CCR representation theory
Definition 0.1
In connection with the Schrödinger
representation, one defines
a Schrödinger d-system as a set

of self-adjoint
operators
on a Hilbert space

(such as the
position
and
momentum
operators, for example) when there exist mutually orthogonal closed subspaces

of

such
that

with the following two properties:
Definition 0.2
A set

of self-adjoint operators on a Hilbert space

is called a
Weyl representation with
degrees of freedom if

and

satisfy the Weyl relations:
with
.
The Schrödinger representation
is a Weyl representation of CCR.
Von Neumann established a uniqueness theorem: if the Hilbert space
is separable, then every Weyl representation of CCR with
degrees of freedom is a Schrödinger
-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).
-
- 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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