The Hamiltonian Approach and Phase Space Path Integration for Nonlinear Sigma Models with and without Fermions

Abstract

Instead of imposing the Schrödinger equation to obtain the configuration space propagator $\csprop$ for a quantum mechanical nonlinear sigma model, we directly evaluate the phase space propagator $\psprop$ by expanding the exponent and pulling all operators $\hat p$ to the right and $\hat x$ to the left. Contrary to the widespread belief that it is sufficient to keep only terms linear in $Δt$ in the expansion if one is only interested in the final result through order $Δt$, we find that all terms in the expansion must be retained. We solve the combinatorical problem of summing the infinite series in closed form through order $Δt$. Our results straightforwardly generalize to higher orders in $Δt$. We then include fermions for which we use coherent states in phase space. For supersymmetric $N{=}1$ and $N{=}2$ quantum mechanics, we find that if the super Van Vleck determinant replaces the original Van Vleck determinant the propagator factorizes into a classical part, this super determinant and the extra scalar curvature term which was first found by DeWitt for the purely bosonic case by imposing the Schrödinger equation. Applying our results to anomalies in $n$-dimensional quantum field theories, we note that the operator ordering in the corresponding quantum mechanical Hamiltonians is fixed in these cases. We present a formula for the path integral action, which corresponds one to one to any given covariant or noncovariant $\hat H$. We then evaluate these path integrals through two loop order, and reobtain the same propagators in all cases.