I understand that they're, in the end, related because they're the same theory. But is there a closer relationship because both are theories of probability distributions on phase space?
I also understand that the phase space path integral is a tool for calculating the evolution of the wavefunction $\psi (x) $, which is a probability distribution on Hilbert space, and not on phase space (which has double the dimension of the configuration space).
But still, it's temtping to see the phase space path integral as describing probability amplitude evolution on phase space. It attributes each path on the phase space with a complex number amplitude. Phase space quantum mechanics describes time evolution of a quasi-probability distribution on phase space. The time evolution laws in both cases are the path integral and Moyal bracket respectively.
I think these formulations might be very closely related because one of the steps of Geometric Quantisation is "Polarisation", which is what's responsible for halving the number of dimensions of the phase space. Is it possible that complex numbers are a trick to encapsulate the Moyal bracket evolution of the Quasiprobability distribution into a path integral evolution?
