# PlanetPhysics/Wigner Weyl Moyal Quantization Procedures
### 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 \htmladdnormallink{operator {http://planetphysics.us/encyclopedia/QuantumOperatorAlgebra4.html} kernels} in representing asymptotic morphisms . A fundamental example is an \emph{asymptotic morphism} Failed to parse (unknown function "\bR"): {\displaystyle C_{0} (T^* \bR^n) \lra \mathcal K(L^2(\bR^n))} as expressed by the \emph{Moyal `deformation'}~:
</math> [T_{\hslash} (a) f](x) := \frac{1}{(2 \pi \hslash)^n} \int_{\bR^n} a (\frac{x+y}{2}, \xi) \exp[\frac{\iota}{\hslash}] f(y)~dy~d \xi~, Failed to parse (unknown function "\bR"): {\displaystyle where <math>a \in C_{0} (T^* \bR^n)} and the operators $T_{\hslash }(a)$ are of trace class. In Connes (1994), it is called the `Heisenberg \htmladdnormallink{deformation {http://planetphysics.us/encyclopedia/CohomologicalProperties.html}'}.
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 \htmladdnormallink{groupoid {http://planetphysics.us/encyclopedia/LocallyCompactGroupoid.html} of a pair Lie groupoid} Failed to parse (unknown function "\xymatrix"): {\displaystyle \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 </math>G'= X \times X \times (0,1]$and$G= TX$,fromwhichitcanbeseen$diag(G') = X \times (0,1]$and<math>diag(G'')=X$~. Then in terms of disjoint unions one has:
</math> \mathcal T X & = G' \bigvee G''\\ diag(\mathcal TX) & = diag(G') \bigvee diag(G'')~.
as well as a manifold $X_{Mb}$ with boundary.
Quantization relative to ${\mathcal {T}}X$ is outlined by V\'arilly (1997) to which the reader is referred for further details. The procedure entails characterizing a function on ${\mathcal {T}}X$ 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 Failed to parse (unknown function "\bR"): {\displaystyle X = \bR^n} as a suitable example. Thus, one can take a function $a(x,\xi )$ on Failed to parse (unknown function "\bR"): {\displaystyle T^*\bR^n} whose inverse Fourier transform
Failed to parse (unknown function "\F"): {\displaystyle \F^{-1}(a(u,v)) = \frac{1}{(2 \pi)^n} \int_{\bR^n} \exp[\iota \xi v] a (u, \xi) ~d \xi~,}
yields a function on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikiversity.org/v1/":): {\displaystyle T \bR^n} ~. Consider next the terms
</math> 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 ~, $whichonsolvingleadsto<math>u={\frac {1}{2}}(x+y)$ and $v={\frac {1}{\hslash }}(x-y)$~. Then, the following family of operator kernels \bigbreak </math> k_a(x,y, \hslash) := \hslash^{-n} \F^{-1}a(u,v) = \frac{1}{(2 \pi \hslash)^n} \int_{\bR^n} a(\frac{x+y}{2}, \xi) \exp[\frac{\iota}{\hslash}(x -y) \xi]~ a (u, \xi) ~d \xi~,$Thismechanismcanbegeneralizedtoquantizeanyfunctionon<math>T^{*}X$ when $X$ is a Riemannian manifold, and produces an asymptotic morphism Failed to parse (unknown function "\lra"): {\displaystyle C^{\infty}_c(T^*X) \lra \mathcal K(L^2(X))} ~. Furthermore, there is the corresponding K--theory map Failed to parse (unknown function "\lra"): {\displaystyle K^0(T^*X) \lra \bZ} , 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 Failed to parse (unknown function "\lra"): {\displaystyle L \lra 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 \emph {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 Failed to parse (unknown function "\a"): {\displaystyle \a \in \Gamma} of the above field satisfying Failed to parse (unknown function "\a"): {\displaystyle \a(0) = a} ~. Choosing such an </math>\a = \a_a$foreach$a \in \mathfrak A$,weset$\vp_t(a) = \a_a (\frac{1}{t}) \in \mathfrak B$,forall<math>t\in [1,\infty )$~.
Given the continuity of norm Failed to parse (unknown function "\a"): {\displaystyle \Vert \a(t) \Vert} for any continuous section Failed to parse (unknown function "\a"): {\displaystyle \a \in \Gamma} , consider the following conditions~: