Weyl Ordering Rule

The basic Weyl ordering property generating all the Weyl ordering identities for polynomial functions is:

$((sq+tp)^n)_W = (sQ+tP)^n$

$(q, p)$ are the commuting phase space variables, $(Q, P)$ are the corresponding noncommuting operators (satisfying $[Q,P] = i\hbar $).

For example for n = 2, the identity coming from the coefficient for the $st$ term is the known basic Weyl ordering identity:

$(qp)_W = \frac{1}{2}(QP+PQ)$

By choosing the classical Hamiltonian as $h(p,q) = (sq+tp)^n$ and carefully performing the Fourier and inverse Fourier transforms, we obtain the Weyl identity:

$\int {dx\over2\pi}{dk\over2\pi} e^{ixP + ikQ} \int dpdqe^{-ixp-ikq} (sq+tp)^n =(sQ+tP)^n $

The Fourier integral can be solved after the change of variables:

$l = sq+tp, m = tq-sp$

and using the identity

$ \int dl e^{-iul} l^n =2 \pi \frac{\partial^n}{\partial v^n} \delta_D(v)|_{v=u}$

Where $ \delta_D$ is the Dirac delta function.

Let the position and momentum operators in $n$ phase-space dimensions be collectively denoted $\hat{Z}^I$, and let the corresponding symbols be denoted $z^{I}$, where $I\in\{1,\ldots,n\}$. The operator $\hat{f}(\hat{Z})$ corresponding to the Weyl-symbol $f(z)$ is

$$ \hat{f}(\hat{Z})~\stackrel{\begin{matrix}\text{symmetri-}\\ \text{zation}\end{matrix}}{=}~ \left.\sum_{m=0}^{\infty}\frac{1}{m!}\left[\hat{Z}^1\frac{\partial}{\partial z^1}+\ldots +\hat{Z}^n\frac{\partial}{\partial z^n} \right]^m f(z)\right|_{z=0} \qquad $$ $$~\stackrel{\begin{matrix}\text{Taylor}\\ \text{expan.}\end{matrix}}{=}~ \left.\exp\left[\sum_{I=1}^n\hat{Z}^I\frac{\partial}{\partial z^I}\right] f(z)\right|_{z=0} \qquad $$ $$~=~\int_{\mathbb{R}^{n}} \! d^{n}z~\delta^{n}(z)~ \exp\left[\sum_{I=1}^n\hat{Z}^I\frac{\partial}{\partial z^I}\right] f(z) $$ $$ ~\stackrel{\delta\text{-fct}}{=}~\int_{\mathbb{R}^{2n}} \! \frac{d^{n}z~d^{n}k}{(2\pi)^{n}} \exp\left[-i\sum_{J=1}^n k_Jz^J\right] \exp\left[\sum_{I=1}^n \hat{Z}^I\frac{\partial}{\partial z^I}\right] f(z)$$ $$~\stackrel{\text{int. by parts}}{=}~\int_{\mathbb{R}^{2n}} \! \frac{d^{n}z~d^{n}k}{(2\pi)^{n}} f(z)~ \exp\left[-\sum_{I=1}^n\hat{Z}^I\frac{\partial}{\partial z^I}\right] \exp\left[-i\sum_{J=1}^n k_Jz^J\right] $$ $$~=~\int_{\mathbb{R}^{2n}} \! \frac{d^{n}z~d^{n}k}{(2\pi)^{n}} f(z)~ \exp\left[i\sum_{I=1}^n k_I\hat{Z}^I\right] \exp\left[-i\sum_{J=1}^n k_Jz^J\right] $$ $$~\stackrel{\text{BCH}}{=}~\int_{\mathbb{R}^{2n}} \! \frac{d^{n}z~d^{n}k}{(2\pi)^{n}} f(z)~ \exp\left[i\sum_{I=1}^n k_I(\hat{Z}^I-z^I)\right].$$

The above manipulations make sense for a sufficiently well-behaved function $z\mapsto f(z)$.

Example: If the Weyl-symbol is of the form $f(z)=g\left(\sum_{I=1}^n k_I z^I\right)$ for some analytic function $g:\mathbb{C}\to \mathbb{C}$, then the corresponding operator is $\hat{f}(\hat{Z})=g\left(\sum_{I=1}^n k_I\hat{Z}^I\right)$.