Phase-Space Conventions

This page fixes the phase-space conventions used in the Dynamics volume. It is a reference page for signs, factors of $2\pi\hbar$, and product rules. The full explanations live in Weyl Transform, Wigner Function, Star Product, and Moyal Bracket.

The conventions are chosen so that

$$\langle x|p\rangle = \frac{1}{\sqrt{2\pi\hbar}}e^{ipx/\hbar},$$

and

$$x\star p-p\star x=i\hbar.$$

Different books may place signs or factors in different places. Translate definitions before comparing Wigner functions, Weyl symbols, or star products.

Fourier Convention

The position-momentum convention is

$$\psi(x) = \frac{1}{\sqrt{2\pi\hbar}} \int_{-\infty}^{\infty}dp\, e^{ipx/\hbar}\phi(p),$$

and

$$\phi(p) = \frac{1}{\sqrt{2\pi\hbar}} \int_{-\infty}^{\infty}dx\, e^{-ipx/\hbar}\psi(x).$$

Consequently,

$$\langle x|p\rangle = \frac{1}{\sqrt{2\pi\hbar}}e^{ipx/\hbar}.$$

This sign choice fixes the sign in the Wigner and Weyl transforms below. The site-wide convention page is Fourier Transform Conventions.

One-Degree-of-Freedom Weyl Symbol

For an operator $A$, its Weyl symbol is

$$A_W(x,p) = \int_{-\infty}^{\infty}dy\, e^{-ipy/\hbar} \left\langle x+\frac{y}{2} \middle| A \middle| x-\frac{y}{2} \right\rangle.$$

Equivalently, if

$$A(x_1,x_2) = \langle x_1|A|x_2\rangle,$$

then

$$A_W(x,p) = \int dy\, e^{-ipy/\hbar} A\left( x+\frac{y}{2}, x-\frac{y}{2} \right).$$

The inverse transform is

$$\left\langle x_1\middle|A\middle|x_2\right\rangle = \int\frac{dp}{2\pi\hbar}\, e^{ip(x_1-x_2)/\hbar} A_W\left( \frac{x_1+x_2}{2}, p \right).$$

The Weyl symbol is tied to symmetric ordering. For example,

$$\left( \frac12(\hat x\hat p+\hat p\hat x) \right)_W = xp.$$

Wigner Function

The Wigner function of a density operator $\rho$ is the normalized Weyl symbol

$$W_\rho(x,p) = \frac{1}{2\pi\hbar}\rho_W(x,p).$$

Explicitly,

$$W_\rho(x,p) = \frac{1}{2\pi\hbar} \int_{-\infty}^{\infty}dy\, e^{-ipy/\hbar} \left\langle x+\frac{y}{2} \middle| \rho \middle| x-\frac{y}{2} \right\rangle.$$

For a pure state,

$$W_\psi(x,p) = \frac{1}{2\pi\hbar} \int dy\, e^{-ipy/\hbar} \psi\left(x+\frac{y}{2}\right) \psi^*\left(x-\frac{y}{2}\right).$$

With this normalization,

$$\int dx\,dp\,W_\rho(x,p) = \operatorname{Tr}\rho.$$

For a normalized state, the integral is $1$.

Marginals

The position marginal is

$$\int dp\,W_\rho(x,p) = \langle x|\rho|x\rangle.$$

The momentum marginal is

$$\int dx\,W_\rho(x,p) = \langle p|\rho|p\rangle.$$

These are ordinary probability densities. The full Wigner function is not an ordinary joint probability density, because it can be negative.

Phase-Space Measure

The trace measure for one degree of freedom is

$$d\Gamma = \frac{dx\,dp}{2\pi\hbar}.$$

For suitable trace-class operators,

$$\operatorname{Tr}A = \int d\Gamma\,A_W(x,p).$$

For products of operators,

$$\operatorname{Tr}(AB) = \int d\Gamma\,A_W(x,p)B_W(x,p).$$

Since $W_\rho=\rho_W/(2\pi\hbar)$, expectation values are written without an extra $1/(2\pi\hbar)$:

$$\operatorname{Tr}(\rho A) = \int dx\,dp\, W_\rho(x,p)A_W(x,p).$$

This formula resembles a classical phase-space average, but the Wigner function may be negative and operator products use the star product.

Many Degrees of Freedom

For $n$ Cartesian degrees of freedom, write

$$\mathbf q=(q^1,\ldots,q^n), \qquad \mathbf p=(p_1,\ldots,p_n).$$

The Wigner function is

$$\begin{aligned} W_\rho(\mathbf q,\mathbf p) &= \frac{1}{(2\pi\hbar)^n} \int d^n\mathbf y\, e^{-i\mathbf p\cdot\mathbf y/\hbar} \\ &\quad\times \left\langle \mathbf q+\frac{\mathbf y}{2} \middle| \rho \middle| \mathbf q-\frac{\mathbf y}{2} \right\rangle. \end{aligned}$$

The trace measure is

$$d\Gamma_n = \frac{d^n\mathbf q\,d^n\mathbf p} {(2\pi\hbar)^n}.$$

Again,

$$\int d^n\mathbf q\,d^n\mathbf p\, W_\rho(\mathbf q,\mathbf p) = \operatorname{Tr}\rho.$$

For non-Cartesian coordinates or constrained phase spaces, this flat measure is not automatically correct.

