I What Do Phase Space Path Integrals Compute?

[Intrastellar]

TL;DR Summary

Looking for a summary of phase space path integrals and answers to followup questions.

I have heard of phase space path integrals, but couldn't find anything in Wikipedia about it, so I am wondering, what does it compute ? In particular, are the endpoints points of definite position and momentum? If so, how does one convert them to quantum states ? Also, how is it related to wigner function ?

 

[vanhees71]

Maybe my short intro to quantum-mechanical path integrals in my QFT notes can help:

https://itp.uni-frankfurt.de/~hees/publ/lect.pdf

The phase-space path integral is the one where you integrate over the trajectories in phase space, i.e., the version of the path integral using the Hamiltonian version of the action and before integrating over the momenta. The latter in many cases leads to a path-integral formula where you integrate over trajectories in configuration space only, and the action in the exponent is in the Lagrange form.

The latter version is nice for relativistic QFT since the Lagrangian formalism for fields is manifestly covariant.

 

[Intrastellar]

Thank you for the answer. How is the phase space path integral related to the wigner function?

 

[Intrastellar]

As an elaboration, the phase space path integral can be cut into pieces made up of an endpoint having a specific position and momentum, and even though the cut pieces wouldn't inherently have a physical meaning. Wigner function is also a function which assigns numbers to specific position and momenta, so I was wondering whether they were related

 

[vanhees71]

In the Hamilton principle of least action, formulated in the Hamiltonian way as variations in phase space the momenta are not fixed at the final positions. That's understandable from the path-integral method, because there you want to calculate the propagator in position space, $\langle x,t|x',t' \rangle$, where $|x,t \rangle$ are the position eigenvectors in the Heisenberg picture. That's why in the path integral you integrate over all paths connecting the fixed points $\vec{x}$ and $\vec{x}'$ in configuration space, but the integrals over the trajectories in momentum space are unrestricted. In my derivation that becomes clear, because there the path integral is derived from the usual Dirac bra-ket formalism through introduction of completeness relations with position and momentum eigenvectors.

 

[Intrastellar]

Sure, but can't you write the path integral as a sum of parts where each part has the final momenta fixed ?

 

[Intrastellar]

Do you possibly run into problems with ordering the path integral if you do that?