# Is there a Schrödinger equation for phase space?

The Schrödinger equation is generally formulated in position space $$i \hbar \frac{\partial}{\partial t}\psi(x,t) = \hat H_x \psi(x,t) = \left [ \frac{-\hbar^2}{2m}\frac{\partial^2}{\partial x^2} + V(x,t)\right ]\psi(x,t)$$ or in momentum space $$i \hbar \frac{\partial}{\partial t}\psi(p,t) = \hat H_p \psi(p,t) = \left[ \frac{p^2}{2m} + V\left(\frac{\hbar}{i}\frac{\partial}{\partial p},t\right)\right ]\psi(p,t).$$

But is it also possible to derive a Schrödinger equation for position $\times$ momentum $=$ phase space? I assume in that case it should be an equation acting on a wave function $\psi = \psi(x,p,t)$ . So I guess simply multiplying the two Schrödinger equations above is not enough. How could one then derive a Schrödinger equation for phase space, if it is possible at all?

There is no "wavefunction in phase space" because the wavefunctions $\psi(x)$ and $\psi(p)$ are obtained from the abstract state vector $\lvert \psi\rangle$ by $\langle x\vert \psi\rangle$ and $\langle p\vert \psi\rangle$, respectively. Since position and momentum don't commute, there are no $\lvert x,p\rangle$ to get a naive $\psi(x,p)$.

However, from any wavefunction we may obtain the Wigner quasiprobability distribution $W(x,p,t)$ on the classical phase space by the Wigner-Weyl transform. It obeys the equation $$\partial_t W(x,p,t) = \{\{H(x,p,t),W(x,p,t)\}\}$$ with $H$ the classical Hamiltonian and the bracket on the r.h.s. as the Moyal bracket.

There actually is a formulation, due to Torres-Vega and Frederick, but let me hasten to add that you really don't want to go there, for about half a dozen good reasons, most of them very technical, and some quite conceptual.

As @ACuriousMind already outlined, the classic satisfactory and powerful formulation of QM in phase space involves the analog of the density matrix (which is traced with observables to yield expectation values in Hilbert space), the Wigner Function (which is integrated with c-number observables in phase space instead). You might consider this introduction Quantum mechanics in phase space, by Curtright, Fairlie & me, if you wished to know more about what this formulation gets you that the other two, Hilbert space, and path integrals, don't. (You might choose to focus on eqn (91) and Exercise 0.11 there for a gateway to the T-V & F reformulation. Remember, all phases needed for interference are already in off-diagonal Wigner Functions!)

The Torres-Vega/Frederick formulation I mentioned finds its meaning by painful projection/reference to the Wigner Function mentioned, but, once you stretch your conceptual horizons to appreciate that formulation, you might realize you really have no use for the Schroedinger equation you are envisioning, but its density matrix analog, the von Neumann equation, instead, much as you derive stat mech expectation values by integrating agains Liouville densities.

Added edit: The Ambiguity function connects to the WF via a 2d Fourier transform, as indicated; basically, all roads go through the WF. For instance, by multiplying the coordinate to the momentum space wave functions, you came close to defining the Mehta distribution function, cf my answer to 233353, Exercise 0.19, but by a phase... what Terletsky and Blokhintsev did in the 30s. The systematic theory of all of these connexions could be accessed in L Cohen's book.

• Your phase space function in your intro book looks like a signal processing ambiguity function. Good for analysis even if no different information than the original wavefunction. Anyway, does the magnitude of the function have a clear physical meaning, or does the phase and the negatives have some interpretation? Just thinking outloud, in comparison to signal processing. Have not read the online book yet but will look more carefully at it. – Bob Bee Sep 19 '16 at 0:36
• Yes, the ambiguity function and signal processing WF are invertible transforms of each other. The math of QM and signal processing are identical to each other, except the effective hbar in EE is 1. In QM the negative values are safe, necessary, and the hallmark of QM interference. They are protected from bservatin by the uncertainty principle. The magnitude of the (real) WF represents increased measure/contribution at a point in phase space. – Cosmas Zachos Sep 19 '16 at 2:24
• Thanks. I've done more ambiguity functions than I ever wanted to, so I'll read your online book. Maybe I'll understand it. – Bob Bee Sep 19 '16 at 4:01
• Maurice de Gosson expands a little on the mathematics (see e. g. arxiv:math.0503709), but as you correctly point out, the whole point of view is a bit pointless. Mathematically speaking, what Gosson presents is equivalent to the usual point of view (the exact nature of the equivalency depends on your mathematical framework of choice), so there are no additional insights to be had. I'd much rather suggest you have a look at the Wigner-Weyl calculus, because this can be used to perform semiclassical limits and construct perturbation expansions. – Max Lein Sep 20 '16 at 4:50

There is an obvious candidate for what you are looking for in the process of geometric quantisation (https://en.wikipedia.org/wiki/Geometric_quantization). This is mostly a mathematical game, and the only books I have ever encountered it in are of a strongly mathematical nature.

Geometric quantisation seeks to generate a quantum Hilbert space by considering wavefunctions on the phase space, together with position and momentum operators. The Schroedinger equation then has the same form, but the wavefunctions and operators are defined on phase space.

However, this approach runs into some serious trouble. Basically, a wavefunction on position or momentum space only already has the correct number of degrees of freedom, so using a wavefunction on phase space has too many. This problem is solved by only considering wavefunctions obeying certain symmetry requirements. The problem is that the operator content of the theory depends on a choice of multiple possible symmetry requirements, and usually the theory contains way too few operators (it is not uncommon for the Hamiltonian not to be part of the theory).

There is, however, a notable exeption. In Chern-Simons theory (https://en.wikipedia.org/wiki/Chern%E2%80%93Simons_theory), the classical phase space has a non-trivial geometry that makes geometric quantisation the only viable candidate. Luckily, this theory is a topological gauge theory, and therefore the operator content is extremely limited. Due to this, geometric quantisation is sufficient to quantise the theory in some simple cases, as Witten showed (Comm. Math. Phys. Volume 121, Number 3 (1989), 351-399). One then uses formal properties of the path integral to extend this to arbitrary closed spacetimes.

This last example might be a bit over the head of anyone not familiar with the topic already, but what one should take away from it is the following: Geometric quantisation is a mathematically heavy procedure that many physicists do not even know about, yet it has at least one useful application in theoretical physics, which makes it potentially useful despite the theoretical difficulties in learning it.

The answers given so far are correct, but I don't think they hit the core of the issue. Quantum mechanics has either a position or a momentum representation where this or is an exclusive or. In general quantum physics is described according to a complete set of commuting operators, and in the case of momentum and space this means one must work in one representation, but not two at the same time.

Hamiltonian mechanics in phase space describes dynamics according to the energy surface in the $6n$ dimensional phase space, for $n$ particles in $3$ dimensional space, and $3$ dimensional momentum space. The condition $E~=~H$ on the energy surface reduces the space of motion to $6n-1$ dimensions. Lagrangian mechanics works in configuration variables that are $3n$ dimensional. This is then "half of phase space." Quantum mechanics then operates in this domain when working with configuration variables. This is even though QM and the Schroedinger equation uses a Hamiltonian.