WignerWeylMoyalQuantizationProcedures 2008-12-14 23:11:40 2009-02-16 05:05:45 Topic quantization Moyal deformation Heisenberg deformation tangent groupoid general quantization procedure asymptotic morphisms Wigner--Weyl--Moyal quantization procedure Heisenberg quantization generalized quantization procedures asymptotic morphisms Wigner--Weyl--Moyal quantization procedures % this is the default PlanetPhysics preamble. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % define commands here \usepackage{amsmath, amssymb, amsfonts, amsthm, amscd, latexsym} \usepackage{xypic} \usepackage[mathscr]{eucal} \theoremstyle{plain} \newtheorem{lemma}{Lemma}[section] \newtheorem{proposition}{Proposition}[section] \newtheorem{theorem}{Theorem}[section] \newtheorem{corollary}{Corollary}[section] \theoremstyle{definition} \newtheorem{definition}{Definition}[section] \newtheorem{example}{Example}[section] %\theoremstyle{remark} \newtheorem{remark}{Remark}[section] \newtheorem*{notation}{Notation} \newtheorem*{claim}{Claim} \renewcommand{\thefootnote}{\ensuremath{\fnsymbol{footnote%%@ }}} \numberwithin{equation}{section} \newcommand{\Ad}{{\rm Ad}} \newcommand{\Aut}{{\rm Aut}} \newcommand{\Cl}{{\rm Cl}} \newcommand{\Co}{{\rm Co}} \newcommand{\DES}{{\rm DES}} \newcommand{\Diff}{{\rm Diff}} \newcommand{\Dom}{{\rm Dom}} \newcommand{\Hol}{{\rm Hol}} \newcommand{\Mon}{{\rm Mon}} \newcommand{\Hom}{{\rm Hom}} \newcommand{\Ker}{{\rm Ker}} \newcommand{\Ind}{{\rm Ind}} \newcommand{\IM}{{\rm Im}} \newcommand{\Is}{{\rm Is}} \newcommand{\ID}{{\rm id}} \newcommand{\GL}{{\rm GL}} \newcommand{\Iso}{{\rm Iso}} \newcommand{\Sem}{{\rm Sem}} \newcommand{\St}{{\rm St}} \newcommand{\Sym}{{\rm Sym}} \newcommand{\SU}{{\rm SU}} \newcommand{\Tor}{{\rm Tor}} \newcommand{\U}{{\rm U}} \newcommand{\A}{\mathcal A} \newcommand{\Ce}{\mathcal C} \newcommand{\D}{\mathcal D} \newcommand{\E}{\mathcal E} \newcommand{\F}{\mathcal F} \newcommand{\G}{\mathcal G} \newcommand{\Q}{\mathcal Q} \newcommand{\R}{\mathcal R} \newcommand{\cS}{\mathcal S} \newcommand{\cU}{\mathcal U} \newcommand{\W}{\mathcal W} \newcommand{\bA}{\mathbb{A}} \newcommand{\bB}{\mathbb{B}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bD}{\mathbb{D}} \newcommand{\bE}{\mathbb{E}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bG}{\mathbb{G}} \newcommand{\bK}{\mathbb{K}} \newcommand{\bM}{\mathbb{M}} \newcommand{\bN}{\mathbb{N}} \newcommand{\bO}{\mathbb{O}} \newcommand{\bP}{\mathbb{P}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bV}{\mathbb{V}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bfE}{\mathbf{E}} \newcommand{\bfX}{\mathbf{X}} \newcommand{\bfY}{\mathbf{Y}} \newcommand{\bfZ}{\mathbf{Z}} \renewcommand{\O}{\Omega} \renewcommand{\o}{\omega} \newcommand{\vp}{\varphi} \newcommand{\vep}{\varepsilon} \newcommand{\diag}{{\rm diag}} \newcommand{\grp}{{\mathbb G}} \newcommand{\dgrp}{{\mathbb D}} \newcommand{\desp}{{\mathbb D^{\rm{es}}}} \newcommand{\Geod}{{\rm Geod}} \newcommand{\geod}{{\rm geod}} \newcommand{\hgr}{{\mathbb H}} \newcommand{\mgr}{{\mathbb M}} \newcommand{\ob}{{\rm Ob}} \newcommand{\obg}{{\rm Ob(\mathbb G)}} \newcommand{\obgp}{{\rm Ob(\mathbb G')}} \newcommand{\obh}{{\rm Ob(\mathbb H)}} \newcommand{\Osmooth}{{\Omega^{\infty}(X,*)}} \newcommand{\ghomotop}{{\rho_2^{\square}}} \newcommand{\gcalp}{{\mathbb G(\mathcal P)}} \newcommand{\rf}{{R_{\mathcal F}}} \newcommand{\glob}{{\rm glob}} \newcommand{\loc}{{\rm loc}} \newcommand{\TOP}{{\rm TOP}} \newcommand{\wti}{\widetilde} \newcommand{\what}{\widehat} \renewcommand{\a}{\alpha} \newcommand{\be}{\beta} \newcommand{\ga}{\gamma} \newcommand{\Ga}{\Gamma} \newcommand{\de}{\delta} \newcommand{\del}{\partial} \newcommand{\ka}{\kappa} \newcommand{\si}{\sigma} \newcommand{\ta}{\tau} \newcommand{\lra}{{\longrightarrow}} \newcommand{\ra}{{\rightarrow}} \newcommand{\rat}{{\rightarrowtail}} \newcommand{\oset}[1]{\overset {#1}{\ra}} \newcommand{\osetl}[1]{\overset {#1}{\lra}} \newcommand{\hr}{{\hookrightarrow}} \subsection{Quantization Procedures} \emph{ 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 \emph{operator kernels} in representing \emph{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 $a \in C_{0} (T^* \bR^n)$ and the operators $T_{\hslash}(a)$ are of trace class. In Connes (1994), it is called the \emph{`Heisenberg deformation'}. An elegant way of generalizing this construction entails the introduction of the \emph{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 \emph{tangent groupoid} $\mathcal T X$ is defined as \emph{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{aligned} \mathcal T X & = G' \bigvee G''\\ diag(\mathcal TX) & = diag(G') \bigvee diag(G'')~. \end{aligned} $ In this way $\mathcal T X$ shapes up both as a smooth groupoid $\mathsf{\G}$, 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 $\mathcal TX$ 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 $ \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 $T \bR^n$~. 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 $u = \frac{1}{2}(x + y)$ and $v = \frac{1}{\hslash}(x - y)$~. 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 $C^{\infty}_c(T^*X) \lra \mathcal K(L^2(X))$~. Furthermore, there is the corresponding K--theory map $K^0(T^*X) \lra \bZ$, 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 $L \lra X$~. Then one can define a \emph{`twisted Dirac operator'}, $D_L$, and a `virtual' Hilbert space given by \subsection{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 \emph{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 $\a \in \Gamma$ 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 $t \in [1, \infty)$~. Given the continuity of norm $\Vert \a(t) \Vert$ for any continuous section $\a \in \Gamma$, consider the following conditions~: \begin{itemize} \item[(1)] For any $a \in \mathfrak A$, the map $t \ra \vp_t(a)$ is norm continuous. \item[(2)] For any $a, b \in \mathfrak A$ and $\lambda \in \bC$, we have $ \begin{aligned} &\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~. \end{aligned} $ \end{itemize} \bigbreak Then an \emph{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.