Journal of Statistical Physics

, Volume 161, Issue 1, pp 1–15 | Cite as

Stochastic Quantization, Reflection Positivity, and Quantum Fields

Article

Abstract

We investigate stochastic quantization as a mathematical tool for quantum field theory. We test the method for the free scalar field. We find that the usual method of stochastic quantization is incompatible with establishing a Hilbert-space interpretation for transition probabilities in quantum theory. In particular, we prove that for any finite stochastic time, the standard probability measure violates reflection positivity. As a consequence, if one desires to use stochastic quantization in constructive quantum field theory, one needs to find a more robust procedure than the standard one.

Keywords

Stochastic quantization Reflection positivity Quantum fields  Stochastic pde 

References

  1. 1.
    Brydges, D., Dimock, J., Hurd, T.: Weak perturbations of Gaussian measures, and Applications of the renormalization group. In: Feldman, J.S., Froese, R., Rosen, L.M. (eds.) Mathematical Quantum Field Theory I: CRM Proceedings & Lecture notes, vol. 7, pp. 1–28 and pp. 171–190. American Mathematics Society, Providence, RI (1994)Google Scholar
  2. 2.
    Brydges, D., Dimock, J., Hurd, T.: The short distance behaviour of \(\phi ^{4}_{3}\). Commun. Math. Phys. 172, 143–186 (1995)MathSciNetADSCrossRefMATHGoogle Scholar
  3. 3.
    Feldman, J., Osterwalder, K.: The Wightman axioms and the mass gap for weakly coupled \(\phi ^{4}_{3}\) quantum field theories. Ann. Phys. 97, 80–135 (1976)MathSciNetADSCrossRefGoogle Scholar
  4. 4.
    Gelfand, I.M., Vilenkin, M.I.: Generalized functions: applications of harmonic analysis, translated by Amiel Feinstein, vol. 4. Academic Press, New York (1964)Google Scholar
  5. 5.
    Guerra, F., Rosen, L., Simon, B.: The \(P(\phi )_{2}\) Euclidean quantum field theory as classical statistical mechanics I. Ann. Math. 101, 111–259 (1975)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Glimm, J., Jaffe, A.: A \(\lambda \phi ^{4}\) quantum field theory without cut-offs. I. Phys. Rev. 176, 1945–1961 (1968)MathSciNetADSCrossRefMATHGoogle Scholar
  7. 7.
    Glimm, J., Jaffe, A.: The \(\lambda (\phi ^4)_2\) quantum field theory without cut-offs: II. The field operators and the approximate vacuum. Ann. Math. 91, 362–401 (1970)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Glimm, J., Jaffe, A.: The \(\lambda (\phi ^4)_2\) quantum field theory without cut-offs. III. The physical vacuum. Acta Math. 125, 203–267 (1970)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Glimm, J., Jaffe, A.: The \(\lambda (\phi ^4)_2\) quantum field theory without cut-offs: IV. Perturbations of the Hamiltonian. J. Math. Phys. 13, 1568–1584 (1972)MathSciNetADSCrossRefGoogle Scholar
  10. 10.
    Glimm, J., Jaffe, A.: Positivity of the \(\phi {^4_3}\) Hamiltonian. Fortschr. Phys. 21, 327–376 (1973)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Glimm, J., Jaffe, A.: A note on reflection positivity. Lett. Math. Phys. 3, 377–378 (1979)MathSciNetADSCrossRefMATHGoogle Scholar
  12. 12.
    Glimm, J., Jaffe, A.: Quantum Physics. Springer, New York (1987)CrossRefGoogle Scholar
  13. 13.
    Glimm, J., Jaffe, A., Spencer, T.: The Particle Structure of the Weakly Coupled \(P(\phi )_2\) Model and Other Applications of High Temperature Expansions, Part I: Physics of Quantum Field Models, in Constructive Quantum Field Theory. In: A.S. Wightman (ed), vol. 25, Heidelberg, Springer Lecture Notes in Physics (1973)Google Scholar
  14. 14.
    Glimm, J., Jaffe, A., Spencer, T.: The Wightman axioms and particle structure in the \(P(\phi )_2\) quantum field model. Ann. Math. 100, 585–632 (1974)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Glimm, J., Jaffe, A., Spencer, T.: A convergent expansion about mean field theory, Part I. The expansion. Ann. Phys. 101, 610–630 (1976)MathSciNetADSCrossRefGoogle Scholar
  16. 16.
    Glimm, J., Jaffe, A., Spencer, T.: A convergent expansion about mean field theory, Part II. Convergence of the expansion. Ann. Phys. 101, 631–669 (1976)MathSciNetADSCrossRefGoogle Scholar
  17. 17.
    Hairer, M.: Introduction to Stochastic Partial Differential Equations, arXiv:0907.4178
  18. 18.
    Hairer, M.: A Theory of Regular Structures, arXiv:1303.5113v4
  19. 19.
    Nelson, E.: The Euclidean Markoc field. J. Funct. Anal. 12, 211–227 (1973)CrossRefMATHGoogle Scholar
  20. 20.
    Nelson, E.: Derivation of the Schrödinger equation from Newtonian mechanics. Phys. Rev. 150, 1079–1085 (1966)ADSCrossRefGoogle Scholar
  21. 21.
    Nelson, E.: Dynamical Theories of Brownian Motion. Mathematical Notes. Princeton University Press, Princeton (1967)Google Scholar
  22. 22.
    Moser, D.: Renormalization of \(\phi ^{4}_{3}\) quantum field theory, E.T.H. Diploma Thesis, 2006Google Scholar
  23. 23.
    Osterwalder, K., Schrader, R.: Axioms for Euclidean Green’s functions. I. Commun. Math. Phys. 31, 83–112 (1973)MathSciNetADSCrossRefMATHGoogle Scholar
  24. 24.
    Osterwalder, K., Schrader, R.: Axioms for Euclidean Green’s functions. II. Commun. Math. Phys. 42, 281–305 (1975)MathSciNetADSCrossRefMATHGoogle Scholar
  25. 25.
    Parisi, G., Wu, Y.: Perturbation theory without gauge fixing. Sci. Sin. 24, 483–496 (1981)MathSciNetGoogle Scholar
  26. 26.
    Symanzik, K.: A Modified Model of Euclidean Quantum Field Theory. Courant Institute of Mathematical Sciences, Report IMM-NYU 327 (June 1964)Google Scholar
  27. 27.
    Wilson, K.G.: Renormalization group and critical phenomena. I. Renormalization group and the Kadanoff scaling picture. Phys. Rev. B 4(9), 3174–3183 (1971)Google Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Harvard UniversityCambridgeUSA

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