Wigner–Weyl transform

The Weyl transform (or Weyl quantization) of the function f is given by the following operator in Hilbert space,[5]

Throughout, ħ is the reduced Planck constant.

It is instructive to perform the p and q integrals in the above formula first, which has the effect of computing the ordinary Fourier transform ${\displaystyle {\tilde {f}}}$ of the function f, while leaving the operator ${\displaystyle e^{i(aQ+bP)}}$. In that case, the Weyl transform can be written as[6]

${\displaystyle \Phi [f]={\frac {1}{(2\pi )^{2}}}\iint {\tilde {f}}(a,b)e^{iaQ+ibP}\,da\,db}$.

We may therefore think of the Weyl map as follows: We take the ordinary Fourier transform of the function ${\displaystyle f(p,q)}$, but then when applying the Fourier inversion formula, we substitute the quantum operators ${\displaystyle P}$ and ${\displaystyle Q}$ for the original classical variables p and q, thus obtaining a "quantum version of f."

A less symmetric form, but handy for applications, is the following,

${\displaystyle \Phi [f]={\frac {2}{(2\pi \hbar )^{3/2}}}\iint \!\!\!\iint \!\!dq\,dp\,d{\tilde {x}}\,d{\tilde {p}}\ e^{{\frac {i}{\hbar }}({\tilde {x}}{\tilde {p}}-2({\tilde {p}}-p)({\tilde {x}}-q))}~f(q,p)~|{\tilde {x}}\rangle \langle {\tilde {p}}|.}$

The Weyl map may then also be expressed in terms of the integral kernel matrix elements of this operator,[7]

${\displaystyle \langle x|\Phi [f]|y\rangle =\int _{-\infty }^{\infty }{{\text{d}}p \over h}~e^{ip(x-y)/\hbar }~f\left({x+y \over 2},p\right).}$

The inverse of the above Weyl map is the Wigner map, which takes the operator Φ back to the original phase-space kernel function f,

For example, the Wigner map of the oscillator thermal distribution operator ${\displaystyle \exp(-\beta (P^{2}+Q^{2})/2)}$ is[5]

${\displaystyle \exp _{\star }\left(-\beta (p^{2}+q^{2})/2\right)=\left(\cosh({\frac {\beta \hbar }{2}})\right)^{-1}\exp \left({\frac {-2}{\hbar }}\tanh({\frac {\beta \hbar }{2}})(p^{2}+q^{2})/2\right).}$

If one replaces ${\displaystyle \Phi [f]}$ in the above expression with an arbitrary operator, the resulting function f may depend on Planck's constant ħ, and may well describe quantum-mechanical processes, provided it is properly composed through the star product, below.[8] In turn, the Weyl map of the Wigner map is summarized by Groenewold's formula,[5]

${\displaystyle \Phi [f]=h\iint \,da\,db~e^{iaQ+ibP}\operatorname {Tr} (e^{-iaQ-ibP}\Phi ).}$

While the above formulas give a nice understanding of the Weyl quantization of a very general observable on phase space, they are not very convenient for computing on simple observables, such as those that are polynomials in ${\displaystyle q}$ and ${\displaystyle p}$. In later sections, we will see that on such polynomials, the Weyl quantization represents the totally symmetric ordering of the noncommuting operators ${\displaystyle Q}$ and ${\displaystyle P}$. For example, the Wigner map of the quantum angular-momentum-squared operator L² is not just the classical angular momentum squared, but it further contains an offset term −3ħ²/2, which accounts for the nonvanishing angular momentum of the ground-state Bohr orbit.

The action of the Weyl quantization on polynomial functions of ${\displaystyle q}$ and ${\displaystyle p}$ is completely determined by the following symmetric formula:[9]

${\displaystyle (aq+bp)^{n}\longmapsto (aQ+bP)^{n}}$

for all complex numbers ${\displaystyle a}$ and ${\displaystyle b}$. From this formula, it is not hard to show that the Weyl quantization on a function of the form ${\displaystyle q^{k}p^{l}}$ gives the average of all possible orderings of ${\displaystyle k}$ factors of ${\displaystyle Q}$ and ${\displaystyle l}$ factors of ${\displaystyle P}$. For example, we have

${\displaystyle 6p^{2}q^{2}~~\longmapsto ~~P^{2}Q^{2}+Q^{2}P^{2}+PQPQ+PQ^{2}P+QPQP+QP^{2}Q.}$

While this result is conceptually natural, it is not convenient for computations when ${\displaystyle k}$ and ${\displaystyle l}$ are large. In such cases, we can use instead McCoy's formula[10]

${\displaystyle p^{m}q^{n}~~\longmapsto ~~{1 \over 2^{n}}\sum _{r=0}^{n}{n \choose r}Q^{r}P^{m}Q^{n-r}={1 \over 2^{m}}\sum _{s=0}^{m}{m \choose s}P^{s}Q^{n}P^{m-s}.}$

This expression gives an apparently different answer for the case of ${\displaystyle p^{2}q^{2}}$ from the totally symmetric expression above. There is no contradiction, however, since the canonical commutation relations allow for more than one expression for the same operator. (The reader may find it instructive to use the commutation relations to rewrite the totally symmetric formula for the case of ${\displaystyle p^{2}q^{2}}$ in terms of the operators ${\displaystyle P^{2}Q^{2}}$, ${\displaystyle QP^{2}Q}$, and ${\displaystyle Q^{2}P^{2}}$ and verify the first expression in McCoy's formula with ${\displaystyle m=n=2}$.)

It is widely thought that the Weyl quantization, among all quantization schemes, comes as close as possible to mapping the Poisson bracket on the classical side to the commutator on the quantum side. (An exact correspondence is impossible, in light of Groenewold's theorem.) For example, Moyal showed the

Theorem: If ${\displaystyle f(q,p)}$ is a polynomial of degree at most 2 and ${\displaystyle g(q,p)}$ is an arbitrary polynomial, then we have ${\displaystyle \Phi (\{f,g\})={\frac {1}{i\hbar }}[\Phi (f),\Phi (g)]}$.

• If f is a real-valued function, then its Weyl-map image Φ[f] is self-adjoint.
• If f is an element of Schwartz space, then Φ[f] is trace-class.
• More generally, Φ[f] is a densely defined unbounded operator.
• The map Φ[f] is one-to-one on the Schwartz space (as a subspace of the square-integrable functions).