2003/06/30 by Cristian Predescu
Chemistry · Mathematics · Physics and Astronomy · #Advanced NMR Techniques and Applications #Algorithm #Applied mathematics #Bounded function #Calculus (dental) #Computation #Computer science #Constant (computer programming) #Convergence (economics) #Feynman diagram #Mathematical analysis #Mathematical physics #Mathematics #Matrix (chemical analysis) #Monte Carlo method #Path (computing) #Path integral formulation #Physics #Polynomial #Quantum mechanics #Quantum, superfluid, helium dynamics #Sign (mathematics) #Spectroscopy and Quantum Chemical Studies #cond-mat.stat-mech #math-ph #math.MP #physics.chem-ph
paper · pdf · doi:10.1103/physreve.69.056701
published as Phys. Rev. E 69, 056701 (2004). · 19 pages, 4 figures; the discrete short-time approximations are now treated as independent from their continuous version; new examples of discrete short-time approximations of order three and four are given; a new appendix containing a short review on Brownian motion has been added; also, some additional explanations are provided here and there; this is the last version; to appear in Phys. Rev. E
arxiv created 2003/12/22 · openalex publication_date 2004/05/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
In this paper I provide significant mathematical evidence in support of the existence of direct 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 Appendix B, 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 paper 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.