  Instead of imposing the Schr\"{o}dinger 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 $\Delta t$ in the expansion if one is only interested in the
final result through order $\Delta t$, 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 $\Delta t$. Our results straightforwardly
generalize to higher orders in $\Delta t$. 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 Schr\"{o}dinger
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.
