Non-perturbative analysis of quantum interaction systems

研究了量子相互作用系统的基态。利用泛函积分表示，从非微扰的角度研究了量子系统。我们证明了在洛伦兹流形上定义的所谓尼尔森模型基态的存在和不存在，我们通过微局部分析消除了紫外截止。建立了定义在不定Hilbert空间上的Pauli-Fierz模型的渐近完备性。我们研究增强绑定。我们指定了Pauli-Fierz模型的无绑定状态，并展示了相对论性Nelson模型的增强绑定。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子产生的热半群用布朗运动、泊松过程和从属过程表示。由此我们可以推导出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函数的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。进一步地，我们成功地构造了以二重随机积分为势的吉布斯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.