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 [[../QuantizationMethods/|canonical quantization]] methods employed in [[../QuantumOperatorAlgebra5/|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 $T_{ℏ} (a)$ are of [[../Trace/|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 , $𝒯 X$ , 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 $𝒯 X$ is defined as the normal \htmladdnormallink{groupoid {http://planetphysics.us/encyclopedia/LocallyCompactGroupoid.html} of a pair [[../LieAlgebroids/|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 $d i a g (X)$ in $X$ . More specifically, if $X$ is a (smooth) [[../NoncommutativeGeometry4/|manifold]], then let </math>G'= X \times X \times (0,1] $a n d$ G= TX $, f r o m w h i c h i t c a n b e s e e n$ diag(G') = X \times (0,1] $a n d < m a t h > d i a g (G^{″}) = X$ ~. Then in terms of disjoint [[../ModuleAlgebraic/|unions]] one has:

</math> \mathcal T X & = G' \bigvee GTemplate:'Template:'\\ diag(\mathcal TX) & = diag(G') \bigvee diag(GTemplate:'Template:')~.

 $I n t h i s w a y < m a t h > 𝒯 X$  shapes up both as a smooth groupoid Failed to parse (unknown function "\G"): {\displaystyle \mathsf{\G}}
,

as well as a manifold $X_{M b}$ with [[../GenericityInOpenSystems/|boundary]].

Quantization relative to $𝒯 X$ is outlined by V\'arilly (1997) to which the reader is referred for further details. The procedure entails characterizing a [[../Bijective/|function]] on $𝒯 X$ 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 [[../FourierTransforms/|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 $a (x, ξ)$ 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 (unknown function "\bR"): {\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 ~, $w h i c h o n s o l v i n g l e a d s t o < m a t h > u = \frac{1}{2} (x + y)$ and $v = \frac{1}{ℏ} (x - y)$ ~. 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~, $T h i s m e c h a n i s m c a n b e g e n e r a l i z e d t o q u a n t i z e a n y f u n c t i o n o n < m a t h > T^{*} X$ 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 [[../QuarkAntiquarkPair/|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' [[../NormInducedByInnerProduct/|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) $𝔄$ and $𝔅$ , together with a continuous [[../CosmologicalConstant/|field]] $(𝔄 (t), Γ)$ of C*--algebras over $[0, 1]$ whose fiber at $0$ is $𝔄 (0) = 𝔄$ ,and whose restriction to $(0, 1]$ is the constant field with fiber $𝔄 (t) = 𝔅$ , for $t > 0$ ~. This may be called a strong 'deformation' from $𝔄$ to $𝔅$ ~.

For any $a \in 𝔄 = 𝔄 (0)$ , it can be shown that there exists a continuous [[../IsomorphicObjectsUnderAnIsomorphism/|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 $f o r e a c h$ a \in \mathfrak A $, w e s e t$ \vp_t(a) = \a_a (\frac{1}{t}) \in \mathfrak B $, f o r a l l < m a t h > t \in [1, \infty)$ ~.

Given the continuity of [[../NormInducedByInnerProduct/|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~:

  \item[(1)] For any  $a \in 𝔄$ , the map Failed to parse (unknown function "\ra"): {\displaystyle t \ra \vp_t(a)}
 is norm continuous.   \item[(2)] For any  $a, b \in 𝔄$  and Failed to parse (unknown function "\bC"): {\displaystyle \lambda \in \bC}
, we have </math>  &\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~.  Failed to parse (unknown function "\bigbreak"): {\displaystyle   \bigbreak  Then an asymptotic morphism from <math>\mathfrak A}
 to  $𝔅$  is given by a family of maps Failed to parse (unknown function "\vp"): {\displaystyle \{ \vp_t \}, t \in [1, \infty)}
, from  $𝔄$  to  $𝔅$  satisfying conditions (1) and (2) above.

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PlanetPhysics/Wigner Weyl Moyal Quantization Procedures

Quantization Procedures

Asymptotic Morphisms

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