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Communications in Mathematical Physics

, Volume 248, Issue 2, pp 305–333 | Cite as

Particle Weights and Their Disintegration II

  • Martin PorrmannEmail author
Article

Abstract

The first article in this series presented a thorough discussion of particle weights and their characteristic properties. In this part a disintegration theory for particle weights is developed which yields pure components linked to irreducible representations and exhibiting features of improper energy-momentum eigenstates. This spatial disintegration relies on the separability of the Hilbert space as well as of the C*-algebra. Neither is present in the GNS-representation of a generic particle weight so that we use a restricted version of this concept on the basis of separable constructs. This procedure does not entail any loss of essential information insofar as under physically reasonable assumptions on the structure of phase space the resulting representations of the separable algebra are locally normal and can thus be continuously extended to the original quasi-local C*-algebra.

Keywords

Hilbert Space Phase Space Irreducible Representation Characteristic Property Reasonable Assumption 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Alfsen, E.M.: Compact Convex Sets and Boundary Integrals. Berlin, Heidelberg, New York: Springer-Verlag, 1971Google Scholar
  2. 2.
    Araki, H., Haag, R.: Collision Cross Sections in Terms of Local Observables. Commun. Math. Phys. 4, 77–91 (1967)Google Scholar
  3. 3.
    Arveson, W.: An Invitation to C*-Algebras. New York, Heidelberg, Berlin: Springer-Verlag, 1976Google Scholar
  4. 4.
    Borchers, H.-J.: Translation Group and Spectrum Condition. Commun. Math. Phys. 96, 1–13 (1984)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Borchers, H.-J., Buchholz, D.: The Energy-Momentum Spectrum in Local Field Theories with Broken Lorentz-Symmetry. Commun. Math. Phys. 97, 169–185 (1985)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Bratteli, O., Robinson, D.W.: Operator Algebras and Quantum Statistical Mechanics 1. 2nd ed. New York, Berlin, Heidelberg: Springer-Verlag, 1987Google Scholar
  7. 7.
    Bratteli, O., Robinson, D.W.: Operator Algebras and Quantum Statistical Mechanics 2. 2nd ed. Berlin, Heidelberg, New York: Springer-Verlag, 1997Google Scholar
  8. 8.
    Buchholz, D., D’Antoni, C., Longo, R.: Nuclear Maps and Modular Structures II: Applications to Quantum Field Theory. Commun. Math. Phys. 129, 115–138 (1990)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Buchholz, D., Porrmann, M.: How Small is the Phase Space in Quantum Field Theory? Ann. Inst. Henri Poincaré - Physique théorique 52, 237–257 (1990)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Buchholz, D., Porrmann, M., Stein, U.: Dirac versus Wigner: Towards a Universal Particle Concept in Local Quantum Field Theory. Phys. Lett. B267, 377–381 (1991)Google Scholar
  11. 11.
    Buchholz, D., Wichmann, E.H.: Causal Independence and the Energy-Level Density of States in Local Quantum Field Theory. Commun. Math. Phys. 106, 321–344 (1986)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Dixmier, J.: Von Neumann Algebras. Amsterdam, New York, Oxford: North-Holland Publishing Co., 1981Google Scholar
  13. 13.
    Dixmier, J.: C*-Algebras. rev. ed. Amsterdam, New York, Oxford: North-Holland Publishing Co., 1982Google Scholar
  14. 14.
    Fell, J. M.G., Doran, R.S.: Representations of *-Algebras, Locally Compact Groups, and Banach *-Algebraic Bundles – Volume 1. San Diego, London: Academic Press, Inc., 1988Google Scholar
  15. 15.
    Fredenhagen, K., Hertel, J.: Zwei Sätze über Kompaktheit, 1979. Unpublished manuscriptGoogle Scholar
  16. 16.
    Guido, D., Longo, R.: Natural Energy Bounds in Quantum Thermodynamics. Commun. Math. Phys. 218, 513–536 (2001)CrossRefMathSciNetzbMATHGoogle Scholar
  17. 17.
    Haag, R.: Local Quantum Physics. 2nd ed. Berlin, Heidelberg, New York: Springer-Verlag, 1996Google Scholar
  18. 18.
    Haag, R., Swieca, J.: When Does a Quantum Field Theory Describe Particles? Commun. Math. Phys. 1, 308–320 (1965)zbMATHGoogle Scholar
  19. 19.
    Halmos, P.R.: Measure Theory. 12th ed., Princeton, New Jersey, Toronto, Melbourne, London: D. Van Nostrand Company, Inc., 1968Google Scholar
  20. 20.
    Heuser, H.: Lehrbuch der Analysis – Teil 2. 8th ed. Stuttgart: B. G. Teubner, 1993Google Scholar
  21. 21.
    Kadison, R.V., Ringrose, J.R.: Fundamentals of the Theory of Operator Algebras – Volume II. Orlando, London: Academic Press, Inc., 1986Google Scholar
  22. 22.
    Köthe, G.: Topological Vector Spaces I. 2nd ed. Berlin, Heidelberg, New York: Springer-Verlag, 1983Google Scholar
  23. 23.
    Pedersen, G.K.: C*-Algebras and their Automorphism Groups. London, New York, San Francisco: Academic Press, Inc., 1979Google Scholar
  24. 24.
    Phelps, R.R.: Lectures on Choquet’s Theorem. New York, Toronto, London, Melbourne: American Book, Van Nostrand, Reinhold, 1966Google Scholar
  25. 25.
    Porrmann, M.: Ein verschärftes Nuklearitätskriterium in der lokalen Quantenfeldtheorie. Master’s thesis, Universität Hamburg, 1988Google Scholar
  26. 26.
    Porrmann, M.: The Concept of Particle Weights in Local Quantum Field Theory. Ph.D. thesis, Universität Göttingen, 2000. http://arxiv.org/ps_cache/hep-th/pdf/0005/0005057.pdf, 2000
  27. 27.
    Porrmann, M.: Particle Weights and their Disintegration I. Commun. Math. Phys. 248, 269–304 (2004)Google Scholar
  28. 28.
    Takesaki, M.: Theory of Operator Algebras I. New York, Heidelberg, Berlin: Springer-Verlag, 1979Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  1. 1.II. Institut für Theoretische PhysikUniversität HamburgHamburgGermany

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