Upon the existence of short-time approximations of any polynomial order for the computation of density matrices by path integral methods

TL;DR

This paper provides mathematical evidence for the existence of arbitrarily high-order short-time approximations of density matrices in path integral methods, with explicit constructions and numerical verification, impacting physics, chemistry, and mathematics.

Contribution

It introduces the first explicit constructions of high-order short-time approximations for density matrices and verifies their convergence, advancing path integral computational techniques.

Findings

Constructed second-order and fourth-order short-time approximations.

Verified convergence orders and constants through numerical simulations.

Discussed implications for real-time path integral simulations.

Abstract

In this article, I provide significant mathematical evidence in support of the existence of short-time approximations of any polynomial order for the computation of density matrices of physical systems described by arbitrarily smooth and bounded from below potentials. While for Theorem 2, which is ``experimental'', I only provide a ``physicist's'' proof, I believe the present development is mathematically sound. As a verification, I explicitly construct two short-time approximations to the density matrix having convergence orders 3 and 4, respectively. Furthermore, in the Appendix, I derive the convergence constant for the trapezoidal Trotter path integral technique. The convergence orders and constants are then verified by numerical simulations. While the two short-time approximations constructed are of sure interest to physicists and chemists involved in Monte Carlo path integral simulations, the present article is also aimed at the mathematical community, who might find the results interesting and worth exploring. I conclude the paper by discussing the implications of the present findings with respect to the solvability of the dynamical sign problem appearing in real-time Feynman path integral simulations.