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Unbounded GNS representations of a *-algebra in a Krein space

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Abstract

Given a *-algebra \(\mathfrak{A}\) and a linear, nonpositive definite, functional on it, the GNS construction yields a representation of \(\mathfrak{A}\) in an indefinite metric space. We show that, if the functional satisfies a suitable condition, expressed in terms of a conditional expectation on \(\mathfrak{A}\), then the representation space is a Krein space. This provides an abstract setup for the description of a massless quantum field theory.

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References

  1. Borchers, H. J., Nuovo Cim. 24, 214 (1962); Algebraic aspects of Wightman field theory, in R. N. Sen and C. Weil (eds.), Statistical Mechanics and Field Theory, Wiley, New York, 1972.

    Google Scholar 

  2. Uhlmann, A., Wiss. Z. Karl-Marx-Univ. Leipzig 11, 213 (1962); Lassner, G and Uhlmann, A., Commun. Math. Phys. 7, 152 (1968).

    Google Scholar 

  3. Wightman, A. S. and Gårding, L., Arkiv Physik 28 129 (1964); Strocchi F. and Wightman, A. S., J. Math. Phys. 15, 2198 (1974); Morchio G. and Strocchi, F. Ann. Inst. Henri Poincaré 33, 251 (1980).

    Google Scholar 

  4. Yngvason, J., Rep. Math. Phys. 12, 57 (1977).

    Google Scholar 

  5. Bongaarts, P. J. M., J. Math. Phys. 18, 1510 (1977).

    Google Scholar 

  6. Dadashyan, K. Yu and Khoruzhii, S. S., Theor. Math. Phys. 54, 35 (1983).

    Google Scholar 

  7. Mintchev, M. and d'Emilio, E., J. Math. Phys. 22, 1267 (1981).

    Google Scholar 

  8. Jakóbczyk, L., J. Math. Phys. 25, 617 (1984).

    Google Scholar 

  9. Völkel, A. H., J. Math. Phys. 26, 2956 (1985); 27, 1113 (1986).

    Google Scholar 

  10. Jakóbczyk, L. and Strocchi, F., J. Math. Phys. 29, 1231 (1988).

    Google Scholar 

  11. Jadczyk, A., Rep. Math. Phys. 2, 263 (1971).

    Google Scholar 

  12. Bognar, J., Indefinite Inner Product Spaces, Springer-Verlag, Berlin, 1974.

    Google Scholar 

  13. Rideau, G., J. Math. Phys. 19, 1627 (1978).

    Google Scholar 

  14. Jakóbczyk, L., Ann. Phys. 161, 314 (1985).

    Google Scholar 

  15. Ôta, S., Ann. Inst. Henri Poincaré 48, 333 (1988).

    Google Scholar 

  16. Morchio, G. and Strocchi, F., Nucl. Phys. B 211, 471 (1983); 232, 547 (1984); Infrared problems, Higgs phenomena and long range interaction, in G. Velo and A. S. Wightman (eds.), Fundamental Problems of Gauge Field Theory (Proc. Erice 1985), Plenum, New York, 1986.

    Google Scholar 

  17. Sakai, S., C *-Algebras and W *-Algebras, Springer-Verlag, Berlin, 1971.

    Google Scholar 

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Author notes

  1. S. Ôta

    Present address: Department of Mathematics, Kyushu Institute of Design, 815, Fukuoka, Japan

Authors and Affiliations

  1. Institut de Physique Théorique, Université Catholique de Louvain, B-1348, Lowvain-la-Neuve, Belgium

    J. -P. Antoine & S. Ôta

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  1. J. -P. Antoine
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  2. S. Ôta
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Antoine, J.P., Ôta, S. Unbounded GNS representations of a *-algebra in a Krein space. Lett Math Phys 18, 267–274 (1989). https://doi.org/10.1007/BF00405258

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  • DOI: https://doi.org/10.1007/BF00405258

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