PlanetPhysics/CCR Representation Theory

In connection with the Schr\"odinger representation, one defines a Schr\"odinger d-system as a set $\left\{Q_{j},P_{j}\right\}_{j=1}^{d}$ 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 $\left\{Q_{j},P_{j}\right\}_{j=1}^{d}$ is, in each ${\mathcal {H}}_{\alpha }$, unitarily equivalent to the Schr\"odinger representation $\left\{Q_{j}^{S},P_{j}^{S}\right\}_{j=1}^{d},$ .

A set $\left\{Q_{j},P_{j}\right\}_{j=1}^{d}$ 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:

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. $e^{itQ_{j}}{\dot {e}}^{isQ_{k}}=e^{isQ_{k}}{\dot {e}}^{itQ_{j}},$
3. $e^{itP_{j}}{\dot {e}}{isP_{k}}=e^{isP_{k}}{\dot {e}}^{itP_{j}},$

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

The Schr\"odinger representation $\left\{Q_{j},P_{j}\right\}_{j=1}^{d}$ 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).