Moyal Bracket

The Moyal bracket is the phase-space version of the quantum commutator. In Hilbert-space language, dynamics is generated by

$$\frac{1}{i\hbar}[H,\rho].$$

In Wigner phase space, the same closed-system dynamics is generated by

$$\{H_W,W_\rho\}_M,$$

where $H_W$ is the Weyl symbol of the Hamiltonian, $W_\rho$ is the Wigner function, and $\{\cdot,\cdot\}_M$ is the Moyal bracket. It reduces to the classical Poisson bracket when the relevant $\hbar$ corrections are negligible.

Star Product Background

The Weyl transform maps operators to phase-space functions. Operator multiplication does not become ordinary multiplication. Instead, it becomes the star product. For one degree of freedom, a common convention is

$$(A\star B)(x,p) = A(x,p) \exp\left[ \frac{i\hbar}{2} \left( \overleftarrow{\partial_x}\overrightarrow{\partial_p} - \overleftarrow{\partial_p}\overrightarrow{\partial_x} \right) \right] B(x,p).$$

The arrows specify which factor the derivative acts on. Equivalently, define the bidifferential operator $\Lambda$ by

$$A\Lambda B = \frac{\partial A}{\partial x} \frac{\partial B}{\partial p} - \frac{\partial A}{\partial p} \frac{\partial B}{\partial x}.$$

Then the star product can be written schematically as

$$A\star B = A\exp\left( \frac{i\hbar}{2}\Lambda \right)B.$$

The lowest terms are

$$A\star B = AB + \frac{i\hbar}{2}\{A,B\}_{\rm PB} + O(\hbar^2),$$

where

$$\{A,B\}_{\rm PB} = \frac{\partial A}{\partial x} \frac{\partial B}{\partial p} - \frac{\partial A}{\partial p} \frac{\partial B}{\partial x}$$

is the classical Poisson bracket.

Definition

The Moyal bracket is the antisymmetric part of the star product:

$$\{A,B\}_M = \frac{1}{i\hbar} \left( A\star B-B\star A \right).$$

Using $\Lambda$, this may be written as

$$\{A,B\}_M = \frac{2}{\hbar} A\sin\left( \frac{\hbar}{2}\Lambda \right)B.$$

Expanding the sine gives

$$\{A,B\}_M = A\Lambda B - \frac{\hbar^2}{24} A\Lambda^3B + O(\hbar^4).$$

The first term is the Poisson bracket. The remaining terms encode quantum corrections due to noncommutativity.

Relation to the Commutator

Let $A_W$ and $B_W$ be Weyl symbols of operators $A$ and $B$. Then

$$\left( \frac{1}{i\hbar}[A,B] \right)_W = \{A_W,B_W\}_M.$$

This is the conceptual core of the Moyal bracket. It is not a new law of motion; it is the commutator written in phase-space language.

For canonical variables, the Moyal bracket reproduces the basic commutator:

$$\{x,p\}_M=1,$$

matching

$$\frac{1}{i\hbar}[\hat x,\hat p]=1.$$

Because $x$ and $p$ are linear functions, there are no higher-derivative corrections in this example.

Wigner-Function Evolution

The closed-system density operator obeys the Liouville–von Neumann equation

$$\frac{\partial\rho}{\partial t} = \frac{1}{i\hbar}[H,\rho].$$

Taking the Wigner transform gives

$$\frac{\partial W}{\partial t} = \{H_W,W\}_M.$$

For

$$H=\frac{p^2}{2m}+V(x),$$

with the standard Weyl symbol

$$H_W(x,p)=\frac{p^2}{2m}+V(x),$$

the equation becomes

$$\begin{aligned} \frac{\partial W}{\partial t} &= - \frac{p}{m}\frac{\partial W}{\partial x} + \sum_{\ell=0}^{\infty} \frac{(-1)^\ell}{(2\ell+1)!} \left( \frac{\hbar}{2} \right)^{2\ell} \\ &\quad \times \frac{d^{2\ell+1}V}{dx^{2\ell+1}} \frac{\partial^{2\ell+1}W}{\partial p^{2\ell+1}}. \end{aligned}$$

The $\ell=0$ term is

$$\frac{dV}{dx}\frac{\partial W}{\partial p}.$$

Together with the kinetic term, it gives the classical Liouville flow:

$$\frac{\partial W}{\partial t} = \{H,W\}_{\rm PB} +O(\hbar^2).$$

Classical Limit

The Moyal bracket has the formal classical limit

$$\lim_{\hbar\to0}\{A,B\}_M = \{A,B\}_{\rm PB},$$

provided the functions are smooth enough on the relevant phase-space scale. More physically, the Poisson bracket dominates when the Wigner function and Hamiltonian vary slowly compared with the $\hbar$-scale oscillations where quantum interference is important.

For quadratic Hamiltonians, all third and higher derivatives vanish, so the Moyal bracket equals the Poisson bracket exactly. This is why free particles, harmonic oscillators, and general quadratic systems move Wigner functions by classical phase-space flows even though the states remain quantum.

For anharmonic systems, higher derivatives of $V(x)$ generate quantum corrections. These corrections are the phase-space version of noncommutative operator dynamics.

