Ordering, symbols, and finite-dimensional approximations of path integrals

TL;DR

This paper develops a general framework for finite-dimensional path integral approximations in quantum systems, showing their independence from operator symbol types up to small errors, and introduces new approximation classes with non-local actions.

Contribution

It derives a general form of finite-dimensional path integral approximations for bosonic and fermionic systems based on operator symbols, revealing their independence from symbol types and introducing new approximation classes.

Findings

Approximations are independent of symbol type up to O(ε) terms.

New classes of non-local approximations are identified.

Explicit analysis of the fermionic oscillator example.

Abstract

We derive general form of finite-dimensional approximations of path integrals for both bosonic and fermionic canonical systems in terms of symbols of operators determined by operator ordering. We argue that for a system with a given quantum Hamiltonian such approximations are independent of the type of symbols up to terms of $O(\epsilon)$, where $\epsilon$ is infinitesimal time interval determining the accuracy of the approximations. A new class of such approximations is found for both c-number and Grassmannian dynamical variables. The actions determined by the approximations are non-local and have no classical continuum limit except the cases of $pq$- and $qp$-ordeeing. As an explicit example the fermionic oscillator is considered in detail.