What Do Phase Space Path Integrals Compute?

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Discussion Overview

The discussion centers on the concept of phase space path integrals, exploring what they compute, their relationship to quantum states, and their connection to the Wigner function. Participants delve into theoretical aspects, including the Hamiltonian formulation and implications for quantum field theory (QFT).

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant questions what phase space path integrals compute, specifically regarding the nature of endpoints in terms of position and momentum.
  • Another participant provides a brief introduction to quantum-mechanical path integrals, noting that phase space path integrals involve integrating over trajectories in phase space using the Hamiltonian action.
  • A participant seeks clarification on the relationship between phase space path integrals and the Wigner function, suggesting that both assign values to specific position and momentum pairs.
  • It is proposed that phase space path integrals can be segmented into pieces with specific position and momentum endpoints, although these segments may lack inherent physical meaning.
  • One participant discusses the Hamilton principle of least action, emphasizing that momenta are not fixed at final positions in the context of path integrals, which aim to calculate propagators in position space.
  • A question is raised about the possibility of expressing the path integral as a sum of parts with fixed final momenta.
  • Another participant warns that fixing final momenta may lead to issues with the ordering of the path integral.

Areas of Agreement / Disagreement

Participants express differing views on the implications of fixing momenta in phase space path integrals and whether such an approach is feasible or problematic. The discussion remains unresolved regarding the relationship between phase space path integrals and the Wigner function.

Contextual Notes

There are limitations regarding the assumptions made about the physical meaning of segments in phase space path integrals and the implications of fixing momenta. The discussion also reflects varying interpretations of the path integral formulation in quantum mechanics.

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TL;DR
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 ?
 
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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.
 
Thank you for the answer. How is the phase space path integral related to the wigner function?
 
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
 
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.
 
Sure, but can't you write the path integral as a sum of parts where each part has the final momenta fixed ?
 
Do you possibly run into problems with ordering the path integral if you do that?
 

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