Weyl Transform

The Weyl transform maps operators on Hilbert space to functions on phase space. It is the dictionary behind expressions such as

$$A \quad\longleftrightarrow\quad A_W(x,p),$$

where $A_W$ is the Weyl symbol of the operator $A$. The map is designed so that symmetrically ordered operator expressions look like ordinary functions of the classical variables $x$ and $p$.

This is not a claim that quantum mechanics secretly has ordinary classical phase-space probabilities. The Weyl transform preserves noncommutativity by changing the product rule: operator multiplication becomes the star product, not ordinary pointwise multiplication. The Wigner function is the corresponding phase-space representation of a density operator.

Throughout this page, use

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

With the opposite Fourier convention, signs in the exponential kernels change.

Operator-Function Correspondence

For one degree of freedom, the Weyl symbol of an operator $A$ is

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

The phase-space variables $x$ and $p$ are ordinary real coordinates. The variable $y$ measures how far off diagonal the operator kernel is in the position representation.

It is often useful to write

$$A(x_1,x_2) = \langle x_1\vert A\vert x_2\rangle.$$

Then the Weyl transform uses center and separation coordinates:

$$x=\frac{x_1+x_2}{2}, \qquad y=x_1-x_2.$$

The symbol is the Fourier transform of the off-diagonal separation:

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

The inverse transform reconstructs the operator kernel:

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

Thus, up to domain and distributional subtleties, the Weyl transform is an invertible way to represent operators as phase-space functions.

Weyl Ordering

The Weyl transform is tied to Weyl ordering: products of $\hat x$ and $\hat p$ are symmetrized over all orderings. For example, the classical-looking monomial $xp$ corresponds to

$$xp \quad\longleftrightarrow\quad \frac12(\hat x\hat p+\hat p\hat x).$$

More generally, a function $a(x,p)$ can be quantized by Fourier decomposition:

$$a(x,p) = \int\frac{dk\,d\xi}{(2\pi)^2}\, \tilde a(k,\xi)e^{i(kx+\xi p)}.$$

The Weyl-ordered operator is

$$\hat A = \int\frac{dk\,d\xi}{(2\pi)^2}\, \tilde a(k,\xi) e^{i(k\hat x+\xi\hat p)}.$$

The exponential contains $k\hat x+\xi\hat p$ as a single operator, so it treats $\hat x$ and $\hat p$ symmetrically. Expanding it reproduces the fully symmetrized ordering of monomials.

Weyl ordering is one convention among several possible quantization prescriptions. Normal ordering, antinormal ordering, and other phase-space orderings lead to different symbols and different quasiprobability distributions. The Weyl choice is special because it pairs naturally with the Wigner function and has a particularly simple star product.

Symbols of Basic Operators

The basic canonical operators have the expected symbols:

$$(\hat x)_W=x, \qquad (\hat p)_W=p.$$

Powers of only one canonical variable also map directly:

$$(\hat x^n)_W=x^n, \qquad (\hat p^n)_W=p^n.$$

For noncommuting products, ordering matters. A useful pair of checks is

$$(\hat x\hat p)_W = xp+\frac{i\hbar}{2}, \qquad (\hat p\hat x)_W = xp-\frac{i\hbar}{2}.$$

Therefore

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

while

$$([\hat x,\hat p])_W=i\hbar.$$

The symbol remembers noncommutativity through these $\hbar$-dependent ordering terms.

Hamiltonian Examples

For the standard one-dimensional Hamiltonian

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

the Weyl symbol is simply

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

This simplicity holds because the kinetic term depends only on $\hat p$ and the potential term depends only on $\hat x$. For Hamiltonians with mixed products, one must check the ordering. For example,

$$\frac12(\hat x\hat p+\hat p\hat x)$$

has Weyl symbol $xp$, but $\hat x\hat p$ alone has Weyl symbol $xp+i\hbar/2$.

For quadratic Hamiltonians written in symmetric form, the Weyl symbol is the corresponding classical quadratic function. This is one reason harmonic oscillators and general quadratic systems have especially transparent phase-space dynamics.

Trace Formulas

For suitable trace-class operators,

$$\operatorname{Tr}A = \int\frac{dx\,dp}{2\pi\hbar}\, A_W(x,p).$$

For a product,

$$\operatorname{Tr}(AB) = \int\frac{dx\,dp}{2\pi\hbar}\, A_W(x,p)B_W(x,p).$$

This second formula is one of the main reasons Weyl symbols are useful. It says that traces of operator products can be computed as phase-space integrals of symbols, provided the symbols and conventions match.

For a density operator $\rho$, the Wigner function used in this volume is

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

Therefore

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

This looks like a classical expectation value, but $W_\rho$ can be negative and the product rule for noncommuting observables is not ordinary multiplication.

Relation to the Wigner Function

Applying the Weyl transform to a density operator gives

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

The Wigner function is the normalized version:

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

This normalization makes

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

For a normalized state, the integral is one. The Wigner Function page is the canonical home for marginals, negativity, and state examples. The present page is the canonical home for the operator-to-symbol map.

