Abstract
After briefly putting algebraic quantum theory into the context of a probabilistic interpretation with emphasis on local measurements, certain general features of the theory are examined. Sectors are defined and shown to be the components of the pure state space in the norm topology. Transition probabilities are defined by a simple algebraic formula and it is shown how superpositions of pure states may be defined. With the aid of these results, symmetries are characterized and the connexion with Wigner's Theorem is established.
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References
Neumann, J. von: Mathematische Grundlagen der Quantenmechanik. Berlin: Springer 1932.
Mackey, G. W.: Mathematical foundations of quantum mechanics. New York: Benjamin 1963.
Segal, I.: Postulates for general quantum mechanics. Ann. Math.48, 930–948 (1947).
Araki, H.: Lecture Notes (Zürich 1962) unpublished.
Haag, R., andD. Kastler: An algebraic approach to field theory. J. Math. Phys.5, 848–861 (1964).
Plymen, R. J.:C*-algebras and Mackey's axioms. Commun. Math. Phys.8, 132–146 (1968).
Davies, E. B.: On the Borel structure ofC*-algebras. Commun. Math. Phys.8, 147–163 (1968).
Wigner, E. P.: Gruppentheorie, pp. 251–254. Braunschweig: Frederick Vieweg u. Sohn 1931; Group theory, pp. 233–236, New York: Academic Press 1959.
Bargmann, V.: Note on Wigner's theorem on symmetry operations. J. Math. Phys.5, 862–868 (1964).
Kadison, R. V.: Transformations of states in operator theory and dynamics. Topology3, Suppl. 2, 177–198 (1965).
Dixmier, J.: LesC*-algèbres et leurs répresentations. Paris: Gauthier-Villars 1964.
Bourbaki, N.: Topologie générale, chap IX, 2nd. ed. Paris: Hermann 1958.
—— Topologie générale, chap. I, 3rd. ed., Paris: Hermann 1961.
Glimm, J., andR. V. Kadison: Unitary operators inC*-algebras. Pacific J. Math.10, 547–556 (1960).
Kaplansky, I.: A theorem on rings of operators. Pacific J. Math.1, 227–232 (1951).
Dixmier, J.: Les algèbres d'opérateurs dans l'espace hilbertien. Paris: Gauthier-Villars 1957.
Schatten, R.: Norm ideals of completely continuous operators. Berlin-Göttingen-Heidelberg: Springer 1960.
Wick, G. C., A. S. Wightman, andE. P. Wigner: The intrinsic parity of elementary particles. Phys. Rev.88, 101–105 (1952).
Kadison, R. V.: Irreducible operator algebras. Proc. Natl. Acad. Sci. US.43, 273–276 (1957).
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Roberts, J.E., Roepstorff, G. Some basic concepts of algebraic quantum theory. Commun.Math. Phys. 11, 321–338 (1969). https://doi.org/10.1007/BF01645853
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DOI: https://doi.org/10.1007/BF01645853