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Integral representations for Schwinger functionals and the moment problem over nuclear spaces

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Abstract

It is shown that a continuous positive linear functional on a commutative nuclear *-algebra has an integral decomposition into characters if and only if the functional is strongly positive, i.e. positive on all positive polynomials. When applied to the symmetric tensor algebra over a nuclear test function space this gives a necessary and sufficient condition for the Schwinger functions of Euclidean quantum field theory to be the moments of a continuous cylinder measure on the dual space. Another application is to the problem of decomposing a Wightman functional into states having the cluster property.

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References

  1. Powers, R.T.: Commun. math. Phys.21, 85–124 (1971)

    Article  MATH  MathSciNet  Google Scholar 

  2. Borchers, H.J., Yngvason, J.: On the algebra of field operators. The Weak Commutant and Integral Decomposition of states. Commun. math. Phys.42, 231–252 (1975)

    MathSciNet  Google Scholar 

  3. Shohat, J.A., Tamarkin, J.D.: The Problem of Moments, Surveys No. 1, Amer. Math. Soc. New York 1943

    Google Scholar 

  4. Choquet, G.: Lectures on Analysis, Vol. II, New York: Benjamin 1969

    Google Scholar 

  5. Berezansky, Ju.M.: Generalized Power moment problems. In: Hilbert space operators and operator algebras, Sz-Nagy, B. (ed.), Amsterdam-London: North Holland 1972

    Google Scholar 

  6. Powers, R.T.: Transactions Amer. Mat. Soc.187, 261–293 (1974)

    ADS  MATH  MathSciNet  Google Scholar 

  7. Lassner, G.: Rep. math. Phys.3, 279–293 (1972)

    MATH  MathSciNet  Google Scholar 

  8. Dixmier, J.: Les algèbres d'operateurs dans l'espace Hilbertien, 2ième édition. Paris: Gauthier-Villars 1969

    Google Scholar 

  9. Maurin, K.: Generalized Eigenfunction Expansion and Unitary Representations of Topological Groups. Warszawa: Polish Scientific Publishers 1968

    Google Scholar 

  10. Gel'fand, I.M., Vilenkin, N.Ya.: Generalized functions, Vol. 4. New York: Academic Press 1968

    Google Scholar 

  11. Pietsch, A.: Nuclear locally convex spaces, Berlin-Heidelberg-New York: Springer 1972

    Google Scholar 

  12. Symanzik, K.: Euclidean Quantum Field Theory. In: Proc. of the International School of Physics “ENRICO FERMI”, Varenna Course XLV, ed. R. Jost, New York: Academic Press 1969

    Google Scholar 

  13. Borchers, H.J.: Commun. math. Phys.1, 49–56 (1965)

    MATH  MathSciNet  Google Scholar 

  14. Borchers, H.J.: Nuovo Cimento24, 1118–1140 (1962)

    MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Institut für Theoretische Physik, Universität Göttingen, Göttingen, Federal Republic of Germany

    H. J. Borchers & J. Yngvason

Authors
  1. H. J. Borchers
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  2. J. Yngvason
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Communicated by H. Araki

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Borchers, H.J., Yngvason, J. Integral representations for Schwinger functionals and the moment problem over nuclear spaces. Commun.Math. Phys. 43, 255–271 (1975). https://doi.org/10.1007/BF02345023

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  • DOI: https://doi.org/10.1007/BF02345023

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