CCR representation theory | Physics Library

Definition 0.1. In connection with the Schrödinger representation, one defines a Schrödinger d-system as a set {Q ,P }
j j _j=1^d of self-adjoint operators on a Hilbert space ℋ (such as the position and momentum operators, for example) when there exist mutually orthogonal closed subspaces ℋ_α of ℋ such that ℋ = ⊕_αℋ_α with the following two properties:

(i) each ℋ_α reduces all Q_j and all P_j ;
(ii) the set _j=1^d is, in each ℋ_α, unitarily equivalent to the Schrödinger representation _j=1^d, [8].

Definition 0.2. A set {Qj,Pj } _j=1^d of self-adjoint operators on a Hilbert space ℋ is called a Weyl representation with d degrees of freedom if Q_j and P_j satisfy the Weyl relations:

$eitQj ˙eisPk = eˆaˆ′istℏjkeisPke˙itQj,$
$itQj isQk isQk itQj e e˙ = e e˙ ,$
$eitPj ˙eisPk = eisPk ˙eitPj,$

with j,k = 1,...,d,s,t ∈ ℝ.

The Schrödinger representation {Qj, Pj} _j=1^d is a Weyl representation of CCR.

Von Neumann established a uniqueness theorem: if the Hilbert space 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).

References

[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.

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