Product Rule Preview

The Weyl transform is linear:

$$(\alpha A+\beta B)_W = \alpha A_W+\beta B_W.$$

But it is not multiplicative under ordinary multiplication:

$$(AB)_W \neq A_WB_W$$

in general. Instead,

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

where $\star$ is the phase-space star product. For one degree of freedom, the common convention is

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

The first terms are

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

Thus ordinary multiplication is recovered only when noncommutative corrections can be neglected. The antisymmetric part of the star product gives the Moyal Bracket, the phase-space version of the commutator.

More Degrees of Freedom

For $n$ canonical pairs $q^a,p_a$, the Weyl symbol is

$$A_W(q,p) = \int d^n y\, e^{-ip_a y^a/\hbar} \left\langle q+\frac{y}{2} \middle| A \middle| q-\frac{y}{2} \right\rangle,$$

with summation over repeated indices. The phase-space trace measure becomes

$$\frac{d^nq\,d^np}{(2\pi\hbar)^n}.$$

The star product uses the symplectic bidifferential operator

$$\Lambda = \sum_{a=1}^{n} \left( \overleftarrow{\partial_{q^a}} \overrightarrow{\partial_{p_a}} - \overleftarrow{\partial_{p_a}} \overrightarrow{\partial_{q^a}} \right).$$

Then

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

This is the natural multidimensional extension of the one-dimensional formulas.

Domains and Distributions

Many important quantum operators are unbounded, and many useful symbols are distributions rather than ordinary functions. Position and momentum eigenstates, projectors onto sharp phase-space-like regions, and polynomial operators all require some care.

For physics calculations, the Weyl transform is often used formally with test functions, wave packets, Gaussian states, or regulated expressions. For rigorous work, one must specify operator domains, trace-class conditions, distribution spaces, and convergence of Fourier transforms. These issues are not cosmetic; they determine when the trace formulas and inversions are legitimate.

Common Mistakes

• Confusing the Weyl symbol of an operator with a measurement probability distribution.
• Forgetting the factor $1/(2\pi\hbar)$ that relates the density-operator symbol to the Wigner function.
• Assuming $(AB)_W=A_WB_W$ for noncommuting operators.
• Ignoring operator ordering when translating mixed products such as $\hat x\hat p$.
• Comparing formulas across books without checking Fourier signs and phase-space measures.
• Treating distributional symbols of unbounded operators as ordinary functions without a regulator.

Cross-Links

• Wigner Function
• Phase-Space Conventions
• Moyal Bracket
• Phase Space
• Fourier Transform Conventions
• Canonical Commutation Relations
• Commutators

References

• H. Weyl, The Theory of Groups and Quantum Mechanics, Dover, 1950.
• E. Wigner, “On the Quantum Correction For Thermodynamic Equilibrium,” Physical Review 40, 749, 1932.
• J. E. Moyal, “Quantum mechanics as a statistical theory,” Proceedings of the Cambridge Philosophical Society 45, 99, 1949.
• M. Hillery, R. F. O’Connell, M. O. Scully, and E. P. Wigner, “Distribution functions in physics: Fundamentals,” Physics Reports 106, 121, 1984.
• C. K. Zachos, D. B. Fairlie, and T. L. Curtright, eds., Quantum Mechanics in Phase Space, World Scientific, 2005.
• M. de Gosson, Symplectic Methods in Harmonic Analysis and in Mathematical Physics, Birkhäuser, 2011.

Exercises

1. Derive the inverse Weyl transform.

Solution

Start from

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

Fourier inversion gives

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

Now set

$$x=\frac{x_1+x_2}{2}, \qquad y=x_1-x_2.$$

This gives

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

2. Use the commutator to check the symbols of $\hat x\hat p$ and $\hat p\hat x$.

Solution

The symmetrized product has Weyl symbol $xp$:

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

The commutator is

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

Let

$$(\hat x\hat p)_W=xp+c.$$

Then

$$(\hat p\hat x)_W=xp-c,$$

because the average must be $xp$. Their difference must be $i\hbar$, so

$$2c=i\hbar, \qquad c=\frac{i\hbar}{2}.$$

Therefore

$$(\hat x\hat p)_W=xp+\frac{i\hbar}{2}, \qquad (\hat p\hat x)_W=xp-\frac{i\hbar}{2}.$$

3. Show that the Wigner function is a normalized Weyl symbol of $\rho$.

Solution

The Weyl symbol of $\rho$ is

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

The Wigner function convention used here is

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

Therefore

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

The $p$ integral gives $2\pi\hbar\,\delta(y)$, leaving

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

4. Explain why the Weyl transform is not an algebra homomorphism under ordinary multiplication.

Solution

If it were an algebra homomorphism, then

$$(\hat x\hat p)_W = (\hat x)_W(\hat p)_W = xp.$$

But the Weyl symbol is actually

$$(\hat x\hat p)_W = xp+\frac{i\hbar}{2}.$$

The missing term encodes noncommutativity. The correct product rule is not ordinary multiplication but the star product:

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

5. For $H=\hat p^2/(2m)+V(\hat x)$, explain why $H_W=p^2/(2m)+V(x)$.

Solution

The term $\hat p^2/(2m)$ contains only powers of $\hat p$, so its Weyl symbol is $p^2/(2m)$. The term $V(\hat x)$ contains only powers or functions of $\hat x$, so its Weyl symbol is $V(x)$. There are no mixed products of $\hat x$ and $\hat p$ whose ordering would generate extra $\hbar$-dependent terms.