# PlanetPhysics/CCR Representation Theory

In connection with the Schr\"odinger representation, one defines a Schr\"odinger d-system as a set ${\displaystyle \left\{Q_{j},P_{j}\right\}_{j=1}^{d}}$ of self-adjoint operators on a Hilbert space ${\displaystyle {\mathcal {H}}}$ (such as the position and momentum operators, for example) when there exist mutually orthogonal closed subspaces ${\displaystyle {\mathcal {H}}_{\alpha }}$ of ${\displaystyle {\mathcal {H}}}$ such that ${\displaystyle {\mathcal {H}}=\oplus _{\alpha }{\mathcal {H}}_{\alpha }}$ with the following two properties:

• (i) each ${\displaystyle {\mathcal {H}}_{\alpha }}$ reduces all ${\displaystyle Q_{j}}$ and all ${\displaystyle P_{j}}$ ;
• (ii) the set ${\displaystyle \left\{Q_{j},P_{j}\right\}_{j=1}^{d}}$ is, in each ${\displaystyle {\mathcal {H}}_{\alpha }}$, unitarily equivalent to the Schr\"odinger representation ${\displaystyle \left\{Q_{j}^{S},P_{j}^{S}\right\}_{j=1}^{d},}$ [1].

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

1. $\displaystyle e^{itQ_j} \dot e^{isP_k} = e^{âˆ’ist} \hbar_{jk} e^{isP_k} \dot e^{itQ_j},$
2. ${\displaystyle e^{itQ_{j}}{\dot {e}}^{isQ_{k}}=e^{isQ_{k}}{\dot {e}}^{itQ_{j}},}$
3. ${\displaystyle e^{itP_{j}}{\dot {e}}{isP_{k}}=e^{isP_{k}}{\dot {e}}^{itP_{j}},}$

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

The Schr\"odinger representation ${\displaystyle \left\{Q_{j},P_{j}\right\}_{j=1}^{d}}$ is a Weyl representation of CCR.

Von Neumann established a uniqueness \htmladdnormallink{theorem {http://planetphysics.us/encyclopedia/Formula.html}: if the Hilbert space ${\displaystyle {\mathcal {H}}}$ is separable, then every Weyl representation of CCR with ${\displaystyle d}$ degrees of freedom is a Schr\"odinger ${\displaystyle d}$-system} ([2]). Since the pioneering work of von Neumann [2] there have been numerous reports published concerning representation theory of CCR (viz. ref. [1] and references cited therein).

## References

1. Putnam C. R., Commutation Properties of Hilbert Space Operators, Springer, Berlin, 1967.
2. von Neumann J., Die Eindeutigkeit der Schr\"odingerschen Operatoren, Math. Ann. , 1931, v.104, 570--578.
3. Arai A., Characterization of anticommutativity of self-adjoint operators in connection with Clifford algebra and applications, Integr. Equat. Oper. Th. , 1993, v.17, 451--463.
4. Arai A., Commutation properties of anticommuting self-adjoint operators, spin representation and Dirac operators, Integr. Equat. Oper. Th. , 1993, v.16, 38--63.
5. Arai A., Analysis on anticommuting self--adjoint operators, Adv. Stud. Pure Math. , 1994, v.23, 1--15.
6. Arai A., Scaling limit of anticommuting self-adjoint operators and applications to Dirac operators, Integr. Equat. Oper. Th., 1995, v.21, 139--173.
7. Arai A., Some remarks on scattering theory in supersymmetric quantum mechanics, J. Math. Phys. , 1987, V.28, 472--476.
8. Pedersen S., Anticommuting self--adjoint operators, J. Funct. Anal., 1990, V.89, 428--443.
9. Reed M. and Simon B., Methods of Modern Mathematical Physics ., vol.I, Academic Press, New York, 1972.