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Infinite Volume Limits in Euclidean Quantum Field Theory via Stereographic Projection

  • Svetoslav Zahariev
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Abstract

We present a general infinite volume limit construction of probability measures obeying the Glimm–Jaffe axioms of Euclidean quantum field theory in arbitrary space–time dimension. In particular, we obtain measures that may be interpreted as corresponding to scalar quantum fields with arbitrary bounded continuous self-interaction. It remains, however, an open problem whether this general construction contains non-Gaussian measures.

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Notes

Acknowledgements

The author is grateful toWojciech Dybalski, Leonard Gross, NikolayM. Nikolov and Yoh Tanimoto for many helpful suggestions and stimulating discussions at various stages of this project.

References

  1. 1.
    Adams, R., Fournier, J.: Sobolev Spaces, 2nd edn. Academic Press, London (2003)MATHGoogle Scholar
  2. 2.
    Barata, J.C.A., Jäkel, C. D., Mund, J.: The \(P(\phi )_{2}\) model on the de Sitter space. arXiv:1311.2905
  3. 3.
    Bogachev, V.I.: Measure Theory. Springer, Berlin (2007)CrossRefMATHGoogle Scholar
  4. 4.
    Bourbaki, N.: General Topology. Chapters 5–10. Springer, Berlin (1989)CrossRefMATHGoogle Scholar
  5. 5.
    Buchholz, D., Verch, R.: Scaling algebras and renormalization group in algebraic quantum field theory. II. Instructive examples. Rev. Math. Phys. 10, 775–800 (1998)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Cannon, J.T.: Continuous sample paths in quantum field theory. Commun. Math. Phys. 35, 215–233 (1974)ADSMathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Dalecky, Y.L., Fomin, S.V.: Measures and Differential Equations in Infinite-Dimensional Space. Kluwer, Dordrecht (1991)CrossRefGoogle Scholar
  8. 8.
    Glimm, J., Jaffe, A.: Quantum Physics: A Functional Integral Point of View, 2nd edn. Springer, Berlin (1987)CrossRefMATHGoogle Scholar
  9. 9.
    Graham, C. R.: Conformal powers of the Laplacian via stereographic projection. SIGMA Symmetry Integrability Geom. Methods Appl. 3, Paper 121, 4 (2007)Google Scholar
  10. 10.
    Hervé, M.: Analyticity in Infinite Dimensional Spaces. Walter de Gruyter, Berlin (1989)CrossRefMATHGoogle Scholar
  11. 11.
    Jaffe, A., Ritter, G.: Reflection positivity and monotonicity. J. Math. Phys. 49, 052301 (2008)ADSMathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Jaffe, A., Witten, E.: Quantum Yang–Mills theory. In: Carlson, J., Jaffe, A., Wiles, A. (eds.) The Millennium Prize Problems, pp. 129–152. Clay Mathematics Institute, Cambridge, MA (2006)Google Scholar
  13. 13.
    Osterwalder, K., Schrader, R.: Axioms for Euclidean Green’s functions. Commun. Math. Phys. 31, 83–112 (1973)ADSMathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Osterwalder, K., Schrader, R.: Axioms for Euclidean Green’s functions. Commun. Math. Phys. 42, 281–305 (1975)ADSMathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Reed, M., Rosen, L.: Support properties of the free measure for Boson fields. Commun. Math. Phys. 36, 123–132 (1974)ADSMathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Schlingemann, D.: Euclidean field theory on a sphere. arXiv:hep-th/9912235
  17. 17.
    Serfozo, R.: Convergence of Lebesgue integrals with varying measures. Indian J. Stat. Ser. A 44, 380–402 (1982)MathSciNetMATHGoogle Scholar
  18. 18.
    Summers, S.: A perspective on constructive quantum field theory. arXiv:1203.3991
  19. 19.
    Streater, R., Wightman, A.: PCT, Spin and Statistics, and all that. W.A. Benjamin, Reading (1964)MATHGoogle Scholar
  20. 20.
    Taylor, M.: Pseudodifferential Operators. Princeton University Press, Princeton (1981)MATHGoogle Scholar
  21. 21.
    Treves, F.: Topological Vector Spaces, Distributions and Kernels. Academic Press, London (1967)MATHGoogle Scholar

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© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.MEC DepartmentLaGuardia Community College of The City University of New YorkLong Island CityUSA

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