The quantum thermal average is a central topic in quantum physics and can be represented by the path integrals. For the computational perspective, the path integral representation (PIR) needs to be approximated in a finite-dimensional space, and the convergence of such approximation is termed as the convergence of the PIR. In this paper, we establish the Trotter product formula in the trace form, which connects the quantum thermal average and the Boltzmann distribution of a continuous loop in a rigorous way. We prove the qualitative convergence of the standard PIR, and obtain the explicit convergence rates of the continuous loop PIR. These results showcase various approaches to approximate the quantum thermal average, which provide theoretical guarantee for the path integral approaches of quantum thermal equilibrium systems, such as the path integral molecular dynamics.

翻译：量子热平均是量子物理中的核心课题，可通过路径积分进行表示。从计算角度而言，路径积分表示需要在有限维空间中进行近似，这种近似的收敛性被称为路径积分表示的收敛性。本文建立了迹形式的Trotter乘积公式，该公式以严格方式将量子热平均与连续环的玻尔兹曼分布联系起来。我们证明了标准路径积分表示的定性收敛性，并获得了连续环路径积分表示的显式收敛速率。这些结果展示了逼近量子热平均的多种方法，为量子热平衡系统（如路径积分分子动力学）的路径积分方法提供了理论保证。