Quantum Projector Method on Curved Manifolds

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A generalized stochastic method for projecting out the ground state of the quantum many-body Schrödinger equation on curved manifolds is introduced. This random-walk method is of wide applicability to any second order differential equation (first order in time), in any spatial dimension. The technique reduces to determining the proper ``quantum corrections'' for the Euclidean short-time propagator that is used to build up their path-integral Monte Carlo solutions. For particles with Fermi statistics the ``Fixed-Phase'' constraint (which amounts to fixing the phase of the many-body state) allows one to obtain stable, albeit approximate, solutions with a variational property. We illustrate the method by applying it to the problem of an electron moving on the surface of a sphere in the presence of a Dirac magnetic monopole.