Schwartz Weyl quantization and Wigner distributions

Let $d\ge 1$, let $0<ϵ$ be a dimensionless parameter, and use the Schwartz space $\mathcal{S}({\mathbb{R}}^d)$ of Schwartz space and its seminorms. For $a\in \mathcal{S}({\mathbb{R}}^{2d})$, define the Weyl quantization on $u\in \mathcal{S}({\mathbb{R}}^d)$ by

$({\mathrm{Op}}_{ϵ}^W(a)u)(x):=(2\pi ϵ{)}^{-d}{\int }_{{\mathbb{R}}^d}{\int }_{{\mathbb{R}}^d}{e}^{i(x-y)\cdot \xi \mathrm{/}ϵ}a\text{\hspace{0.17em}⁣}(\frac{x+y}{2},\xi )u(y)d\xi dy\mathrm{.}$

The integral is absolutely convergent for each fixed $x$, since its integrand is Schwartz in $(y,\xi )$. The Wigner integral below is absolutely convergent for each $(x,\xi )$ for the same reason.

For $u\in \mathcal{S}({\mathbb{R}}^d)$ satisfying ${\int }_{{\mathbb{R}}^d}\mathrm{\mid }u(x){\mathrm{\mid }}^{2}dx=1,$ define its Wigner distribution by

$W_{ϵ}[u](x,\xi ):=(2\pi ϵ{)}^{-d}{\int }_{{\mathbb{R}}^d}{e}^{-iy\cdot \xi \mathrm{/}ϵ}u(x+y\mathrm{/}2)\overline{u(x-y\mathrm{/}2)}dy\mathrm{.}$

Use the inner product $⟨f,g⟩=\int f(x)\overline{g(x)}dx$, linear in its first argument (Complex Lp classes and Euclidean test-function conventions). Define the scaled Fourier transform on $\mathcal{S}({\mathbb{R}}^d)$ by

$({\mathcal{F}}_{ϵ}u)(\xi ):=(2\pi ϵ{)}^{-d\mathrm{/}2}{\int }_{{\mathbb{R}}^d}{e}^{-ix\cdot \xi \mathrm{/}ϵ}u(x)dx\mathrm{.}$

For the coordinate and momentum observables define $Q_{j}u=x_{j}u$ and $P_{j}u=-iϵ{\mathrm{\partial }}_{x_j}u$ on $\mathcal{S}({\mathbb{R}}^d)$. Applying the real product rule (Sums, scalar multiples, products and quotients: $(f+g{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(c)+g^{\mathrm{\prime }}(c)$, $(\alpha f{)}^{\mathrm{\prime }}(c)=\alpha f^{\mathrm{\prime }}(c)$, $(fg{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(c)g(c)+f(c)g^{\mathrm{\prime }}(c)$, and $(f\mathrm{/}g{)}^{\mathrm{\prime }}(c)=(f^{\mathrm{\prime }}(c)g(c)-f(c)g^{\mathrm{\prime }}(c))\mathrm{/}g(c{)}^2$ when $g(c)\mathrm{\ne }0$) to the real and imaginary components, with the componentwise derivative convention of Complex Lp classes and Euclidean test-function conventions, gives $Q_j{P}_{k}u=-iϵx_j{\mathrm{\partial }}_{x_k}u,P_k{Q}_{j}u=-iϵ({\delta }_{jk}u+x_j{\mathrm{\partial }}_{x_k}u),[Q_j,P_k]u:=Q_j{P}_{k}u-P_k{Q}_{j}u=iϵ{\delta }_{jk}u\mathrm{.}$ This fixes the Poisson and Moyal signs used on this page.

Schwartz space and its seminorms

Complex Lp classes and Euclidean test-function conventions

For a physical interpretation, after choosing a length $L_0$ and momentum $P_0$, put $S_0=L_0{P}_0$ and $ϵ=\mathrm{\hslash }\mathrm{/}{S}_0$. The formulas here use dimensionless variables $x=x_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}\mathrm{/}{L}_0$ and $\xi =p_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}\mathrm{/}{P}_0$; dimensional variables must be restored before interpreting an application. No physical measurement or postulate is part of this mathematical definition.