  Dirac's approach to the canonical quantization of constrained systems is
applied to $N = 1$ supergravity, with or without gauged supermatter. Two
alternative types of boundary condition applicable to quantum field theory or
quantum gravity are contrasted. The first is the `coordinate' boundary
condition as used in quantum cosmology; the second type is scattering boundary
conditions, as used in Feynman diagrams, applicable to asymptotically flat
space-time. The first yields a differential-equation form of the theory, dual
to the integral version appropriate to the second. Here, the first (Dirac)
approach is found to be extremely streamlined for the calculation of loop
amplitudes in these locally supersymmetric theories. By contrast,
Feynman-diagram methods have led to calculations which are typically so large
as to be unmanageable. Remarkably, the Riemannian quantum amplitude for
coordinate boundary conditions in $N = 1$ supergravity (without matter) is
exactly semi-classical, being of the form $exp(-I/\hbar)$, where $I$ is the
classical action, allowing for the presence of fermions as well as gravity on
the boundaries. Even when supermatter is included, typical one-loop amplitudes
are often very simple, sometimes not even involving an infinite sum or
integral. Specifically, the boundary conditions considered for a number of
concrete one-loop examples are set on a pair of concentric 3-spheres in
Euclidean 4-space. In the non-trivial cases the amplitudes appear to be
exponentially convergent.
