Relation between phase space and path integral formulation?

[Matta Tanning]

I am trying to conceptually connect the two formulations of quantum mechanics.

The phase space formulation deals with quasi-probability distributions on the phase space and the path integral formulation usually deals with a sum-over-paths in the configuration space.

I see how they both lead to non-classical physics but I can't see how they relate. I would love some insight into this.

 

[naima]

Feynman path formulation is equivalent to QM. Phase space formulation is also equivalent to QM.
This second point is very well explained in Bobbah's favourite textbook (Ballentine) Skip to chapter 15.
From a given Wigner Function on phase space you can construct the correponding density matrix state in QM.

 

[Matta Tanning]

I guess that I am trying to get a direct link so that I don't have to go near Hilbert spaces. The phase space is the tangent bundle of the configuration space so I was hoping for some direct path between the two ideas.

My intuition comes from the fact that the phase space variables pop up in the weighting in the path integral. I wanted to explore this link and see if the Wigner function pops out anywhere.

 

[naima]

How do they pop out? Paths have the same weight.

 

[stevendaryl]

In this paper, http://arxiv.org/pdf/quant-ph/0308119v2.pdf, the relationship between the Wigner function and path integrals is described.

I haven't read it. I'm a little curious about how complex-conjugates come into a path-integral derivation.

 

[naima]

Path integrals are in space time
Here the author shows how a Wigner function, at a given moment, can be computed with paths in the phase space. It is closer to the density matrix <-> Wigner function equivalence than Feynman paths integrals.

 

[atyy]

The path integrals are just a convenient way of doing some calculations. They are not an independent formulation of quantum mechanics. The Hilbert space formulation is primary, because it determines which path integrals are acceptable. For example, one set of conditions that determines an acceptable path integral formulation are the Osterwalder-Schrader conditions.

 

[Matta Tanning]

In this paper, http://arxiv.org/pdf/quant-ph/0308119v2.pdf, the relationship between the Wigner function and path integrals is described.

I haven't read it. I'm a little curious about how complex-conjugates come into a path-integral derivation.

Path integrals are in space time
Here the author shows how a Wigner function, at a given moment, can be computed with paths in the phase space. It is closer to the density matrix <-> Wigner function equivalence than Feynman paths integrals.

HI naima. Thanks for you responses. Just to double check, you are referring to the Samson paper given by stevendaryl right? I looked at it but it didn't feel to be space-timey so I left it. (cheer for putting it on here though stevendaryl)

The path integrals are just a convenient way of doing some calculations. They are not an independent formulation of quantum mechanics. The Hilbert space formulation is primary, because it determines which path integrals are acceptable. For example, one set of conditions that determines an acceptable path integral formulation are the Osterwalder-Schrader conditions.

Hi atyy, thank you for your response too. I am unfamiliar the Osterwalder-Schrader conditions and I don't know if I am going to be able to understand exactly what you are saying. This is my understanding:

We have a configuration space based on space-time. On this configuration space we have a tangent and cotangent bundle, related by the Legendre transform. These bundles give a sense of change in the configuration space by applying (co)tangent vectors. The Wigner function on the cotangent bundle (phase space) leads the way to one form of quantisation. A second way is to take the Lagrangian as a map from the tangent bundle and have a path integral quantisation. A third way would be in some way adding commutation relations to the axes of the phase space and having these as operators on some Hilbert space which is presumably related to the configuration space.

It seems me that I could probably find places that give me the link between Feynman and Hilbert and Wigner and Hilbert, but not Feynman and Wigner.

I would be grateful if anyone could give a clarification of my picture (which may be woefully misguided), or an idea of how the Wigner function and the path integral connect. The paper provided by stevendaryl shows that they look similar in being included as weights in an integral. Do I just take the logarithm of e^S and then Legendre transform? (my guess is "No!").

 

[naima]

One more time Wigner functions are on phase space not on on phase space time. How can you introduce time to get paths in space time? You need an hamiltonian for time to appear.

 

[Matta Tanning]

One more time Wigner functions are on phase space not on on phase space time. How can you introduce time to get paths in space time? You need an hamiltonian for time to appear.

Are you saying that Wigner functions are on phase space where phase space is the cotangent bundle of space, whereas path integrals are on spacetime, and this is the reason we can't get straight between them? Are you saying there is no direct link or that we just need the extra structure of a Hamiltonian? In which case can you paint the picture?

Sorry for being slow.

 

[stevendaryl]

Are you saying that Wigner functions are on phase space where phase space is the cotangent bundle of space, whereas path integrals are on spacetime, and this is the reason we can't get straight between them? Are you saying there is no direct link or that we just need the extra structure of a Hamiltonian? In which case can you paint the picture?

Sorry for being slow.

I would not say that path integrals are particularly about spacetime. They are about PATHS, which are parametrized one-dimensional curves through whatever space is being discussed, but the parameter is not necessarily time. For example, in quantum statistical mechanics, the partition function can be calculated as a path integral, where the path variable is the temperature (the reciprocal of the temperature, actually), rather than time.

 

[naima]

It is right.But in statistical physics the sum gives you a probability not an amplitude. I read that every QM problem has an equivalent problem in thermodynamics. I do not know what is the equivalent of entanglement.

 

[vanhees71]

I'd say, path integrals are first of all (somewhat imprecise) defined integrals over trajectories in phase space, and it is very important to keep that in mind in applications. Very often the canonical momenta occur only up to quadratic order in the Hamiltonian version of the action, and thus one can take the momentum path integral explicitly (up to an undetermined constant, if not somehow regularized, usually in terms of a discretized "lattice" version of the integral or, more elegantly, using the heat-kernel approach), very often leading to a pathintegral over paths in configuration space with the Lagrangian form of the action in the exponential.

The exception is, when one of the coefficients in the quadratic momentum dependence in the Hamiltonian action become dependent on the configuration-space degrees of freedom. Then one has to do the momentum path-integral with great care of each such case. An nice example is the relativistic charged scalar (Klein-Gordon) field for the canonical statistical operator. When calculating the partition sum of the free (ideal) gas or when one likes to establish perturbation theory for the interacting case, one has to do the momentum path integral, and one ends up with a modified Lagrangian. This is, because the configuration-space version of the conserved-charge density contains a time derivative of the field, and the corresponding piece is linearly dependent on the canonical field momenta, which are $\dot{\phi}$ and $\dot{\phi}^*$, with the fields as coefficient. This leads to a modification of the naive Lagrangian in the term involving the chemical potential.

In vavuum QFT such a case occurs for scalar QED (for the same reason), but there one can find a gauge, so that the naive Lagrangian occurs, and this holds for all other gauges due to gauge invariance. See, e.g.,

http://fias.uni-frankfurt.de/~hees/publ/lect.pdf