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

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

1995-11-20

arxiv.org/abs/hep-th/9511141v1

Physics

High Energy Physics

High Energy Physics - Theory

17 pages, LaTeX, with one figure (needs psfig). To appear in proceedings of workshop on gauge theories, applied supersymmetry

Scientific paper

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.

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