PlanetPhysics/CCR Representation Theory

In connection with the Schr\"odinger representation, one defines 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:


 * (i) each $$\mathcal{H}_{\alpha}$$ reduces all $$Q_j$$ and all $$P_j$$ ;
 * (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 $$\left\{Q^S_j,P^S_j\right\} ^d_{j =1},$$.

A set $$\left\{Q_j,P_j\right\} ^d_{j =1}$$ of self-adjoint operators on a Hilbert space $$\mathcal{H}$$ is called a Weyl representation with $$d$$ degrees of freedom if $$Q_j$$ and $$P_j$$ satisfy the Weyl relations:

e^{itQ_j},$$
 * 1) $$e^{itQ_j} \dot e^{isP_k} = e^{âˆ’ist} \hbar_{jk} e^{isP_k} \dot
 * 1) $$e^{itQ_j} \dot e^{isQ_k} = e^{isQ_k} \dot e^{itQ_j},$$
 * 2) $$ e^{itP_j} \dot e{isP_k} = e^{isP_k} \dot e^{itP_j} ,$$

with $$j, k = 1,..., d, s, t \in \mathbb{R}$$.

The Schr\"odinger representation $$\left\{Q_j,P_j\right\} ^d_{j =1}$$ is a Weyl representation of CCR.

Von Neumann established a uniqueness \htmladdnormallink{theorem {http://planetphysics.us/encyclopedia/Formula.html}: 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} . Since the pioneering work of von Neumann there have been numerous reports published concerning representation theory of CCR (viz. ref. and references cited therein).