This booklet addresses your questions in Chapter 0.9 and specifically in equation (6). The Moyal equation is just the von Neumann evolution equation for the density matrix in a different, phase-space, representation. The von Neumann equation amounts to the Schrödinger equation, i.e. they specify time development of functions, with absolutely no information on what such functions are. If the initial condition inputs are physical wavefunctions $\psi(x,t_0)$ or Wigner distributions (WF) $f(x,p,t_0)$ with all the requisite requirements for physicality (normalization, reality for the WF, eqn (6), etc), the evolution output is a physical function at time t. If the inputs are are garbage, the outputs are most likely garbage: the Hamiltonian bears no relation to the physicality of the solutions.
You are thus probably asking about the requisite conditions for a phase space function to be physical at a specific moment, e.g. the origin of time. For a real normalized phase-space function to be of the form
$$
f(x,p)=\frac{1}{2\pi}\int\! dy~\psi^* \left (x-\frac{\hbar}{2}
y \right )~e^{-iyp} ~ \psi \left (x+\frac{\hbar}{2} y \right ),
$$
so bona-fide physical and normalized, you need to ensure its cross-spectral density, its Fourier transform, "left-right" factorizes,
$$
\tilde{f}(x,y)=\int dp ~e^{ipy} f(x,p) ~ = ~
g^{*}_L (x-\hbar y/2) ~g_R (x+\hbar y/2)~.
$$
That is,
$$
{\partial^2 ~~~\ln \tilde{f} \qquad \qquad \phantom{a} \over
\partial(x-\hbar y/2)~\partial(x+\hbar y/2)}
=0 ~, \tag{6}
$$
so that, for real $f$, ~ $g_L=g_R$. You may thus work out the Schrödinger wavefunction from this cross-spectral density; the map is bijective.
This all holds for pure states; for mixed states, you have to parse and reorganize the above argument.
Further comments as per question edit
Could Moyal's evolution equation predict results as Schrödiger's equation does? What it means : for a given system, investigating Scroedinger's equation give us a 𝜓 and hence we know all about the system in QM. This question is for the case of Moyal's evolution equation.
Yes! Just as TDSE tells you everything about the future development of a complex normalized function as IC, just so, the Moyal equation tells you everything about a real normalized f satisfying (6) as IC, now for all times in the future.
And is this process independent of Schroedinger's equation? What this means: Could we get the same physics without knowing 𝜓
and not solving Schroedinger's equation, but just from solving Moyal's evolution equation of given system?
A qualified yes: as emphasized in Lemma 0.3 of Ch 0.10, much more important than Moyal's equation is the fundamental "stargenfunction" equation of the *-spectrum of the Hamiltonian!
The *-spectral resolution of the Hamiltonian essentially solves the problem, and its determination does not rely on wavefunctions 𝜓!
Once this star spectrum is known, (which is explicitly illustrated for the oscillator in Ch 0.13), the future development of a bona-fide WF at the origin of time is known completely for all points in the future. The IC WF is *-resolvable in these stargenfunctions, assuming it obeys all the above conditions, and propagates trivially as illustrated by Moyal's equation, just as 𝜓 does for the TDSE.
Indeed, this is the celebrated autonomy of the formulation: the formulation is self-standing, and equivalent, to the Hilbert space formulation (TDSE).
No knowledge of Schrödinger's equation is assumed. It would yield the same physics results on a planet where this formulation arose first, and then the TDSE of Hilbert space after it, the opposite of what happened on our plant!
(Truth be told, Hilbert space and the TDSE are easier to handle than Moyal's equation, so chemistry would be less advanced on that planet than in ours, unless these aliens were really smarter than us-- hardly a surprise...)
- If you were mathematically inclined, all you have to appreciate is that the WF is the (invertible) Winger transform of the density matrix, and the Moyal equation of the von Neumann equation. You are then solving the von Neumann equation in an unfamiliar representation (phase space), Feynman's point of view.