Star Product

The Weyl symbol of an operator product is the star product of Weyl symbols:

$$(AB)_W = A_W\star B_W.$$

For one degree of freedom,

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

The first terms are

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

The basic check is

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

Therefore,

$$x\star p-p\star x=i\hbar.$$

For $n$ degrees of freedom, set

$$\mathbf z=(q^1,\ldots,q^n,p_1,\ldots,p_n),$$

and

$$\Omega = \begin{pmatrix} 0&I\\ -I&0 \end{pmatrix}.$$

Then the compact convention is

$$A\star B = A \exp\left[ \frac{i\hbar}{2} \overleftarrow{\partial_{\mathbf z}}^{\,T} \Omega \overrightarrow{\partial_{\mathbf z}} \right] B.$$

Poisson and Moyal Brackets

The Poisson bracket convention is

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

For many Cartesian degrees of freedom,

$$\{A,B\}_{\rm PB} = \sum_{a=1}^{n} \left( \frac{\partial A}{\partial q^a} \frac{\partial B}{\partial p_a} - \frac{\partial A}{\partial p_a} \frac{\partial B}{\partial q^a} \right).$$

This sign convention gives Hamilton’s equations in the form

$$\dot q^a=\{q^a,H\}_{\rm PB} = \frac{\partial H}{\partial p_a},$$

and

$$\dot p_a=\{p_a,H\}_{\rm PB} = -\frac{\partial H}{\partial q^a}.$$

The Moyal bracket is

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

It obeys

$$\{A,B\}_M = \{A,B\}_{\rm PB} +O(\hbar^2)$$

for smooth functions in the formal small-$\hbar$ expansion.

Dynamics Convention

For a closed system with density operator $\rho$,

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

With

$$W_\rho = \frac{1}{2\pi\hbar}\rho_W,$$

the phase-space evolution equation is

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

For

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

the leading classical part is

$$\frac{\partial W}{\partial t} = - \frac{p}{m}\frac{\partial W}{\partial x} + V'(x)\frac{\partial W}{\partial p} + O(\hbar^2).$$

This equals the classical Liouville equation with the Poisson-bracket sign convention above:

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

For quadratic Hamiltonians, the Moyal bracket equals the Poisson bracket exactly.

Translation Warnings

Item	Convention here	Common alternative
Wigner prefactor	$1/(2\pi\hbar)^n$	$1/h^n$ with $h=2\pi\hbar$
Separation phase	$e^{-i\mathbf p\cdot\mathbf y/\hbar}$	opposite sign if $\langle x
Trace measure	$d^nq\,d^np/(2\pi\hbar)^n$	absorb measure into symbol definitions
Star product	$\exp[(i\hbar/2)\Lambda]$	opposite sign with opposite Fourier convention
Moyal bracket	$(A\star B-B\star A)/(i\hbar)$	definitions that absorb $i\hbar$
Wigner function	$W_\rho=\rho_W/(2\pi\hbar)^n$	calling $\rho_W$ itself the Wigner function

Common Mistakes

• Forgetting the prefactor in the Wigner transform.
• Mixing a Weyl symbol from one convention with a Wigner function from another.
• Treating $W(x,p)$ as a positive joint probability distribution.
• Using ordinary multiplication for symbols of noncommuting operator products.
• Reversing the Poisson-bracket sign when translating Hamilton’s equations.
• Dropping the phase-space trace measure $1/(2\pi\hbar)^n$.
• Assuming flat Cartesian formulas apply unchanged to spin, angle variables, curved spaces, constrained systems, or gauge systems.

Cross-Links

• Weyl Transform
• Wigner Function
• Star Product
• Moyal Bracket
• Phase-Space Dynamics
• Gaussian States and Wigner Functions
• Formula Sheet
• Path Integral Conventions
• Fourier Transform Conventions

References

• H. Weyl, The Theory of Groups and Quantum Mechanics, Dover, 1931.
• E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Physical Review 40, 749-759, 1932.
• J. E. Moyal, “Quantum mechanics as a statistical theory,” 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. P. Schleich, Quantum Optics in Phase Space, Wiley-VCH, 2001.

Exercises

1. Starting from the Wigner definition, show that $\int dx\,dp\,W_\rho(x,p)=\operatorname{Tr}\rho$.

Solution

Use

$$W_\rho(x,p) = \frac{1}{2\pi\hbar} \int dy\, e^{-ipy/\hbar} \left\langle x+\frac{y}{2} \middle| \rho \middle| x-\frac{y}{2} \right\rangle.$$

Integrating over $p$ gives

$$\int\frac{dp}{2\pi\hbar}e^{-ipy/\hbar} = \delta(y).$$

Therefore

$$\int dp\,W_\rho(x,p) = \langle x|\rho|x\rangle.$$

Integrating over $x$ gives

$$\int dx\,\langle x|\rho|x\rangle = \operatorname{Tr}\rho.$$

2. Use the star product to compute $x\star p-p\star x$.

Solution

For $x\star p$, only the first derivative term contributes:

$$x\star p = xp + \frac{i\hbar}{2} \left( \frac{\partial x}{\partial x} \frac{\partial p}{\partial p} \right) = xp+\frac{i\hbar}{2}.$$

Similarly,

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

Subtracting gives

$$x\star p-p\star x=i\hbar.$$

3. Check the Poisson-bracket sign for $H=p^2/(2m)+V(x)$.

Solution

With

$$\{A,B\}_{\rm PB} = A_xB_p-A_pB_x,$$

one finds

$$\{x,H\}_{\rm PB} = 1\cdot\frac{p}{m}-0 = \frac{p}{m},$$

and

$$\{p,H\}_{\rm PB} = 0- 1\cdot V'(x) = -V'(x).$$

These are Hamilton’s equations for the particle.

4. A text defines the Wigner function with $e^{+ipy/\hbar}$ while keeping the same $1/(2\pi\hbar)$ prefactor. What changed?

Solution

The separation Fourier sign changed. This is usually tied to using the conjugate position-momentum convention, or equivalently to replacing $p$ by $-p$ relative to the convention used here. Star-product signs and formulas involving momentum arguments must be translated consistently; one cannot mix the two conventions term by term.