Interpretation of Wigner function in optics I work in the field of synchrotron radiation sources where radiation (often x-rays) is produced from an electron beam going through magnetic fields.  The quality of the resulting x-ray beam is determined by a parameter called the brightness, which is formally computed by the Wigner function for the radiation.  The Wigner function is claimed to be a representation of the photon flux density in the radiation.  It can sometimes be negative, which would be unphysical, but people say this is somehow due to Quantum Mechanics.  There is this analogy between light optics and quantum mechanics, where $\hbar$ is replaced by the wavelength of the light.  But I don't think that there is actual quantum mechanics involved, though I could be wrong.  I think it is just an analogy.
So my question is: When one represents radiation via a Wigner function, is this really quantum mechanics? (A kind of semi-classical approximation?) Can someone point me to good references on understanding this from a slightly deeper perspective?  I'm interested if there may be some real mystery here, or if its actually well-understood.
A reference by Kim:
http://www.osti.gov/energycitations/product.biblio.jsp?osti_id=6202594
 A: Citations are from here on Wikipedia.
Is this really quantum mechanics?

In the modelling of optical systems such as telescopes or fibre telecommunications devices, the Wigner function is used to bridge the gap between simple ray tracing and the full wave analysis of the system. Here p/ħ is replaced with k = |k|sinθ ≈ |k|θ in the small angle (paraxial) approximation. In this context, the Wigner function is the closest one can get to describing the system in terms of rays at position x and angle θ while still including the effects of interference.
Seems that answer is "no". While I have a feeling that there is no strict boundary. I usually think of optics as of a "bridge" between quantum and classical mechanics...

It can sometimes be negative

If it becomes negative at any point then simple ray-tracing will not suffice to model the system.

Good references
In the end of the Wikipedia article. For example this one:
http://scripts.mit.edu/~raskar/lightfields/index.php?title=An_Introduction_to_The_Wigner_Distribution_in_Geometric_Optics
A: You're effectively doing signal analysis. It's neither quantum mechanics nor a semi-classical approximation. Signal processing is (very) often not stochastic (random), unless thermal noise is an issue for accurate modeling of the apparatus. It's often easier to find a way to eliminate the effects of thermal noise than to calculate the effects of thermal noise. If quantum noise has a significant effect, then one has to use quantum mechanics.
The mathematics of signal processing can almost always be presented in terms of Hilbert spaces, just because Fourier analysis of the signals is so important, even when the signals being processed are noise-free, which gives a specific mathematical link between deterministic signal processing and quantum mechanics, however the interpretation of the observables is generally quite different. I would say that the ideas that the "Wigner function is claimed to be a representation of the photon flux density in the radiation", and that negative values are unphysical, are not helpful, that it's better to see the negativity as a consequence of attempting to measure the frequency of a signal over a very small time period, whereas measuring the frequency of a signal precisely in principle requires the signal to be measured as a function of time for all time, because the Fourier analysis of a signal requires us to take the integral over all time, $\int_{-\infty}^\infty f(t)\mathrm{e}^{\mathrm{i}t\omega}\mathrm{d}t$.
A good reference for Wigner functions is Leon Cohen, "Time-Frequency Distributions-A Review", PROCEEDINGS OF THE IEEE, VOL. 77, NO. 7, JULY 1989, DOI: 10.1109/5.30749. Alternatively, a book by the same author, Leon Cohen, "Time Frequency Analysis: Theory and Applications". Sorry to say that either of these will need access to an academic library or cash. I didn't know of any on-line references before I searched for them now --- the Wikipedia page you want is cited on the page cited by Kostya, http://en.wikipedia.org/wiki/Cohen%27s_class_distribution_function; you should certainly read it as well because it is much more relevant to your application. There is a fairly strong sense in which this is well understood, but quantum mechanics, and particularly quantum field theory, is not so well-understood, which makes it correspondingly difficult to say that the relationship between classical signal processing and quantum mechanics is well-understood.
A: The Wigner function is used to describe joint probabilities between two sets of observables that do not commute. I could elaborate a bit more, but there is a lot of literature available where the authors explain this better than I can. 
A couple of useful articles from arXiv:
Probabilistic aspects of Wigner function
Negativity of the Wigner function as an indicator of nonclassicality
Photon viewed from Wigner Phase Space
Google Book result of the above paper
What is a photon?: OPN-Trends special series by experts in Quantum Optics -pdf file
Edit: 11/2/2011
Maxwell Wavefunction of a Photon
Photon Wavefunction
Interaction between Light and Matter, a wavefunction approach-pdf file
A: The Wigner function is an equivalent representation of the quantum state used in the Wigner & Moyal formulation of quantum mechanics. Sometimes named the phase space formulation.
In fact the Wigner function is more general than the wavefunction, because also represents quantum mixed states whereas wavefunctions only can represent pure states. This recent formulation is rather popular in quantum optics because the equation of motion takes a simple form for quantum harmonic oscillators (a quantum field is a collection of such oscillators) and because of other reasons.
Yes, the Wigner function is a pseudo-probability distribution and can take on negative values. The negative values are a consequence of quantum interference.
Therefore the answer to your question is yes when one represents radiation via a Wigner function this is just quantum mechanics.
Any advanced textbook in quantum mechanics (e.g. Ballentine) discusses the Wigner function. A classic reference in quantum optics using the Wigner function is http://www.amazon.com/Quantum-Optics-Phase-Wolfgang-Schleich/dp/352729435X
