The equivalence between the operator approach and the path integral approach for quantum mechanical non-linear sigma models

TL;DR

This paper demonstrates the equivalence of operator and path integral approaches in quantum mechanical non-linear sigma models, providing exact derivations, clarifying regularization issues, and confirming results through two-loop calculations.

Contribution

It establishes the precise relationship between operator and path integral formulations, including correct measure and Feynman rules, for supersymmetric and nonsupersymmetric sigma models in quantum mechanics.

Findings

Discretized path integral and propagators derived explicitly.

Mode regularization can lead to incorrect results.

Final covariant results are obtained when Hamiltonian is covariant.

Abstract

We give background material and some details of calculations for two recent papers [1,2] where we derived a path integral representation of the transition element for supersymmetric and nonsupersymmetric nonlinear sigma models in one dimension (quantum mechanics). Our approach starts from a Hamiltonian $H(\hat{x}, \hat{p}, \hat{\psi}, \hat{\psi}^\dagger)$ with a priori operator ordering. By inserting a finite number of complete sets of $x$ eigenstates, $p$ eigenstates and fermionic coherent states, we obtain the discretized path integral and the discretized propagators and vertices in closed form. Taking the continuum limit we read off the Feynman rules and measure of the continuum theory which differ from those often assumed. In particular, mode regularization of the continuum theory is shown in an example to give incorrect results. As a consequence of time-slicing, the action and Feynman rules, although without any ambiguities, are necessarily noncovariant, but the final results are covariant if $\hat{H}$ is covariant. All our derivations are exact. Two loop calculations confirm our results.