# Wigner functions, symmetry

I'm trying to get more insight into quasiprobability distributions, as for example the Wigner function.

There are some Wigner functions, which are symmetric.

Symmetric:

Fock state

Thermal states

Vacuum states

Non symmetric:

Coherent states

What's the interpretation of that? Does every state which is diagonal in the Fock basis has a symmetric Wigner function?

• What do you mean by symmetric? And also quasi-distribution functions are defined for operators not for states. Could give some example of what you have in mind. – WoofDoggy Feb 7 '16 at 12:59

## 1 Answer

You might firm up your intuition from this gallery of WFs. The idea for Fock space stationary states is that they are expansions around the vacuum, the ground state of the axially (x-p) symmetric oscillator Hamiltonian so they are symmetric. That is, if by "diagonal in Fock space" you mean eigenstates of the number operator, in phase space that simply amounts to $\star$-genstates of the oscillator hamiltonian, so, then, axially symmetric by eqn (43) in our book, Thomas L. Curtright, David B. Fairlie, & Cosmas K. Zachos, A Concise Treatise on Quantum Mechanics in Phase Space, World Scientific, 2014 on the subject.

A coherent state, by contrast, not a Hamiltonian eigenstate, amounts to translating the center of these configurations, to a desired new center, by a phase-space translation operator, so they are defined to be asymmetric about the vacuum; although, of course, they have a new center of axial symmetry (Behold!).

Likewise, as the name implies, squeezed states are asymmetric.

Other configurations will, in general, lack axial symmetry, unless they are eigenstates of an x-p symmetric Hamiltonian.

Note, however, in general, a non-stationary configuration of the oscillator, an oscillating particle, so then a rigidly rotating configuration, is not symmetric——except when time-averaged.