Abstract
We prove that a given representation of the canonical commutation relations can be extended uniquely by continuity to larger test function spaces which are maximal in the sense that no further extension is possible. For irreducible tensor product representations of the canonical commutation relations we give a necessary and a sufficient condition for the admissible test functions. We consider the problem of finding topologies on the test function spaces such that this extension can be obtained by a topological completion. Various examples are discussed.
Similar content being viewed by others
References
Araki, H.: J. Math. Phys.1, 492 (1960).
—— Woods, E. J.: J. Math. Phys.4, 637 (1963).
—— —— Publ. RIMS, Kyoto Univ. Ser. A,2, 157 (1966).
-- -- Topologies on Test Function Spaces induced by Representations of the Canonical Commutation Relations.
Bourbaki, N.: Topologie Générale, 3rd Ed. Paris: Hermann 1960.
Chaiken, J. M.: Ann. Phys. (N. Y.)42, 23 (1967).
—— Commun. Math. Phys.8, 164 (1968).
Cook, J. M.: Trans. Am. Math. Soc.74, 222 (1953).
Friedrichs, K. O.: Mathematical aspects of the quantum theory of fields. New York: Interscience 1953.
Gårding, L., and Wightman, A. S.: Proc. Natl. Acad. Sci. U.S.40, 622 (1954).
Gel'fand, I. M., Vilenkin, N. Ya.: Generalized functions, Vol. 4. New York: Academic Press 1964.
Hegerfeldt, G. C., Klauder, J. R.: Metrics on test function spaces for canonical field operators. Commun. Math. Phys.16, 329–346 (1970).
Kelley, J. L.: General topology. New York: van Nostrand 1955.
Klauder, J. R., McKenna, J., Woods, E. J.: J. Math. Phys.7, 822 (1966).
Lew, J. S.: unpublished thesis, Princeton, 1960.
von Neumann, J.: Math. Ann.104, 570 (1931).
—— Compos. Math.6, 1 (1938).
Reed, M.: On the self-adjointness of quantum fields and hamiltonians (Stanford Ph. D. thesis, 1968).
—— A Gårding domain for quantum fields. Commun. Math. Phys.14, 336–346 (1969).
Segal, I. E.: Trans. Am. Math. Soc.88, 12 (1958).
—— Mathematical problems of relativistic physics. Providence, R. I.; Am. Math. Soc. 1963.
Streit, L.: Commun. Math. Phys.4, 22 (1967).
Author information
Authors and Affiliations
Additional information
Supported in part by the National Research Council of Canada.
An earlier version of the present work was distributed as a preprint entitled “Topologies for Test Function Spaces for Representations of the Canonical Commutation Relations”.
Rights and permissions
About this article
Cite this article
Woods, E.J. Continuity properties of the representations of the canonical commutation relations. Commun.Math. Phys. 17, 1–20 (1970). https://doi.org/10.1007/BF01649580
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01649580