How does the Weyl transform take into account which quasiprobability distribution was used?

Question

I'm trying to get a better understanding of the Weyl correspondence which, as described e.g. on Wikipedia, gives "an invertible mapping between functions in the quantum phase space formulation and Hilbert space operators in the Schrödinger picture".

In particular, the Weyl transform of a function $f(x,p)$ is given by $$\Phi[f] = \frac{1}{(2\pi)^2}\int \mathrm dq \,\mathrm d p\,\mathrm d a\,\mathrm d b f(q,p)\exp[i(a(Q-q)+b(P-p))].\tag1$$ I'm not entirely clear how to match this with the more "standard" approach of defining quasiprobability distributions without leveraging the phase-space formalism.

In particular, different quasiprobability distributions can be defined for a given state $\rho$ (e.g. Wigner, $P$, $Q$, etc.). If I'm understanding the phase-space formulation correctly, it is a generalised approach that is supposed to work for all such quasiprobability/phase-space distributions. However, how do we put into (1) the information on which phase-space representation is being used? The function $f$ could be the Wigner function of the same state $\rho$, or maybe its $Q$ function. How is this taken into account into (1)?

From the description in Section 0.18 of (Curtright, Fairlie, Zachos 2014) I think the answer might lie in different operator orderings of $Q$ and $P$, but I'm not entirely positive about it.

A simple example of this in action (e.g. how one can get back the same state $\rho$ from two of its phase-space representations) would be helpful.

Answer 1

But...you are in the wrong chapter! The equivalence inter-relations of the various ordering prescriptions are in Chapter 0.19 of that link! (Chapter 0.18 merely details the obvious Weyl transform (134)/(140) and its inverse, the Wigner transform (138) for all well-behaved phase-space functions, and all sensible Hilbert-space operators, of which the density matrix is but one.)

And it is assumed you have understood and verified to dexterity their inverse relationship. So, for Weyl ordering, it is assumed you are completely unconflicted about (143), the well-trodden bridge to the original definition of the Wigner function and hence the Wigner transform.

Here, I will illustrate the connection of the two most popular orderings, the Weyl and the Husimi ordering, for a trivial phase-space function, the oscillator hamiltonian, $f_W(x,p)=(p^2+x^2)/2$. Using capitals for Hilbert space operators, as you do, the Weyl-ordering quantization of it is essentially your eqn (1), our $$ F = \frac{1}{(2\pi)^2}\int \mathrm dx \,\mathrm d p\,\mathrm d\tau \,\mathrm d \sigma ~f_W(x,p)\exp[i(\sigma (X-x)+\tau(P-p))].\tag {134} $$ For this particular symmetric ordering of X and P the phase space symbol is specified to be Wigner's.

As we detail and illustrate in the next chapter, the Husimi "symbol" $f_H$ for the same operator (not merely quasi-probability images of density matrices $|\psi\rangle \langle \psi| /h$ !) is specified implicitly by $$ F = \frac{1}{(2\pi)^2}\int \mathrm dx \,\mathrm d p\,\mathrm d\tau \,\mathrm d \sigma ~f_H(x,p)\exp[i(\sigma (X-x)+\tau(P-p))+\hbar(\tau^2+\sigma^2)/4]\tag {152} \\ = \frac{1}{(2\pi)^2}\int \mathrm dx \,\mathrm d p\,\mathrm d\tau \,\mathrm d \sigma ~(e^{-\hbar(\partial_x^2+\partial_p^2)/4}f_H(x,p))\exp[i(\sigma (X-x)+\tau(P-p))], $$ after integrating by parts.

So, comparison with the above, readily yields the celebrated double Weierstrass filter cornerstone bridge, $$ f_H(x,p)= e^{\hbar(\partial_x^2+\partial_p^2)/4} f_W .\tag{155} $$ For example, it should be easy to confirm that, from the classical-looking Weyl symbol of the oscillator Hamiltonian, the Husimi-symbol hamiltonian is shifted, $$ h_H(x,p)= e^{\hbar(\partial_x^2+\partial_p^2)/4} (p^2+x^2)/2 \\ = (p^2+x^2+\hbar )/2 = h_W + \hbar/2 .\tag{156} $$ A stark reminder that such symbols may, and routinely do!, involve $\hbar$s, which naive quantizers are vulnerable to missing.

So this is an illustration, in action of how the same operator follows from phase-space kernels ("symbols") of different prescriptions: they are designed to do that, really!

I did not bother to invert (152) for you, but, you realize now its inversion is no different than the inversion of the Weyl map (132): in fact, it may go through it!

Now, can you repeat this for a Gaussian, the oscillator ground state Wigner function? Do you see how Weierstrass low-pass filtering widens a Gaussian?

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NB: Explicit ordering

In case you were wondering what the above Husimi prescription has to do with "ordering" of operators, it really amounts to an implicit elaborate prescription when you look directly at operators--the ordering whose CBH unravelling would result in the exponential written.

You may analogize it to a different one, the "standard ordering" prescription of Blokhintsev (1940) (which some call "the Mehta prescription"), that puts all momentum operators P to the right, after all coordinate operators X.

The corresponding expression is analogous, utilizing Campbell-Baker-Hausdorff rearrangement, $$ F = \frac{1}{(2\pi)^2}\int \mathrm dx \,\mathrm d p\,\mathrm d\tau \,\mathrm d \sigma ~f_S(x,p)\exp[i\sigma (X-x)]\exp[i\tau(P-p))] \\ = \frac{1}{(2\pi)^2}\int \mathrm dx \,\mathrm d p\,\mathrm d\tau \,\mathrm d \sigma ~f_S(x,p)\exp[i(\sigma (X-x) +\tau(P-p)-\hbar \tau \sigma/2)] \\ = \frac{1}{(2\pi)^2}\int \mathrm dx \,\mathrm d p\,\mathrm d\tau \,\mathrm d \sigma ~(e^{i\hbar\partial_x \partial_p /2}f_S(x,p))\exp[i(\sigma (X-x)+\tau(P-p))], $$ that is, $$ f_S=e^{-i\hbar\partial_x \partial_p /2}f_W(x,p). $$ (Interestingly, Moyal utilized this very connection in 1949, to address his readers, who were come comfortable with the standard ordering than the Weyl ordering.)