Quantitative Convergence Analysis of Path Integral Representations for Quantum Thermal Average

Abstract: 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.