Abstract
Starting from an algebra of fields\(\mathfrak{F}\) and a compact gauge group of the first kind ℊ, the observable algebra\(\mathfrak{A}\) is defined as the gauge invariant part of\(\mathfrak{F}\). A gauge group of the first kind is shown to be automatically compact if the scattering states are complete and the mass and spin multiplets have finite multiplicity. Under reasonable assumptions about the structure of\(\mathfrak{F}\) it is shown that the inequivalent irreducible representations of\(\mathfrak{A}\) (“sectors”) which occur are in one-to-one correspondence with the inequivalent irreducible representations of ℊ and that all of them are “strongly locally equivalent”. An irreducible representation of\(\mathfrak{A}\) satisfies the duality property only if the sector corresponds to a 1-dimensional representation of ℊ. If ℊ is Abelian the sectors are connected to each other by localized automorphisms.
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References
Haag, R., andD. Kastler: An algebraic approach to quantum field theory. J. Math. Phys.5, 848–861 (1964).
Borchers, H. J.: Local rings and the connection of spin with statistics. Commun. Math. Phys.1, 281–307 (1965).
Borchers, H. J.: The vacuum state in quantum field theory II. Commun. Math. Phys.1, 57–79 (1965).
Borchers, H. J.: On the converse of the Reeh-Schlieder theorem. Commun. Math. Phys.10, 269–273 (1968).
Araki, H., andR. Haag: Collision cross sections in terms of local observables. Commun. Math. Phys.4, 77–91 (1967).
Bourbaki, N.: Topologie générale, Ch I, 3rd. ed. Paris:Hermann 1961.
Dixmier, J.: Les algèbres d'opérateurs dans l'espace hilbertien. Paris: Gauthier-Villars 1957.
Kovács, I., andJ. Szücs: Ergodic type theorems in von Neumann algebras. Acta Sc. Math.27, 233–246 (1966).
Doplicher, S., D. Kastler, andE. Størmer: Invariant states and asymptotic Abelianness. To appear J. Funct. Analysis.
Dixmier, J.: Les C*-algèbres et leurs répresentations. Paris: Gauthier-Villars 1964.
Doplicher, S., andD. Kastler: Ergodic states in a non-commutative ergodic theory. Commun. Math. Phys.7, 1–20 (1968).
Wightman, A. S.: Recent achievements of axiomatic field theory in “Theoretical Physics” IAEA, Vienna (1963).
Stone, M. H.: Applications of the theory of Boolean rings to general topology. Trans. Am. Math. Soc.41, 375–481 (1937).
Naimark, M. A.: Normed rings. Groningen: P. Noordhoff N. V. 1959.
Knight, J. M.: Strict localization in quantum field theory. J. Math. Phys.2, 459–471 (1961).
Dell'Antonio, G. F.: Structure of the algebras of some free systems. Commun. Math. Phys.9, 81–117 (1968).
Araki, H., andE. J. Woods: A classification of factors. Publ. RIMS, Kyoto Univ. Ser. A3, 51–130 (1968).
Jadczyk, A. Z.: On the spectrum of internal symmetries in algebraic quantum field theory. Commun. Math. Phys.12, 58–63 (1969).
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On leave of absence from Instituto di Fisica G. Marconi, Università di Roma.
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Doplicher, S., Haag, R. & Roberts, J.E. Fields, observables and gauge transformations I. Commun.Math. Phys. 13, 1–23 (1969). https://doi.org/10.1007/BF01645267
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DOI: https://doi.org/10.1007/BF01645267