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[parent] CCR representation theory (Topic)
Definition 0.1   In connection with the Schrödinger representation, one defines a Schrödinger 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 Schrödinger representation $\left\{Q^S_j,P^S_j\right\} ^d_{j =1},$ [8].
Definition 0.2   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:

  1. $\displaystyle e^{itQ_j} \dot e^{isP_k} = e^{\^aˆ’ist} \hbar_{jk} e^{isP_k} \dot e^{itQ_j},$

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

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

Von Neumann established a uniqueness theorem: if the Hilbert space $\mathcal{H}$ is separable, then every Weyl representation of CCR with $d$ degrees of freedom is a Schrödinger $d$-system ([6]). Since the pioneering work of von Neumann [6] there have been numerous reports published concerning representation theory of CCR (viz. ref. [8] and references cited therein).

Bibliography

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



"CCR representation theory" is owned by bci1.
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Other names:  representation theory of canonical commutation relations
Also defines:  Schroedinger d-system
Keywords:  representation theory of canonical commutation relations, CCR

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Cross-references: work, theorem, CCR, relations, momentum, position, Hilbert space, operators, representation

This is version 11 of CCR representation theory, born on 2009-02-21, modified 2009-02-21.
Object id is 548, canonical name is CCRRepresentationTheory.
Accessed 819 times total.

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Physics Classification: 00. (GENERAL)
 02. (Mathematical methods in physics)
 03. (Quantum mechanics, field theories, and special relativity )
 03.65.Fd (Algebraic methods )
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