the analytical index of a Dirac type operator on ℝn (Witten index) associated with the supersymmetric quantum mechanical model.
I Infinite Renormalization of the Hamiltonian Is Necessary.- II Quantum Field Theory Models.- I. The ?22n Model.- Fock space.- Q space.- The Hamiltonian H(g).- Removing the space cutoff.- Lorentz covariance and the Haag-Kastler axioms.- II. The Yukawa Model.- Preliminaries.- First and second order estimates.- Resolvent convergence and self adjointness.- The Heisenberg picture.- III Boson Quantum Field Models.- I. General Results.- Hermite operators.- Gaussian measures and the Schrodinger representation.- Hermite expansions and Fock space.- II. The Solution of Two-Dimensional Boson Models.- The interaction Hamiltonian.- The free Hamiltonian.- Self-adjointness of H(g).- The local algebras and the Lorentz group automorphisms.- IV Boson Quantum Field Models.- III. Further Developments.- Locally normal representations of the observables.- The construction of the physical vaccum.- Formal perturbation theory and models in three space-time dimensions.- V The Particle Structure of the Weakly Coupled P(?)2 Model and Other Applications of High Temperature Expansions.- I. Physics of Quantum Field Models.- Five years of models.- From estimates to physics.- Bound states and resonances.- Phase space localization and renormalization.- VI The Particle Structure of the Weakly Coupled P(?)2 Model and Other Applications of High Temperature Expansions.- II. The Cluster Expansion.- The main results.- The cluster expansion.- Clustering and analyticity: proof of the main results.- Convergence: the main ideas.- An equation of Kirkwood-Salsburg type.- Covariance operators.- Derivatives of covariance operators.- Gaussian integrals.- Convergence: the proof completed.- VII Particles and Bound States and Progess Toward Unitarity and Scaling.- VIII Critical Problems in Quantum Fields.- IX Existence of Phase Transitions for ?24 Quantum Fields.- X Critical Exponents and Renormalization in the ?4 Scaling Limit.- Formulation of the problem.- The scaling and critical point limits.- Renormalization of the ?2(x) field.- Existence of the scaling limit.- The Josephson inequality.- XI A Tutorial Course in Constructive Field Theory.- e?tH as a functional integral.- Examples.- Applications of the functional integral representation.- Ising, Gaussian and scaling limits.- Main results.- Correlation inequalities.- Absence of even bound states.- Bound on g.- Bound on dm2/d? and particles.- The conjecture ?(6) ? 0.- Cluster expansions.- The region of convergence.- The zeroth order expansion.- The primitive expansion.- Factorization and partial resummation.- Typical applications.
Some aspects of Golden-Thompson type inequalities for a boson system with finite degrees of freedom are reviewed.Then a general class of boson-fermion systems with finite degrees of freedom, including supersymmetric ones, is considered.Functional integral representations for the partition function as well as related objects of a boson- fermion system are derived and applied to obtain Golden-Thompson type inequalities.The present article is intended to be a review paper, but, includes new results which are extensions of some results obtained in a previous paper (A.Arai, Rev. Math.
This note is devoted to Feynman formulas (i.e., representations of semigroups by limits of n -fold iterated integrals as n → ∞) and their connections with phase space Feynman path integrals. Some pseudodifferential operators corresponding to different types of quantization of a quadratic Hamiltonian function are considered. Lagrangian and Hamiltonian Feynman formulas for semigroups generated by these operators are obtained. Further, a construction of Hamiltonian (phase space) Feynman path integrals is introduced. Due to this construction, the Hamiltonian Feynman formulas obtained here and in our previous papers do coincide with Hamiltonian Feynman path integrals. This connects phase space Feynman path integrals with some integrals with respect to probability measures. These connections enable us to make a contribution to the theory of phase space Feynman path integrals, to prove the existence of some of these integrals, and to study their properties by means of stochastic analysis. The Feynman path integrals thus obtained are different for different types of quantization. This makes it possible to distinguish the process of quantization in the language of Feynman path integrals.
This article presents functional integral representations for the heat semigroups with the infinitesimal generators given by self-adjoint Hamiltonians describing an interaction of a non-relativistic charged particle and a quantized radiation field in the Coulomb gauge without the dipole approximation.Special attention is paid to definition of the "time-ordered Hilbert space-valued stochastic integrals associated with a family of isometries from a Hilbert space t,o another one" and semigroup techniques.Some inequalities are derived, which are infinite degree versions of those known for finite dimensional Schr\"odinger operators with classical vector potentials.
The Fock space of bosons and fermions and its underlying superalgebra are represented by algebras of functions on a superspace. We define Gaussian integration on infinite-dimensional superspaces, and construct super-analogs of the classical function spaces with a reproducing kernel — including the Bargmann–Fock representation — and of the Wiener–Segal representation. The latter representation requires the investigation of Wick ordering on Z2-graded algebras. As application we derive a Mehler formula for the Ornstein–Uhlenbeck semigroup on the Fock space.