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} $$C_{0} (T^* \bR^n) \lra \mathcal K(L^2(\bR^n))$$ as expressed by the \emph{Moyal `deformation'}~:

[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~, $$ where and the operators  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, , 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  is defined as the normal \htmladdnormallink{groupoid {http://planetphysics.us/encyclopedia/LocallyCompactGroupoid.html} of a pair Lie groupoid} 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 ~. Then in terms of disjoint unions one has:

$$ \mathcal T X & = G' \bigvee G''\\ diag(\mathcal TX) & = diag(G') \bigvee diag(G'')~. $$

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

Quantization relative to is outlined by V\'arilly (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 = \bR^n as a suitable example. Thus, one can take a function a(x,\xi) on T^*\bR^n whose inverse Fourier transform

yields a function on T \bR^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 and ~. Then, the following family of operator kernels \bigbreak $$ k_a(x,y, \hslash) := \hslash^{-n} \F^{-1}a(u,v) = \frac{1}{(2 \pi \hslash)^n} \int_{\bR^n} a(\frac{x+y}{2}, \xi) \exp[\frac{\iota}{\hslash}(x -y) \xi]~ a (u, \xi) ~d \xi~,$$

This mechanism can be generalized to quantize any function on T^*X when X is a Riemannian manifold, and produces an asymptotic morphism ~. Furthermore, there is the corresponding K--theory map, 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 \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 of C*--algebras over [0,1] whose fiber at 0 is  ,and whose restriction to (0,1] is the constant field with fiber, for t > 0~. This may be called a strong 'deformation' from \mathfrak A to \mathfrak B~.

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

Given the continuity of norm for any continuous section , consider the following conditions~:

\item[(1)] For any, the map is norm continuous. \item[(2)] For any and, we have $$  &\lim_{t \to \infty} (\vp_t(a) + \lambda \vp_t(b) - \vp_t(a + \lambda b)) = 0 \\ &\lim_{t \to \infty} (\vp_t(ab) - \vp_t(a) \vp_t(b)) = 0 \\ &\lim_{t \to \infty} (\vp_t(a^*) - \vp_t(a)^*) = 0~.

\bigbreak

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