Quantum phase space

Question

Classical phase space is defined as a space in which all possible states are represented. Every state corresponds to a unique point in the phase space.

On the other hand, in quantum mechanics every quantum mechanical observable corresponds to a unique function (distribution) on phase space.

In quantum mechanics we usually deal with Wigner function in $(x,p)$ space, but there also exist Glauber-Sudarshan $P$ or Husimi $Q$ distribution in parameter space of coherent states. This concept was firstly applied to Hamiltonians with Heisenberg-Weyl dynamical group, but other hamiltonians with different dynamical group (e.g. $SU(2)$) have formulation in phase space - parametric space of generalized coherent space (Perelomov).

In former case, both phase space $(x,p)$ and coherent state is flat - it is just a plane. In the concept of generalized coherent states for e.g. $SU(2)$ group, parameter space can be identified with sphere.

1. What is the topology of the associated phase space with Wigner function?

2. Can we only determine topology of the phase space, or the geometry is also possible?

3. Are all those formulations with Wigner, Glauber, Husimi etc. equivalent?

Answer 1

1 & 2. The geometry and topology of the relevant phase space is identical for both classical and quantum problems: it is the very same phase space. It is the theory acting on the phase space that may differ. The scale of the former is the small $\hbar$ limit of the latter.

Extended WFs appear like δ-fctn spikes ("points") in the small $\hbar$ limit, once the phase space-variables are suitably rescaled by $\sqrt{\hbar}$. That's what you probably mean by classical states. You might monitor the "morphing" of static oscillator WFs as $\hbar$ decreases, in Ref. 1.

3. All these representations (Wigner, Glauber, Husimi, Mehta...) are equivalent, that is, there are standard invertible maps from each to each other, cf. Ref. 1.

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References:

1. Thomas L. Curtright, David B. Fairlie, & Cosmas K. Zachos, A Concise Treatise on Quantum Mechanics in Phase Space, World Scientific, 2014. The PDF file is available here.