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 TMF, 1984, Volume 59, Number 1, Pages 28–48 (Mi tmf4703)

Algebras of unbounded operators and vacuum superselection rules in quantum field theory. I. Some properties of Op*-algebras and vector states on them

A. V. Voronin, V. N. Sushko, S. S. Horuzhy

Abstract: In connection with the physical problem of describing vacuum superselection rules in quantum field theory, a study is made of some properties of Op* algebras, namely, the structure of their commutants and invariant and reducing subspaces and vector states on such algebras. For this, a formalism is developed that uses intertwining operators of Hermitian representations of a * algebra. The formalism is used to obtain a number of new properties of the commutants of Op* algebras, and a description is given of classes of subspaces the projection operators onto which lie in the strong or weak commutant. A study is made of the correspondence between vector states on the Op* algebra $\mathscr P$ and on its associated yon Neumann algebra $R=({\mathscr P_w}^{'})^{'}$; generalizations are found of the class of self-adjoint Op* algebras for which a detailed investigation of vector states can be made. Classes of weakly regular, strongly regular, and completely regular vectors for which the properties of states on $\mathscr P$ approach closer and closer to states on $R$ are identified and studied.

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English version:
Theoretical and Mathematical Physics, 1984, 59:1, 335–350

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Document Type: Article

Citation: A. V. Voronin, V. N. Sushko, S. S. Horuzhy, “Algebras of unbounded operators and vacuum superselection rules in quantum field theory. I. Some properties of Op*-algebras and vector states on them”, TMF, 59:1 (1984), 28–48; Theoret. and Math. Phys., 59:1 (1984), 335–350

Citation in format AMSBIB
\Bibitem{VorSusHor84} \by A.~V.~Voronin, V.~N.~Sushko, S.~S.~Horuzhy \paper Algebras of unbounded operators and vacuum superselection rules in quantum field theory. I. Some properties of Op*-algebras and vector states on them \jour TMF \yr 1984 \vol 59 \issue 1 \pages 28--48 \mathnet{http://mi.mathnet.ru/tmf4703} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=749003} \zmath{https://zbmath.org/?q=an:0559.47033} \transl \jour Theoret. and Math. Phys. \yr 1984 \vol 59 \issue 1 \pages 335--350 \crossref{https://doi.org/10.1007/BF01028511} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=A1984TR03500002} 

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This publication is cited in the following articles:
1. A. V. Voronin, V. N. Sushko, S. S. Horuzhy, “Algebras of unbounded operators and vacuum superselection rules in quantum field theory II. Mathematical structure of vacuum superselection rules”, Theoret. and Math. Phys., 60:3 (1984), 849–862
2. A. V. Voronin, “Discrete vacuum superselection rule in Wightman theory with essentially self-adjoint field operators”, Theoret. and Math. Phys., 66:1 (1986), 8–19
3. A. V. Voronin, S. S. Horuzhy, “$\mathrm{Op}^*$ and $\mathrm C^*$ dynamical systems I. Structural parallels”, Theoret. and Math. Phys., 82:2 (1990), 113–123
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