Phase space path integral approaches to real time Monte Carlo quantum dynamics

Two new formulations of real time Monte Carlo path integration are studied which include the derivation of two phase space path integrals using coherent states.; In the first formalism, presented in Chapter 2, the path integral is derived in the standard fashion by repeated insertions of a resolution of the identity along the quantum propagator, but the identity resolution used is not the standard coherent-state resolution. Because the coherent states are over-complete, we show that the identity can be written in terms of coherent states with different width parameters which can be any complex numbers as long as they have definite positive real part. Using a short time propagator based on a linear Taylor expansion of the potential, we find that for suitably chosen width parameters the basic transition matrix element reduces to the overlap of two Gaussian wavepackets with common, real width parameters. Thus, we achieve a nice form, for the phase space path integral, allowing of straight-forward Monte Carlo integration, because the integrand is separated into a well-defined probability density and an oscillatory part. In this formulation, the path integral is proportional to the average of the phase {dollar}esp{lcub}iover h{rcub} intsp{lcub}(p dq-H dt){rcub}{dollar} over a Markov process containing phase space paths which are classical trajectories interrupted at intervals by Gaussian "quantum jumps". Our numerical tests, however, reveal that despite the convenient form we have attained, the path integral derived is problematic from a practical point of view because the paths execute arbitrary excursions into the phase space leading to very poor Monte Carlo statistics.; The "diffusion" of the paths is attributed to the overcompleteness of the coherent states which do not provide the most compact representation possible. To avoid overexploiting the overcompleteness of the coherent states, we introduce a representation, presented in Chapter 3, which includes expansion of a wavefunction in terms of coherent states distributed on manifolds embedded in the phase space. We show that when certain conditions are satisfied the representation is unique, meaning that the basic Gaussians distributed on suitable manifolds form a complete set. The path integral that ensues from this compact representation is, again, proportional to the average of the same phase as above. The numerical tests we performed show that this second path integral is far more efficient than the first one, since with this representation, for suitably chosen Gaussians, the "diffusion" of the paths is better controlled and thus we can achieve better statistics and faster convergence.