Weyl and Wigner Representations of Quantum States/Operators

Preface

In classical mechanics, a physical quantity is a function on phase space, while the state of a system is a point in phase space (or, for an ensemble, a probability distribution on phase space). Upon quantizing this language, one obtains the phase-space representation of quantum states/operators, namely the Wigner representation. The Weyl representation is its Fourier transform.

Most authors define the Wigner representation as:

\[F_W(x,p)=\int \mathrm{d}y \langle x+\frac{y}{2}\mid F \mid x - \frac{y}{2} \rangle e^{\mathrm{i} p y}\]

and then derive its various properties. However, from a physicist’s perspective, the physical meaning of this expression is unclear, and \((x,p)\) do not have equal status, which is somewhat uncomfortable to look at.

This article introduces the Wigner and Weyl representations from a more natural perspective. This article is only an introduction and does not involve rigorous proofs or derivations.

Unitary Basis in the Discrete Case

Given a \(N\)-dimensional Hilbert space \(\mathcal{H}\) and one of its orthonormal bases \(|v_l\rangle\,\,( l=1,...,N)\). Define

\[U = \sum_{l=1}^N |v_{l+1}\rangle\langle v_{l}|\]

where \(|v_{N+1}\rangle := |v_1\rangle\).

Let the eigenvectors of \(U\) be denoted by \(|u_k\rangle\); then one can define

\(V = \sum_{k=1}^N |u_k\rangle\langle u_{k+1} |\),

and \(\langle u_k| v_l\rangle=e^{\frac{2\pi}{N}kl}\); proof omitted.

Any operator on \(\mathcal{H}\) can be written as a polynomial in \(U\) and \(V\) (proof omitted):

\[F = \sum_{kl} f_{kl}U^k V^l\]\[f_{kl} = \frac{1}{N} \text{tr}[U^{-k}FV^{-l}]\]

Example: On \(\mathbb{C}^2\), let \(U = \sigma_x\), \(V=\sigma_y\), then \(UV = \mathrm{i}\sigma_z\). Then
\(F = \frac{1}{2}\sum_{k=0}^3 \text{tr}[F\sigma_k] \sigma_k\)
where \(\sigma_0 :=\mathbb{I}\). Note that \(\sigma_k^{-1} = \sigma_k^\dagger = \sigma_k\).

Thus, given a finite-dimensional Hilbert space, we can always find a pair of unitary operators \(U,V\) and the \(U^k V^l\) generated by them as generators, which serve as a basis for operators, with a total of \(N^2\) elements.

Unitary Basis in the Continuous Case (Weyl Basis)

The above steps can be generalized to an infinite-dimensional Hilbert space. In this case, we use \(X,P\) as generators to generate \(e^{\mathrm{i}pX}e^{\mathrm{i}xP}\) as a basis for operators.

\[F=\int\frac{\mathrm{d}x\mathrm{d}p}{2\pi} f(x,p) e^{\mathrm{i}pX} e^{\mathrm{i}xP}\]\[f(x,p)=\text{tr}[e^{-\mathrm{i}pX}Fe^{-\mathrm{i}xP}]\]

However, \(X,P\) do not have equal status here. This is because in the expression \(e^{\mathrm{i}pX}e^{\mathrm{i}xP}\), all \(X\) appear before \(P\). If we instead place \(P\) before \(X\), we can likewise obtain another \(\tilde{f}(x,p)\).

A way to give \(X,P\) equal status is to define \(U(x,p)=e^{\mathrm{i}(xP-pX)}\). This is in fact the commonly known displacement operator (Displacement operator). We also call it the Weyl operator. We will next use it as a basis for operators and call it the Weyl basis. It can be proven that:

\[F=\int\frac{\mathrm{d}x\mathrm{d}p}{2\pi} f_W(x,p) U(x,p)\]\[f_W(x,p) = \text{tr}[FU^\dagger(x,p)]\]

where \(f_W(x,p)\) is the Weyl representation of the operator \(F\); it is the expansion of the operator in the Weyl basis.

Hermitian Basis in the Continuous Case (Wigner Basis)

The Weyl basis introduced above is unitary. Is there a Hermitian basis? The answer is yes: it is called the Wigner basis.

Let us first define the inversion operator:

\[\frac{1}{2}W:=\int \mathrm{d}x \mid \!\!-x\rangle\langle x| = \int \mathrm{d}p \mid\!\! -p\rangle\langle p|\]

The reason we introduce the factor \(\frac{1}{2}\) is that \(\text{tr}[W]=1\).

The action of \(\frac{1}{2}W\) is to rotate the \((x,p)\) plane by 180 degrees about the origin. Clearly, it is a Hermitian operator.

It can be proven that it can be written as

\(\frac{1}{2}W=e^{-\mathrm{i}2X;P} = \sum_{k=0}^\infty \frac{1}{k}(-2\mathrm{i})^k X^kP^k = e^{\mathrm{i}2P;X} \).

Here \(X;P\) means that, in the polynomial, all \(X\) are arranged before \(P\).

We further define the operator for a 180-degree rotation about the point \((x,p)\):

\(\frac{1}{2}W(x,p):=e^{-\mathrm{i}2(X-x);(P-p)}=e^{\mathrm{i}2(P-p);(X-x)}\).

It can be proven that \(W(x,p)\) can also serve as a basis for operators, called the Wigner basis:

\[F=\int \frac{\mathrm{d}x\mathrm{d}p}{2\pi} F_W(x,p)W(x,p)\]\[F_W(x,p) = \text{tr}[FW(x,p)]\]

where \(F_W(x,p)\) is the Wigner representation of the operator \(F\).

Relationship Between the Weyl and Wigner Representations

The Weyl basis is unitary, while the Wigner basis is Hermitian.

It can be proven that the Wigner representation is the Fourier transform of the Weyl representation:

\[F_W(x,p)=\int \frac{\mathrm{d}x'\mathrm{d}p'}{2\pi} e^{-\mathrm{i}(xp'-px')}f_W(x',p')\]

Epilogue

It is worth mentioning that none of the above steps has any physical motivation; they have only mathematical motivation, namely, to find a corresponding representation for operators on a Hilbert space in a convenient basis.