CCRRepresentationTheory 2009-02-21 17:28:25 2009-02-21 17:44:29 Topic canonical commutation and anti-commutation relations: their representations Schroedinger d-system representation theory of canonical commutation relations CCR % this is the default PlanetPhysics preamble. % 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}} {\bf [Entry in progress]} \begin{definition} In connection with the Schr\"odinger representation, one defines \emph{a Schr\"odinger d-system} as a set $\left\{Q_j,P_j\right\} ^d_{j =1}$ of self-adjoint operators on a Hilbert space $\mathcal{H}$ (such as the position and momentum operators, for example) when there exist mutually orthogonal closed subspaces $\mathcal{H}_{\alpha}$ of $\mathcal{H}$ such that $\mathcal{H} = \oplus_{\alpha} \mathcal{H}_{\alpha}$ with the following two properties: \begin{itemize} \item (i) each $\mathcal{H}_{\alpha}$ reduces all $Q_j$ and all $P_j$ ; \item (ii) the set $\left\{Q_j,P_j\right\} ^d_{j =1}$ is, in each $\mathcal{H}_{\alpha}$, unitarily equivalent to the Schr\"odinger representation $\left\{Q^S_j,P^S_j\right\} ^d_{j =1},$ \cite{PCR67}. \end{itemize} \end{definition} \begin{definition} A set $\left\{Q_j,P_j\right\} ^d_{j =1}$ of self-adjoint operators on a Hilbert space $\mathcal{H}$ is called a \emph{Weyl representation with $d$ degrees of freedom} if $Q_j$ and $P_j$ satisfy the Weyl relations: \begin{enumerate} \item $$e^{itQ_j} \dot e^{isP_k} = e^{−ist} \hbar_{jk} e^{isP_k} \dot e^{itQ_j},$$ \item $$e^{itQ_j} \dot e^{isQ_k} = e^{isQ_k} \dot e^{itQ_j},$$ \item $$ e^{itP_j} \dot e{isP_k} = e^{isP_k} \dot e^{itP_j} ,$$ \end{enumerate} with $j, k = 1,..., d, s, t \in \mathbb{R}$. \end{definition} The Schr\"odinger representation $\left\{Q_j,P_j\right\} ^d_{j =1}$ is a Weyl representation of CCR. Von Neumann established a {\em uniqueness theorem: if the Hilbert space $\mathcal{H}$ is separable, then every Weyl representation of CCR with $d$ degrees of freedom is a Schr\"odinger $d$-system} (\cite{JVN31}). Since the pioneering work of von Neumann \cite{JVN31} there have been numerous reports published concerning representation theory of CCR (viz. ref. \cite{PCR67} and references cited therein). \begin{thebibliography}{9} \bibitem{JVN31} von Neumann J., Die Eindeutigkeit der Schr\"odingerschen Operatoren, Math. Ann., 1931, V.104, 570–578. \bibitem{PS90} Pedersen S., Anticommuting self-adjoint operators, J. Funct. Anal., 1990, V.89, 428–443. \bibitem{PCR67} Putnam C. R., Commutation Properties of Hilbert Space Operators, Springer, Berlin, 1967. \bibitem{RM-SB72} Reed M. and Simon B., {\em Methods of Modern Mathematical Physics}., vol.I, Academic Press, New York, 1972. \end{thebibliography}