CanonicalCommutationAndAntiCommutationRepresentations 2009-02-21 15:27:02 2009-02-21 16:08:45 Definition CCR CAR Schroedinger representation representation of the canonical commutation relations Schwartz space of rapidly decreasing $C_{\infty}$ functions non-Abelian gauge theory representation of canonical commutation relations in a non-Abelian gauge theory Aharonov-Bohm effect % this is the default PlanetPhysics preamble. as your % 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}} This is a contributed topic on representations of canonical commutation and anti-commutation relations. \subsection{Representations of Canonical Commutation Relations (CCR)} \subsubsection{Canonical Commutation Relations:} Consider a Hilbert space $\mathcal{H}$. For a linear operator {\bf O} on $\mathcal{H}$, we denote its domain by {\bf $D(O)$}. With \PMlinkexternal{Arai's notation}{http://planetphysics.org/?op=getobj&from=lec&id=124}, a set $\left\{Q_j,P_j\right\} ^d_{j =1}$ of self-adjoint operators on $\mathcal{H}$ (such as the position and monetum operators, for example) is called a \emph{representation of the canonical commutation relations (CCR) with $d$ degrees of freedom} if there exists a dense subspace $\mathcal{D}$ of $\mathcal{H}$ such that: \begin{itemize} \item (i) $\mathcal{D} \subset \bigcap^d_{j,k=1}[D(Q_jP_k) \bigcap D(P_kQ_j)\bigcap D(Q_jQ_k) \bigcap D(P_jP_k)],$ and \item (ii) $Q_j$ and $P_j$ satisfy the CCR relations: $$[Q+j,P_k] = i\hbar \delta_{jk},$$ $$[Q_j,Q_k] = 0, \, [P_j,P_k] = 0, \, j, k = 1,...,d,$$ on $\mathcal{D}$, where $\hbar$ is the Planck constant $h$ divided by $2 \pi$. \end{itemize} A standard representation of the CCR is the well-known Schr\"odinger representation $\left\{Q^S_j,P_j^S \right\}^d_j=1 $ which is given by: $$\mathcal{H} = L^2(\mathbb{R}^d), \, Q^S_j= x_j, $$ the multiplication operator by the j-th coordinate $x_j$ , with $P^S_j = (-1) i \hbar D_j$ , with $D_j$ being the generalized partial differential operator in $x_j$ , and with $J\mathcal{D} = \mathcal{S}(\mathbb{R}^d)$ being the Schwartz space of rapidly decreasing $C_{\infty}$ functions on $\mathbb{R}^d$, or $\mathcal{D} = C_0^{\infty}(\mathbb{R}^d)$, that is the space of $C^{\infty}$ functions on $\mathbb{R}^d$ with compact support. \subsubsection{CCR Representations in a Non-Abelian Gauge Theory} One can provide a representation of canonical commutation relations in a \PMlinkexternal{non-Abelian gauge theory}{http://planetphysics.org/?op=getobj&from=lec&id=124} defined on a non-simply connected region in the two-dimensional Euclidean space. Such representations were shown to provide also a mathematical expression for the non-Abelian, Aharonov-Bohm effect. \subsection{Canonical Anti --Commutation Relations (CAR)}