Non-perturbative analysis of quantum interaction systems

研究了量子相互作用系统的基态。借助于泛函积分表示，用非微扰方法研究了量子系统。我们证明了定义在Lorentz流形上的所谓Nelson模型的基态的存在和不存在，我们通过微观局域分析去除了UV截止。证明了定义在不定Hilbert空间上的Pauli-Fierz模型的渐近完备性。我们研究增强的绑定。我们具体说明了Pauli-Fierz模型的无结合区域，并展示了相对论Nelson模型的增强结合。接下来，我们考虑功能集成。由自旋为1/2的相对论性薛定谔算符生成的热半群用布朗运动、泊松过程和从属项来表示。由此我们可以得到一个非平凡的能量比较不等式。此外，我们将这种路径积分表示推广到拉普拉斯量的Bernstein函数，并将其应用于自旋玻色子模型，以显示基态的唯一性。此外，我们还成功地构造了以双重随机积分为势的Gibbs测度的无限体积极限。

Ground states of quantum interaction systems are investigated. By means of functional integral representations quantum systems are studied in non-perturbative way. We show the existence and absence of ground states of the so-called Nelson model defined on a Lorentz manifold, we remove the UV cutoff by means of micro-local analysis. The asymptotic completeness of the Pauli-Fierz model defined on an indefinite Hilbert space is also established. We investigate enhanced bindings. We specify the no-binding regime of the Pauli-Fierz model and show the enhanced binding of the relativistic Nelson model. We next consider functional integrations. The heat semi-group generated by the relativistic Schroedinger operator with spin 1/2 is represented in terms of Brownian motions, Poisson processes and subordinators. By this we can derive a non-trivial energy comparison inequality. Moreover we extend this path integral representation to the Bernstein functions of the Laplacian, and apply this to a spin-boson model to show the uniqueness of the ground state. We furthermore succeeded to construct an infinite volume limit of the Gibbs measure with double stochastic integral as potential.