The quantum Rabi model (QRM) is widely recognized as a particularly important model in quantum optics and beyond. It is considered to be the simplest and most fundamental system describing quantum light-matter interaction. The objective of the paper is to give an analytical formula of the heat kernel of the Hamiltonian explicitly by infinite series of iterated integrals. The derivation of the formula is based on the direct evaluation of the Trotter–Kato product formula without the use of Feynman–Kac path integrals. More precisely, the infinite sum in the expression of the heat kernel arises from the reduction of the Trotter–Kato product formula into sums over the orbits of the action of the infinite symmetric group $\mathfrak{S}_\infty$ on the group $\mathbb{Z}^\infty_2$, and the iterated integrals are then considered as the orbital integral for each orbit. Here, the groups $\mathbb{Z}^\infty_2$ and $\mathfrak{S}_\infty$ are the inductive limit of the families ${\lbrace \mathbb{Z}^n_2 \rbrace} n \geq 0$ and ${\lbrace \mathfrak{S}_n \rbrace} n \geq 0$, respectively. In order to complete the reduction, an extensive study of harmonic (Fourier) analysis on the inductive family of abelian groups ${\lbrace \mathbb{Z}^n_2 \rbrace} (n \geq 0)$ together with a graph theoretical investigation is crucial. To the best knowledge of the authors, this is the first explicit computation for obtaining a closed formula of the heat kernel for a non-trivial realistic interacting quantum system. The heat kernel of this model is further given by a two-by-two matrix valued function and is expressed as a direct sum of two respective heat kernels representing the parity ($\mathbb{Z}_2$-symmetry) decomposition of the Hamiltonian by parity.

中文翻译：

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量子 Rabi 模型的热核

量子拉比模型 (QRM) 被广泛认为是量子光学及其他领域特别重要的模型。它被认为是描述量子光-物质相互作用的最简单和最基本的系统。本文的目的是通过无穷级数的迭代积分明确地给出哈密顿量热核的解析公式。公式的推导基于 Trotter-Kato 乘积公式的直接评估，未使用 Feynman-Kac 路径积分。更准确地说，热核表达式中的无穷和来自于将 Trotter–Kato 乘积公式化简为无限对称群 $\mathfrak{S}_\infty$ 在群上的作用轨道上的和$\mathbb{Z}^\infty_2$, 然后将迭代积分视为每个轨道的轨道积分。这里，群 $\mathbb{Z}^\infty_2$ 和 $\mathfrak{S}_\infty$ 是族 ${\lbrace \mathbb{Z}^n_2 \rbrace} n \geq 0 的归纳极限$ 和 ${\lbrace \mathfrak{S}_n \rbrace} n \geq 0$，分别。为了完成归约，广泛研究阿贝尔群的归纳族 ${\lbrace \mathbb{Z}^n_2 \rbrace} (n \geq 0)$ 的调和（傅里叶）分析以及图论研究至关重要。据作者所知，这是获得非平凡现实相互作用量子系统热核封闭公式的第一个显式计算。