Communications in Mathematical Physics

, Volume 329, Issue 1, pp 1–28 | Cite as

Complex Classical Fields: A Framework for Reflection Positivity

  • Arthur Jaffe
  • Christian D. Jäkel
  • Roberto E. MartinezII


We explore a framework for complex classical fields, appropriate for describing quantum field theories. Our fields are linear transformations on a Hilbert space, so they are more general than random variables for a probability measure. Our method generalizes Osterwalder and Schrader’s construction of Euclidean fields. We allow complex-valued classical fields in the case of quantum field theories that describe neutral particles.

From an analytic point-of-view, the key to using our method is reflection positivity. We investigate conditions on the Fourier representation of the fields to ensure that reflection positivity holds. We also show how reflection positivity is preserved by various spacetime compactifications of \({\mathbb{R}^{d}}\) in different coordinate directions.


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  1. 1.
    Balaban, T., Feldman, J., Knörrer, H., Trubowitz, E.: The temporal ultraviolet limit. In: Fröhlich, J., Salmhofer, M., Mastropietro, V., De Roeck, W., Cugliandolo, L.F. (eds.) Quantum Theory from Small to Large Scales. Lecture Notes of the Les Houches Summer School: vol. 5, August 2010. Oxford University Press, Oxford (2012)Google Scholar
  2. 2.
    Biskup, M.: Reflection positivity and phase transitions for lattice models. In: Roman, K. (ed.) Methods of Contemporary Mathematical Statistical Physics. Lecture Notes in Mathematics, Springer, Berlin (2009)Google Scholar
  3. 3.
    Borchers H., Yngvason J.: Necessary and sufficient conditions for integral representations of Wightman functionals at Schwinger points. Commun. Math. Phys. 47, 197–213 (1976)ADSCrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Brydges D.C., Imbrie J.Z.: Green’s function for a hierarchical self-avoiding walk in four dimensions. Commun. Math. Phys. 239, 549–584 (2003)ADSCrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Gelfand I.M., Vilenkin N.Ya.: Generalized Functions, vol. 4. Academic Press (HBJ Publishers), New York (1964)Google Scholar
  6. 6.
    Glimm J., Jaffe A.: Quantum Physics, A Functional Point of View. Springer, New York (1981)MATHGoogle Scholar
  7. 7.
    Glimm J., Jaffe A., Spencer T.: The Wightman axioms and particle structure in the P(ϕ)2 quantum field model. Ann. Math. 100, 585–632 (1974)CrossRefMathSciNetGoogle Scholar
  8. 8.
    Heifets E.P., Osipov E.P.: The energy-momentum spectrum in the P(φ)2 quantum field theory. Commun. Math. Phys. 56, 161–172 (1977)ADSCrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Jaffe A., Jäkel C., Martinez R.: Complex classical fields: an example. J. Funct. Anal. 266, 1833–1881 (2014)CrossRefMathSciNetGoogle Scholar
  10. 10.
    Jørgensen, P.E.T., Ólafsson, G.: Unitary representations and Osterwalder–Schrader duality. In: Doran, R., Varadarajan, V. (eds.) The Mathematical Legacy of Harish–Chandra: A Celebration of Representation Theory and Harmonic Analysis, Proceedings of the Symposium in Pure Mathematics, vol. 68, pp. 333–401. American Mathematical Society, Providence (2000)Google Scholar
  11. 11.
    Minlos R.A.: Generalized stochastic processes and their extension to a measure. Trudy Mosk. Mat. Obs. 8, 497–518 (1959)MathSciNetGoogle Scholar
  12. 12.
    Nelson E.: Construction of quantum fields from Markoff fields. J. Funct. Anal. 12, 97–112 (1973)CrossRefMATHGoogle Scholar
  13. 13.
    Nelson E.: The free Markoff field. J. Funct. Anal. 12, 211–227 (1973)CrossRefMATHGoogle Scholar
  14. 14.
    Nelson E.: Analytic vectors. Ann. Math. 70, 572–615 (1959)CrossRefMATHGoogle Scholar
  15. 15.
    Osterwalder K., Schrader R.: Axioms for Euclidean Green’s functions. Commun. Math. Phys. 31, 83–112 (1973)ADSCrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Osterwalder K., Schrader R.: Axioms for Euclidean Green’s functions II. Commun. Math. Phys. 42, 281–305 (1975)ADSCrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Osterwalder K., Schrader R.: Euclidean Fermi fields and a Feynman-Kac formula for boson-fermion interactions. Helv. Phys. Acta 46, 227–302 (1973)MathSciNetGoogle Scholar
  18. 18.
    Symanzik, K.: A modified model of Euclidean quantum field theory, Courant Institute of Mathematical Sciences Report IMM-NYU 327, June 1964, and Euclidean quantum field theory. In: Jost, R. (ed.) Local Quantum Theory, Varenna Lectures 1968, pp. 152–226. Academic Press, New York (1969)Google Scholar
  19. 19.
    Yngvason J.: Euclidean invariant integral representations for Schwinger functionals. J. Math. Phys. 27(1), 311–320 (1986)ADSCrossRefMathSciNetGoogle Scholar
  20. 20.
    Zinoviev Y.: Equivalence of Euclidean and Wightman field theories. Commun. Math. Phys. 174, 1–27 (1995)ADSCrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Arthur Jaffe
    • 1
  • Christian D. Jäkel
    • 2
  • Roberto E. MartinezII
    • 1
  1. 1.Harvard UniversityCambridgeUSA
  2. 2.School of MathematicsCardiff UniversityWalesUK

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