Nonperturbative analysis of charged-particle-system interacting with a quantum field

描述带电粒子与量子场相互作用系统的哈密顿谱进行了非摄动研究。它可以看作是无限维Hilbert空间张量积上的自伴随算子的谱分析。主要用泛函分析和泛函积分的方法研究了电子与光子的最小耦合，以及所谓的纳尔逊模型。本课题的目的如下：(1)估计基态玻色子的数目；(2)证明吉布斯测度的紧密性；(3)估计无截止的哈密顿量的基态的多重性；(4)研究某自旋模型的格林函数的衰减。我们的成就如下。(1)利用谱散射理论的渐近场在量子场论中的应用，得到了量子场模型中基态的规则性的一些结果（与Arai和Hirokawa合著）；(2)利用热半群的泛函积分表示得到了Pauli-Fierz模型的极化子型模型的一些结果。结果将尽快提交到某个地方。对于(3)用二次形式定义的哈密顿函数基态多重性的上界，这是我们所知的新概念。我们发现这种方法可以应用于广义自旋玻色子模型。对于所谓的O(N)自旋模型，得到了一个关于(4)的结果，现在我们正在为此写一篇论文。在整个项目中，我们可以研究非相对论性QED的质量重整化，我们得到了一些与传统物理主张相反的结果，这些结果以题为“自旋为1/2的非相对论性QED的质量重整化”的论文提交（与K.R.Ito）。

The spectrum of Hamiltonians describing a system of charged particles interacting with a quantum field is nonpertubatively studied. It can be regarded as the spectral analysis of a self-adjoint operator on a tensor product of infinite dimensional Hilbert spaces. Mainly an electron minimally coupled with photons, and the so-called Nelson model are investigated by means of functional analysis and functional integrals. The purpos of the present project is as follows : (1)estimates of the number of bosons in ground states, (2)to prove the tightness of Gibbs measures, (3)estimates of the multiplicity of ground states of a Hamiltonian without cutoffs, (4)the decay of the Green functions of a certain spin model. Our achievements are as follows.For (1)by virtute of an application of asymptotic fields of spectral scattering theory in the quantum field theory we got some results which are submitted as the paper entitledRegularities of ground states in quantum field models (with Arai and Hirokawa)For (2)it is obtained some results for a polaron type model of the Pauli-Fierz model by means of a functional integral representation of a heat semigrounp. The result will be submitted somewhere as soon as possible.For (3)an upper bound of the multiplicity of ground states of a Hamiltonian defined through a quadratic form, which is new as far as we know. It is submitted as the paper entitledMultiplicity of ground states in quantum field modelsWe find that this method can be applied for a generalized spin-boson model.A result concerning (4)is obtained for the so-called O(N)-spin model, and now we are writing a paper for this.Throughout this project we can investigate a mass renormalization of the nonrelativistic QED, and we got some results contrary to a conventional physical claim, which is submitted as the paper entitledMass renormalization in nonrelativistic QED with spin 1/2 (with K.R.Ito)