We won’t dive deep into the analysis problem and will rather present the basic idea behind Weyl quantization, then we will mention some further developments such as Moyal product and deformation quantization.

## Idea

On a flat phase space $\mathbb{R}^{2n}$ with the standard symplectic form $\omega = \sum dq^i \wedge dp_i$, to any reasonable function $a(q,p)\in C^\infty(\mathbb{R}^{2n})$, the *Weyl calculus* assigns an operator
$$
A = \hat{a} = \text{Op}(a(q,p)) = q(q,-ih\partial_q),
$$
that is defined on functions $u(q) \in S(\mathbb{R}^n)$ by an oscillatory integral
$$
\begin{equation}
Au (q) = (2\pi)^{-n} \int_{\mathbb{R}^{2n}} \exp(\frac{i}{h}p(q-q’)) a(\frac{q+q’}{2},p) u(q’) dq’ dp.
\label{weyl}
\end{equation}
$$
The function $a(q,p)$ is called the *Weyl symbol* of the operator $A$. $h \in (0,1]$ is a parameter.

### Physical motivation

See Stein’s book for more.

On a flact phase space every classical observable should be a function $\sigma(q,p)$. The quantum observable corresponding to $\sigma(Q,P)$ where $Q$ is the position operator and $P=-i\partial_q$. However we have a ordering problem arising from $[Q,P]\neq 0$, so we need to be more careful on the definition of $\sigma(Q,P)$.

When $\sigma(q,p) = \sigma(q)$, *i.e.* no dependence on $p$, then we should have the multiplication operator $f\mapsto \sigma(\cdot)f(\cdot)$. Similarly for $\sigma(q,p)=\sigma(p)$ we should have the multiplier operator $\hat{f}\mapsto \sigma(\cdot)\hat{f}(\cdot)$.

We can define $\sigma(Q,P) = A$ where $A$ is the pseudodifferential operator with symbol $\sigma$. This assignment has the effect that $A = A_1 A_2$ if $\sigma(q,p) = \sigma_1(q)\sigma_1(p)$, i.e. this amounts to putting the $P$ operations to the right of the $Q$ operations.

We can formally treat the $P$ and $Q$ operations on a more symmetric footing and with a Fourier transformation put $$ Op(\sigma) = \iint \exp(i x\cdot Q + y\cdot P) \hat{\sigma}(x,y) dx dy, $$ this is done a lot in physics. Some manipulations on $\hat{\sigma}$ gives $$ Op(\sigma) u (q)= \iint \sigma(\frac{q+q’}{2}) \exp(i(q-q’)\cdot \xi) u(q’) dq’ d\xi. $$

### Mathematical motivation

So far this is motivated by quantum mechanics, but it can also be motivated in a purely mathematical point of view, if we have already motivated the definition of pseudodifferential operators. See 10.11/cpa.3160320304 for more. In fact they are quite similar: both of them arise from the ordering of operators.

Recall from the entry on Fourier integral operators that a pseudodifferential operator on $\mathbb{R}^n$ can be defined through
$$
Au (x) = \iint e^{i(x-x’)\cdot\xi}\sigma_A(x,x’,\xi) u(x’) dx’ d\xi
$$
now suppose that $\sigma$ doesn’t depend on $x’$ and integrate $x’$ out
$$
\begin{equation}
Au (x) = (2\pi)^{-n}\int e^{ix\cdot\xi}\sigma_A(x,\xi) \hat{u}(\xi) d \xi.
\label{def_pdo}
\end{equation}
$$
where $\hat{\cdot}$ denotes the Fourier transform. If $\sigma(x,\xi)$ is a polynomial in $\xi$, then $A$ can be seen as a linear differential operator $\sigma(x,D)$ with $D=-i\partial_x$ put to the *right* of the coefficients. However one could equally well have decided to put all differentiation to the *left* of the coefficients.

The two definitions agree if $\sigma$ is a real linear function of $(x,\xi)\in \mathbb{R}^{2n}$. Any two such functions can be transformed into each other by a linear symplectic transformation $\chi$ with the symplectic form given by $\omega = \sum dx^i \wedge d\xi_i$.

Segal proved that corresponding to $\chi$ there is a unitary transformation $U$ in $L^2(\mathbb{R}^n)$ such that
$$
\begin{equation}
U^{-1} \sigma(x,D) U = (\sigma\circ\chi) (x,D)
\label{cov}
\end{equation}
$$
for all *linear* $\sigma$ and $U$ is unique up to a constant factor.

*covariant representation*, with $\chi$ seen as a group action on an element $\alpha_g$ of a group $G$, of a dynamical system $(Q,G,\alpha)$ in $B$.

- The group $G$ should be seen as a subgroup of $\text{Sp}(2n,\mathbb{R})$, the symplectomorphism group of $\mathbb{R}^{2n}$.
- $B$ should be the algebra of pseudodifferential operators (is this a $C^\ast$-algebra?)
- $Q$ should be (some completion of) the algebra of the coordinates $(x,\xi)$ hence a subset of $C^\infty(\mathbb{R}^{2n})$.

There is a unique way to modify the definition given by $\eqref{def_pdo}$ so that $\eqref{cov}$ remains valid for general $\sigma$, and $\sigma(x,D)$ is a multilication by $\sigma(x,\xi)$ *(i.e. a multiplier)* if $\sigma(x,\xi)=\sigma(x)$ id independent of $\xi$. This can be found, by considering when $\sigma$ is bounded exponential by Fourier decomposition of a general $\sigma\in L(\mathbb{R}^{2n})$, to be
$$
Op(\sigma) u (x) = (2\pi)^{-n}\iint \sigma(\frac{(x+y)\cdot\xi}{2})\exp(i(x-y)\cdot\xi)u(y) dyd\xi,
$$
which agrees with $\eqref{weyl}$.

## Moyal bracket and the star product

There is a inverse formula (or the formula for recovering the symbol) for Weyl quantization $\sigma\mapsto Op(\sigma)$ given by a kind of trace formula. Moyal, when studying the interpretation of the symbol $u$ of agiven a quantum operator $Op(u)$, found what is now called the *Moyal bracket*:
$$
M(u,v) = \frac{2}{i\hbar} \sinh(\frac{i\hbar}{2}P)(u,v) = P(u,v) + \sum_{r=1}^{\infty}(\frac{i\hbar}{2})^{2r} P^{2r+1}(u,v)
$$
where
$$
P^r(u,v) = \Lambda^{i_1 j_1} … \Lambda^{i_r j_r} (\partial_{i_1\ldots i_r} u) (\partial_{j_1\ldots j_r}v)
$$
the $r$-th power of the Poisson bracket $P$. $i_k,j_k = 1,\ldots,2n$ with $k=1,\ldots,r$, $\Lambda = \begin{pmatrix}0 & -I \\ I & 0 \end{pmatrix}$.

We might assume $u,v\in C^\infty(\mathbb{R}^{2n})$ and the sum can be taken as a formal series.

A similar formula for the symbol of a product $Op(u)Op(v)$ is called *star product*:
$$
u \star_M v = \exp(\frac{i\hbar}{2}P)(u,v) = uv + \sum_{r=1}^\infty (\frac{i\hbar}{2})^r P^r(u,v).
$$