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When we "cut" an ordinary path integral, we obtain a state in the position representation. That is, if we fix some initial position $x_i$, then the path integral $$\int_{x_i}^{x_f}Dx e^{-S}$$ is just a complex function of the final position $x_f$ - i.e. it's just a wavefunction $\psi(x_f)$.

The phase space path integral is given by $$\int DxDp e^{i\int(p\dot{q}-H)dt}.$$ Usually, this path integral is thought of as a function of the initial and final positions, which are fixed, whilst the initial and final momenta are not fixed: all values of $p_i$ and $p_f$ are integrated over. But one could instead choose to think of it as a function of both the final position and final momentum, giving us some function $f(x_f,p_f)$.

Clearly $f$ isn't a wavefunction, since it depends on too many variables. I'd like to know: is $f$ equal to some well-known function of $x$ and $p$ (e.g. the Wigner distribution)? And does it carry any physical meaning?

Note that a totally valid answer might be "no, $f$ doesn't appear in the literature, and as far as we know it has no physical meaning".

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    It can't literally be the Wigner distribution because this path integral isn't, in general, real-valued. (But I expect some similar thing is true because the Wigner distribution behaves like a probability distribution so there must be a way to build it up from time slices giving a path integral of some sort.) – Dan Piponi Mar 07 '23 at 19:48
  • It's a long story, and it's been beaten to a pulp. You might like to read the paragraph around eqn (120) of this booklet, which provides the propagator for Wigner functions, and references in that paragraph... – Cosmas Zachos Mar 07 '23 at 23:35

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  1. In the Hamiltonian phase space path integral on a symplectic manifold the initial and final boundary conditions are typically given by a choice of 2 Lagrangian submanifolds.

  2. This rules out OP's example where OP specifies both $x_f$ and $p_f$ but has no initial conditions.

  3. Note however that the coherent state path integral is an important exception with overcomplete boundary conditions, cf. e.g. this related Phys.SE post.

  4. Often the path integral as a function of the final condition can be interpreted as a wavefunction that satisfies the Schrödinger equation, see e.g. this related Phys.SE post for the standard case.

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