Example: Free Particle

For

$$H=\frac{p^2}{2m},$$

the Moyal equation reduces to

$$\frac{\partial W}{\partial t} = - \frac{p}{m}\frac{\partial W}{\partial x}.$$

Equivalently,

$$\frac{\partial W}{\partial t} + \frac{p}{m}\frac{\partial W}{\partial x} =0.$$

The solution is transport along classical straight-line trajectories:

$$W(x,p,t) = W\left( x-\frac{pt}{m}, p, 0 \right).$$

The Wigner function may still have quantum features, such as negative regions or interference fringes, but the free-particle flow translates them classically in phase space.

Example: Harmonic Oscillator

For

$$H=\frac{p^2}{2m}+\frac12m\omega^2x^2,$$

the Moyal bracket again equals the Poisson bracket exactly:

$$\frac{\partial W}{\partial t} = - \frac{p}{m}\frac{\partial W}{\partial x} + m\omega^2x\frac{\partial W}{\partial p}.$$

This is rigid rotation in oscillator phase space. Coherent-state Wigner functions therefore rotate like classical oscillator ellipses without changing shape.

Energy eigenstate Wigner functions are stationary because their density operators commute with $H$. In phase-space language,

$$\{H,W_n\}_M=0.$$

Example: Quartic Correction

For an anharmonic potential

$$V(x)=\frac{\lambda}{4}x^4,$$

the first quantum correction comes from $V'''(x)=6\lambda x$. The Wigner evolution contains

$$\frac{\partial W}{\partial t} = - \frac{p}{m}\frac{\partial W}{\partial x} + \lambda x^3\frac{\partial W}{\partial p} - \frac{\hbar^2\lambda x}{4} \frac{\partial^3 W}{\partial p^3} +\cdots.$$

The first two terms are the classical Liouville flow. The third term is the leading Moyal correction.

Common Mistakes

• Replacing the star product by ordinary multiplication in commutators.
• Assuming the Moyal bracket is always just the Poisson bracket.
• Forgetting that the Hamiltonian must be represented by the correct Weyl symbol.
• Ignoring sign and normalization conventions in the Wigner transform.
• Treating exact classical-looking flow for quadratic Hamiltonians as proof that the state is classical.
• Applying the $\hbar$ expansion where the Wigner function has sharp interference features on the $\hbar$ scale.

Cross-Links

• Wigner Function
• Phase-Space Conventions
• Liouville–von Neumann Equation
• Commutator Dynamics
• Phase Space
• Commutators
• Quantum Harmonic Oscillator
• Free Particle
• Formula Sheet

References

• H. J. Groenewold, “On the Principles of elementary quantum mechanics,” Physica 12, 405-460, 1946.
• J. E. Moyal, “Quantum mechanics as a statistical theory,” Mathematical Proceedings of the Cambridge Philosophical Society 45, 99-124, 1949.
• M. Hillery, R. F. O’Connell, M. O. Scully, and E. P. Wigner, “Distribution functions in physics: Fundamentals,” Physics Reports 106, 121-167, 1984.
• C. K. Zachos, D. B. Fairlie, and T. L. Curtright, eds., Quantum Mechanics in Phase Space, World Scientific, 2005.
• W. B. Case, “Wigner functions and Weyl transforms for pedestrians,” American Journal of Physics 76, 937-946, 2008.

Exercises

1. Show that $\{x,p\}_M=1$.

Solution

Because $x$ and $p$ are linear functions, the star product terminates after the first derivative term:

$$x\star p = xp+\frac{i\hbar}{2},$$

and

$$p\star x = px-\frac{i\hbar}{2}.$$

Therefore

$$\{x,p\}_M = \frac{1}{i\hbar} \left( x\star p-p\star x \right) =1.$$

2. Derive the free-particle Wigner equation from the Moyal bracket.

Solution

For

$$H=\frac{p^2}{2m},$$

the Hamiltonian depends only on $p$. Since it is quadratic, the Moyal bracket equals the Poisson bracket:

$$\{H,W\}_M = \{H,W\}_{\rm PB}.$$

Compute

$$\{H,W\}_{\rm PB} = \frac{\partial H}{\partial x} \frac{\partial W}{\partial p} - \frac{\partial H}{\partial p} \frac{\partial W}{\partial x} = - \frac{p}{m}\frac{\partial W}{\partial x}.$$

Thus

$$\frac{\partial W}{\partial t} = - \frac{p}{m}\frac{\partial W}{\partial x}.$$

3. Why do quadratic Hamiltonians have exact classical phase-space flow in the Wigner representation?

Solution

The Moyal bracket expansion is

$$\{A,B\}_M = \{A,B\}_{\rm PB} - \frac{\hbar^2}{24}A\Lambda^3B + O(\hbar^4).$$

If the Hamiltonian is at most quadratic in $x$ and $p$, all third and higher derivatives of the Hamiltonian vanish. Therefore the higher Moyal terms vanish, and

$$\{H,W\}_M=\{H,W\}_{\rm PB}.$$

The state may still be nonclassical; the statement is about the form of the phase-space flow.

4. Find the leading Moyal correction for $V(x)=\lambda x^4/4$.

Solution

For

$$V(x)=\frac{\lambda}{4}x^4,$$

one has

$$V'(x)=\lambda x^3, \qquad V'''(x)=6\lambda x.$$

The potential part of the Wigner equation begins

$$V'(x)\frac{\partial W}{\partial p} - \frac{\hbar^2}{24} V'''(x) \frac{\partial^3W}{\partial p^3} +\cdots.$$

Substituting $V'''(x)=6\lambda x$ gives

$$\lambda x^3\frac{\partial W}{\partial p} - \frac{\hbar^2\lambda x}{4} \frac{\partial^3W}{\partial p^3} +\cdots.$$