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%%% Primary Title: canonical commutation and anti-commutation relations: their representations
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\begin{document}
This is a contributed topic on \htmladdnormallink{representations}{http://planetphysics.us/encyclopedia/CategoricalGroupRepresentation.html} of canonical commutation and \htmladdnormallink{anti-commutation relations}{http://planetphysics.us/encyclopedia/AntiCommutationRelations.html}.
\subsection{Representations of Canonical Commutation Relations (CCR)}
\subsubsection{Canonical Commutation Relations:}
Consider a \htmladdnormallink{Hilbert space}{http://planetphysics.us/encyclopedia/NormInducedByInnerProduct.html} $\mathcal{H}$. For a \htmladdnormallink{linear operator}{http://planetphysics.us/encyclopedia/Commutator.html} {\bf O} on $\mathcal{H}$, we denote its \htmladdnormallink{domain}{http://planetphysics.us/encyclopedia/Bijective.html} by {\bf $D(O).$} With
\htmladdnormallink{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 \htmladdnormallink{operators}{http://planetphysics.us/encyclopedia/QuantumOperatorAlgebra4.html} on $\mathcal{H}$ (such as the \htmladdnormallink{position}{http://planetphysics.us/encyclopedia/Position.html} and \htmladdnormallink{momentum}{http://planetphysics.us/encyclopedia/Momentum.html} 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 \htmladdnormallink{relations}{http://planetphysics.us/encyclopedia/Bijective.html}:
$$[Q_j,P_k] = i\hbar \delta_{jk},$$
$$[Q_j,Q_k] = 0, \, [P_j,P_k] = 0, \, j, k = 1,...,d,$$
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 \htmladdnormallink{operator}{http://planetphysics.us/encyclopedia/QuantumSpinNetworkFunctor2.html} 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 $C_{\infty}$ \htmladdnormallink{functions}{http://planetphysics.us/encyclopedia/Bijective.html} 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
\htmladdnormallink{non-Abelian gauge theory}{http://planetphysics.org/?op=getobj&from=lec&id=124} defined on a non-simply connected region in the \htmladdnormallink{two-dimensional}{http://planetphysics.us/encyclopedia/CoriolisEffect.html} Euclidean space. Such representations were shown to provide also a mathematical expression for the \htmladdnormallink{non-Abelian}{http://planetphysics.us/encyclopedia/AbelianCategory3.html}, Aharonov-Bohm effect
(\cite{GMS81}).
\subsection{Canonical Anticommutation Relations (CAR)}
\begin{thebibliography}{9}
\bibitem{AA93a}
Arai A., Characterization of anticommutativity of self-adjoint operators in connection with Clifford algebra and applications,
{\em Integr. Equat. Oper. Th.}, 1993, v.17, 451--463.
\bibitem{AA93b}
Arai A., Commutation properties of anticommuting self-adjoint operators, spin representation and Dirac operators, {\em Integr. Equat. Oper. Th.}, 1993, v.16, 38--63.
\bibitem{AA94}
Arai A., Analysis on anticommuting self--adjoint operators, {\em Adv. Stud. Pure Math.}, 1994, v.23, 1--15.
\bibitem{AA95}
Arai A., Scaling limit of anticommuting self-adjoint operators and applications to Dirac operators, Integr. Equat. Oper. Th., 1995, v.21, 139--173.
\bibitem{AA87}
Arai A., Some remarks on scattering theory in supersymmetric quantum mechanics, {\em J. Math. Phys.}, 1987, V.28, 472--476.
\bibitem{GMS81}
Goldin G.A., Menikoff R. and Sharp D.H., Representations of a local current algebra in nonsimply connected space and the Aharonov--Bohm effect, J. Math. Phys., 1981, v.22, 1664--1668.
\bibitem{JVN31}
von Neumann J., Die Eindeutigkeit der Schr\"odingerschen Operatoren,
{\em 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}
\end{document}
