PlanetPhysics/CCR Representation Theory

In connection with the Schr\"odinger [[../CategoricalGroupRepresentation/|representation]], one defines a Schr\"odinger d-system as a set ${\{Q_j,P_j\}}_{j=1}^d$ of self-adjoint [[../QuantumOperatorAlgebra4/|operators]] on a [[../NormInducedByInnerProduct/|Hilbert space]] $\mathscr{H}$ (such as the [[../Position/|position]] and [[../Momentum/|momentum]] operators, for example) when there exist mutually orthogonal closed subspaces ${\mathscr{H}}_{\alpha }$ of $\mathscr{H}$ such that $\mathscr{H}={\oplus }_{\alpha }{\mathscr{H}}_{\alpha }$ with the following two properties:

• (i) each ${\mathscr{H}}_{\alpha }$ reduces all $Q_j$ and all $P_j$ ;
• (ii) the set ${\{Q_j,P_j\}}_{j=1}^d$ is, in each ${\mathscr{H}}_{\alpha }$, unitarily equivalent to the Schr\"odinger representation ${\{Q_j^S,P_j^S\}}_{j=1}^d,$ .

A set ${\{Q_j,P_j\}}_{j=1}^d$ of self-adjoint operators on a Hilbert space $\mathscr{H}$ is called a Weyl representation with $d$ degrees of freedom if $Q_j$ and $P_j$ satisfy the Weyl [[../Bijective/|relations]]:

1. Failed to parse (syntax error): {\displaystyle e^{itQ_j} \dot e^{isP_k} = e^{−ist} \hbar_{jk} e^{isP_k} \dot e^{itQ_j},}
2. $e^{it{Q}_j}{\dot{e}}^{is{Q}_k}=e^{is{Q}_k}{\dot{e}}^{it{Q}_j},$
3. $e^{it{P}_j}\dot{e}is{P}_k=e^{is{P}_k}{\dot{e}}^{it{P}_j},$

with $j,k=1,...,d,s,t\in ℝ$.

The Schr\"odinger representation ${\{Q_j,P_j\}}_{j=1}^d$ is a Weyl representation of [[../SchwartzSpaceOfRapidlyDecreasingC_inftyFunctions/|CCR]].

Von Neumann established a uniqueness \htmladdnormallink{theorem {http://planetphysics.us/encyclopedia/Formula.html}: if the Hilbert space $\mathscr{H}$ is separable, then every Weyl representation of CCR with $d$ degrees of freedom is a Schr\"odinger $d$-system} (). Since the pioneering [[../Work/|work]] of von Neumann there have been numerous reports published concerning representation theory of CCR (viz. ref. and references cited therein).