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On the algebra of test functions for field operators

Decomposition of linear functionals into positive ones

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Abstract

It is shown that every continuous linear functional on the field algebra can be defined by a vector in the Hilbert space of some representation of the algebra. The functionals which can be written as a difference of positive ones are characterized. By an example it is shown that a positive functional on a subalgebra does not always have an extension to a positive functional on the whole algebra.

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References

  1. Wightman, A.: Phys. Rev.101, 860 (1956)

    Google Scholar 

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

    Google Scholar 

  3. Borchers, H.-J.: Algebraic aspects of Wightman field theory. In: Sen, R. N., Weil, C. (Eds.): Statistical mechanics and field theory. Lectures given at the 1971 Haifa Summer School, New York: Halsted Press 1972

    Google Scholar 

  4. Borchers, H.-J.: On the algebra of test functions. Prépublications de la RCPn° 25, Vol. 15, Strasbourg, 1973

  5. Wyss, W.: On Wightman's theory of quantized fields. Lectures in theoretical physics. University of Colorado, Boulder 1968. New York: Gordon and Breach Sci. Publ. 1969

    Google Scholar 

  6. Wyss, W.: Commun. math. Phys.27, 223–234 (1972)

    Google Scholar 

  7. Lassner, G., Uhlmann, A.: Commun. math. Phys.7, 152–159 (1968)

    Google Scholar 

  8. Köthe, G.: Topologische lineare Räume, 2. Aufl. Berlin-Heidelberg-New York: Springer 1966

    Google Scholar 

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

    Google Scholar 

  10. Treves, F.: Topological vector spaces. Distributions and kernels. New York-London: Academic Press 1967

    Google Scholar 

  11. Gel'fand, I. M., Vilenkin, N. Ya.: Generalized functions, Vol. 4, New York: Academic Press 1964

    Google Scholar 

  12. Peressini, A. L.: Ordered topological vector spaces. New York, Evanston, London: Harper & Row 1967

    Google Scholar 

  13. Boas, R. P.: Bull. Am. Math. Soc.45, 399–404 (1939)

    Google Scholar 

  14. Naimark, M. A.: Normed rings, Groningen: Nordhoff 1964

    Google Scholar 

  15. Hustad, O.: Math. Scand.11, 63–68 (1962)

    Google Scholar 

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Authors and Affiliations

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

    Jakob Yngvason

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  1. Jakob Yngvason
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Yngvason, J. On the algebra of test functions for field operators. Commun.Math. Phys. 34, 315–333 (1973). https://doi.org/10.1007/BF01646476

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  • DOI: https://doi.org/10.1007/BF01646476

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