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TMF, 1970, Volume 4, Number 2, Pages 171–195 (Mi tmf4142)  

This article is cited in 14 scientific papers (total in 14 papers)

Vector states on algebras of observables and superselection rules I. Vector states and Hilbert space

V. N. Sushko, S. S. Horuzhy


Abstract: A detailed investigation is made of vector states on an arbitrary reducible $W^*$-algebra of observables $R$. The properties of vector states (purity, subordination, etc.) are reformulated and studied in terms of their “preimages”, i.e., the sets of vectors in the Hilbert space $\mathscr H$ corresponding to one and the same vector state. The properties of preimages of pure vector states are described exhaustively. A special class of quantum theories is studied for which $\mathscr H$ coincides with the closure $\mathscr H$ of the linear hull of the set of all vectors representing pure states. It is proved that a theory belongs to this class if and only if $R$ is a direct sum of type I factors. The structure of $R$ and $\mathscr H$ is analyzed exhaustively for this class of theories, i.e., different representations of $\mathscr H$ are given; the number of pure vector states and the number of subspaces that are irreducible under $R$ are determined. The connection between the results of the present paper and the formalism of the abstract algebraic approach is established.

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English version:
Theoretical and Mathematical Physics, 1970, 4:2, 758–774

Bibliographic databases:

Received: 09.04.1970

Citation: V. N. Sushko, S. S. Horuzhy, “Vector states on algebras of observables and superselection rules I. Vector states and Hilbert space”, TMF, 4:2 (1970), 171–195; Theoret. and Math. Phys., 4:2 (1970), 758–774

Citation in format AMSBIB
\Bibitem{SusHor70}
\by V.~N.~Sushko, S.~S.~Horuzhy
\paper Vector states on algebras of observables and superselection rules I.~Vector states and Hilbert space
\jour TMF
\yr 1970
\vol 4
\issue 2
\pages 171--195
\mathnet{http://mi.mathnet.ru/tmf4142}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=464976}
\transl
\jour Theoret. and Math. Phys.
\yr 1970
\vol 4
\issue 2
\pages 758--774
\crossref{https://doi.org/10.1007/BF01066486}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. V. N. Sushko, S. S. Horuzhy, “Vector states on algebras of observables and superselection rules II. Algebraic theory of superselection rules”, Theoret. and Math. Phys., 4:3 (1970), 877–889  mathnet  crossref  mathscinet
    2. S. S. Horuzhy, “Fields and local observables in an axiomatic algebraic theory with superselection rules”, Theoret. and Math. Phys., 5:1 (1970), 942–952  mathnet  crossref  zmath
    3. V. N. Sushko, S. S. Horuzhy, “Properties of $H$ images of vector states on algebras of observables”, Theoret. and Math. Phys., 8:3 (1971), 862–864  mathnet  crossref  mathscinet  zmath
    4. A. V. Bulinski, “On the classes of $C^*$ algebras that satisfy the Haag–Kastler axioms”, Theoret. and Math. Phys., 8:3 (1971), 865–869  mathnet  crossref  mathscinet  zmath
    5. V. N. Sushko, S. S. Horuzhy, “Local and asymptotic structure of quantum systems with superselection rules”, Theoret. and Math. Phys., 13:3 (1972), 1147–1160  mathnet  crossref  mathscinet
    6. V. N. Sushko, S. S. Horuzhy, “$H$-images of vector states and causal properties of local algebras”, Theoret. and Math. Phys., 15:2 (1973), 460–466  mathnet  crossref  zmath
    7. S. G. Kharatyan, “Von neumann algebras of observables with non-Abelian commutator algebra and superselection rules”, Theoret. and Math. Phys., 14:3 (1973), 227–232  mathnet  crossref  mathscinet  zmath
    8. V. M. Maksimov, “Macroscopic observables in algebraic statistical physics”, Theoret. and Math. Phys., 20:1 (1974), 632–638  mathnet  crossref  mathscinet
    9. Yu. M. Zinoviev, V. N. Sushko, “Physical symmetries in a theory of local observables of the $P$-class”, Theoret. and Math. Phys., 18:1 (1974), 9–18  mathnet  crossref  mathscinet  zmath
    10. S. S. Horuzhy, “Superposition principle in Algebraic quantum theory”, Theoret. and Math. Phys., 23:2 (1975), 413–421  mathnet  crossref  mathscinet  zmath
    11. K. Yu. Dadashyan, “Causality properties in nonrelativistic algebraic models”, Theoret. and Math. Phys., 31:3 (1977), 492–496  mathnet  crossref  mathscinet  zmath
    12. K. Yu. Dadashyan, S. S. Horuzhy, “Algebras of observables of the free Dirac field”, Theoret. and Math. Phys., 36:2 (1978), 665–675  mathnet  crossref  mathscinet
    13. 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”, Theoret. and Math. Phys., 59:1 (1984), 335–350  mathnet  crossref  mathscinet  zmath  isi
    14. 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  mathnet  crossref  mathscinet  isi
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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