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Wigner--Weyl--Moyal quantization procedures

(Topic)

Quantization Procedures

Wigner–Weyl–Moyal quantization procedures and asymptotic morphisms are described as general quantization procedures, beyond first, second or canonical quantization methods employed in quantum theories.

The more general quantization techniques beyond canonical quantization revolve around using operator kernels in representing asymptotic morphisms. A fundamental example is an asymptotic morphism $C_{0} (T^* \mathbb{R}^n) {\longrightarrow}\mathcal K(L^2(\mathbb{R}^n))$ as expressed by the Moyal `deformation' :

$[T_{\hslash} (a) f](x) := \frac{1}{(2 \pi \hslash)^n} \int_{\mathbb{R}^n} a (\frac{x+y}{2}, \xi) \exp[\frac{\iota}{\hslash}] f(y)~dy~d \xi~,$ where $a \in C_{0} (T^* \mathbb{R}^n)$ and the operators $T_{\hslash}(a)$ are of trace class. In Connes (1994), it is called the `Heisenberg deformation'.

An elegant way of generalizing this construction entails the introduction of the tangent groupoid, $\mathcal T X$ , of a suitable space $X$ and using asymptotic morphisms. Putting aside a number of technical details which can be found in either Connes (1994) or Landsman (1998), the tangent groupoid $\mathcal T X$ is defined as the normal groupoid of a pair Lie groupoid $\xymatrix{X \times X \ar@<1ex>[r] \ar[r]& X }$ which is obtained by `blowing up' the diagonal $diag(X)$ in $X$ . More specifically, if $X$ is a (smooth) manifold, then let $G'= X \times X \times (0,1]$ and $G''= TX$ , from which it can be seen $diag(G') = X \times (0,1]$ and $diag(G'') = X$ . Then in terms of disjoint unions one has:

$\begin{equation*}\begin{aligned}\mathcal T X & = G' \bigvee G''\\ diag(\mathcal TX) & = diag(G') \bigvee diag(G'')~. \end{aligned} \end{equation*}$

In this way $\mathcal T X$ shapes up both as a smooth groupoid $\mathsf{\mathcal G}$ , as well as a manifold $X_{Mb}$ with boundary.

Quantization relative to $\mathcal T X$ is outlined by Várilly (1997) to which the reader is referred for further details. The procedure entails characterizing a function on $\mathcal TX$ in terms of a pair of functions on $G'$ and $G''$ respectively, the first of which will be a kernel and the second will be the inverse Fourier transform of a function defined on $T^*X$ . It will be instructive to consider the case $X = \mathbb{R}^n$ as a suitable example. Thus, one can take a function $a(x,\xi)$ on $T^*\mathbb{R}^n$ whose inverse Fourier transform

$\mathcal F^{-1}(a(u,v)) = \frac{1}{(2 \pi)^n} \int_{\mathbb{R}^n} \exp[\iota \xi v] a (u, \xi) ~d \xi~,$ yields a function on $T \mathbb{R}^n$ . Consider next the terms

$x := \exp_u[\frac{1}{2} \hslash v] = u + \frac{1}{2} \hslash v~,~ y := \exp_u[-\frac{1}{2} \hslash v] = u - \frac{1}{2} \hslash v ~,$

which on solving leads to $u = \frac{1}{2}(x + y)$ and $v = \frac{1}{\hslash}(x - y)$ . Then, the following family of operator kernels $k_a(x,y, \hslash) := \hslash^{-n} \mathcal F^{-1}a(u,v) = \frac{1}{(2 \pi \hsla... ...(\frac{x+y}{2}, \xi) \exp[\frac{\iota}{\hslash}(x -y) \xi]~ a (u, \xi) ~d \xi~,$

This mechanism can be generalized to quantize any function on $T^*X$ when $X$ is a Riemannian manifold, and produces an asymptotic morphism $C^{\infty}_c(T^*X) {\longrightarrow}\mathcal K(L^2(X))$ . Furthermore, there is the corresponding K–theory map $K^0(T^*X) {\longrightarrow}\mathbb{Z}$ , which is the analytic index map of Atiyah–Singer (see Berline et al., 1991, Connes, 1994). As an example, suppose $X$ is an even dimensional spin manifold together with a `prequantum' line bundle $L {\longrightarrow}X$ . Then one can define a `twisted Dirac operator', $D_L$ , and a `virtual' Hilbert space given by

Asymptotic Morphisms

This subsection defines the important notion of an asymptotic morphism following Connes (1994). Suppose we have two C*–algebras (see below) $\mathfrak{A}$ and $\mathfrak{B}$ , together with a continuous field $(\mathfrak{A}(t), \Gamma)$ of C*–algebras over $[0,1]$ whose fiber at 0 is $\mathfrak{A}(0)= \mathfrak{A}$ ,and whose restriction to $(0,1]$ is the constant field with fiber $\mathfrak{A}(t) = \mathfrak{B}$ , for $t > 0$ . This may be called a strong 'deformation' from $\mathfrak{A}$ to $\mathfrak{B}$ .

For any $a \in \mathfrak{A} = \mathfrak{A}(0)$ , it can be shown that there exists a continuous section $\alpha \in \Gamma$ of the above field satisfying $\alpha (0) = a$ . Choosing such an $\alpha = \alpha _a$ for each $a \in \mathfrak{A}$ , we set $\varphi _t(a) = \alpha _a (\frac{1}{t}) \in \mathfrak{B}$ , for all $t \in [1, \infty)$ .

Given the continuity of norm $\Vert \alpha (t) \Vert$ for any continuous section $\alpha \in \Gamma$ , consider the following conditions :

(1)	For any $a \in \mathfrak{A}$ , the map $t {\rightarrow}\varphi _t(a)$ is norm continuous.
(2)	For any $a, b \in \mathfrak{A}$ and $\lambda \in \mathbb{C}$ , we have $\begin{aligned}&\lim_{t \to \infty} (\varphi _t(a) + \lambda \varphi _t(b) - \v... ...\ &\lim_{t \to \infty} (\varphi _t(a^) - \varphi _t(a)^) = 0~. \end{aligned}$

Then an asymptotic morphism from $\mathfrak{A}$ to $\mathfrak{B}$ is given by a family of maps $\{ \varphi _t \}, t \in [1, \infty)$ , from $\mathfrak{A}$ to $\mathfrak{B}$ satisfying conditions (1) and (2) above.

"Wigner--Weyl--Moyal quantization procedures" is owned by bci1.