Skip to main content
Log in

Local relativistic invariant flows for quantum fields

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

For quantum fields with trigonometric interaction in arbitrary space dimension we construct a representation of the Lorentz group by automorphisms on a Banach space generated by the Weyl algebra.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Albeverio, S., Høegh-Krohn, R.: Uniqueness of the physical vacuum and the Wightman functions in the infinite volume limit for some non polynomial interactions. Commun. Math. Phys.30, 171–200 (1973)

    Google Scholar 

  2. Albeverio, S., Høegh-Krohn, R.: The scattering matrix for some non-polynomial interactions, I and II. Helv. Phys. Acta46, 504–534, 536–545 (1973)

    Google Scholar 

  3. Albeverio, S., Høegh-Krohn, R.: Uniqueness and the global Markov property for Euclidean fields. The case of trigonometric interactions. Commun. Math. Phys.68, 95–128 (1979)

    Google Scholar 

  4. Albeverio, S., Høegh-Krohn, R.: The Wightman axioms and the mass gap for strong interactions of exponential type in two-dimensional space time. J. Funct. Anal.16, 39–82 (1974)

    Google Scholar 

  5. Albeverio, S., Høegh-Krohn, R.: Mathematical theory of Feynman path integrals. In: Lecture Notes in Mathematics, Vol. 535. Berlin, Heidelberg, New York: Springer 1976

    Google Scholar 

  6. Albeverio, S., Høegh-Krohn, R.: Feynman path integrals and the corresponding method of stationary phase. In: Proceedings of the Conference on Feynman Path Integral, Marseille (1978). Lecture Notes in Physics, Vol. 106. Berlin, Heidelberg, New York: Springer 1979

    Google Scholar 

  7. Araki, H.: Hamiltonian formalism and the canonical commutation relations in quantum field theory. J. Math. Phys.1, 492–504 (1960)

    Google Scholar 

  8. Araki, H.: Expansional in Banach algebras. Ann. Sci. Ec. Norm. Sup.6, 67–84 (1973)

    Google Scholar 

  9. Bałaban, T., Raczka, R.: Second quantization of classical nonlinear relativistic field theory. I. Canonical formalism. J. Math. Phys.16, 1475–1481 (1975)

    Google Scholar 

  10. Bałaban, T., Jezuita, K., Raczka, R.: Second quantization of classical nonlinear relativistic field theory. Part II. Construction of relativistic interacting local quantum field. Commun. Math. Phys.48, 291–311 (1976)

    Google Scholar 

  11. Bargman, V.: On a Hilbert space of analytic functions and an associated integral transform. Part I. Commun. Pure Appl. Math.14, 187–214 (1961)

    Google Scholar 

  12. Beaume, R., Manuceau, J., Pellet, A., Sirugue, M.: Translation invariant states in quantum mechanics. Commun. Math. Phys.38, 29–45 (1974)

    Google Scholar 

  13. Coester, F., Haag, R.: Representation of states in a field theory with canonical variables. Phys. Rev.117, 1137–1145 (1960)

    Google Scholar 

  14. Combe, Ph., Rodriguez, R., Sirugue-Collin, M.: A uniqueness theorem for anticommutation relations and commutation relations of quantum spin systems. Commun. Math. Phys.63, 219–235 (1978)

    Google Scholar 

  15. Combe, Ph., Høegh-Krohn, R., Rodriguez, R., Sirugue, M., Sirugue-Collin, M.: Poisson processes on groups and Feynman path integrals. Commun. Math. Phys.77, 269–288 (1980)

    Google Scholar 

  16. Combe, Ph., Høegh-Krohn, R., Rodriguez, R., Sirugue, M., Sirugue-Collin, M.: Feynman path integrals with piecewise classical paths. J. Math. Phys.23, 405–411 (1982), and Generalized Poisson processes in quantum mechanics and field theory. Phys. Rep.77, 221–233 (1981)

    Google Scholar 

  17. Combe, Ph., Høegh-Krohn, R., Rodriguez, R., Sirugue, M., Sirugue-Collin, M.: Zero-mass, 2-dimensional real sine-Gordon model without ultraviolet cut-off. Ann. Inst. H. Poincaré37, 115–127 (1982)

    Google Scholar 

  18. Chebotarev, A.M., Maslov, V.P.: Jump processes and their applications in quantum mechanics, Viniti. Itogi Nauk Techn.15, 5–78 (1978)

    Google Scholar 

  19. Dirac, P.A.M.: The quantum theory of the emission and absorption of radiation. Proc. Soc. A114, 243–265 (1927)

    Google Scholar 

  20. Feldman, J.S., Osterwalder, K.: In: International symposium on mathematical problems in theoretical physics, Araki, H. (ed.). Berlin, Heidelberg, New York: Springer 1975, and The Wightman axioms and the mass gap for weakly coupled (φ4)3 quantum field theories. Ann. Phys.97, 80–135 (1976)

    Google Scholar 

  21. Friedrichs, K.O.: Mathematical aspects of the quantum theory of fields. New York: Interscience 1953

    Google Scholar 

  22. Friedrichs, K.O., Schapiro, L.: Integration over Hilbert space and outer extensions. Proc. Natl. Acad. Sci.43, 336–338 (1957)

    Google Scholar 

  23. Fröhlich, J.: Classical and quantum statistical mechanics in one and two dimensions: two-component Yukawa- and Coulomb systems. Commun. Math. Phys.47, 233–268 (1976), and In: Constructive field theory, Velo, G., Wightman, A.S. (eds.). Berlin, Heidelberg, New York: Springer 1975

    Google Scholar 

  24. Garding, L., Wightman, A.S.: Representations of the commutation relations. Proc. Natl. Acad. Sci.40, 622–626 (1954)

    Google Scholar 

  25. Gel'fand, I.M., Yaglom, A.M.: Integration in functional spaces and its applications in quantum physics. J. Math. Phys.1, 48–49 (1960)

    Google Scholar 

  26. Glimm, J., Jaffe, A.: Quantum physics. Berlin, Heidelberg, New York: Springer 1981

    Google Scholar 

  27. Gross, L.: In: Proceedings of conference on theory and application of analysis in function spaces, Martin, W.Ted., Segal, I.E. (eds.). Cambridge, MA: MIT Press 1972, and in Proceedings of the Vth Berkeley Symposium on Mathematical Statistics and Probability, University of California, Berkeley 1968

    Google Scholar 

  28. Haag, R.: On quantum field theories. Kgl. Danske Videnskab Selskab. Mat. Fys. Medd.29, No. 12 (1955)

    Google Scholar 

  29. Heisenberg, W., Pauli, W.: Zur Quantendynamik der Wellenfelder. Z. Phys.56, 1–61 (1929); Zur Quantentheorie der Wellenfelder. II. Z. Phys.59, 168–190 (1930)

    Google Scholar 

  30. van Hove, L.: Les difficultés de divergence pour un modèle particulier de champ quantifié. Physica18, 145–159 (1957)

    Google Scholar 

  31. Høegh-Krohn, R.: On the spectrum of the space cut-offP(φ): Hamiltonian in two space-time dimensions. Commun. Math. Phys.21, 244–255 (1971)

    Google Scholar 

  32. Klauder, J.R.: Continuous-representation theory. I. Postulates of continuous-representation theory, and II. Generalized relation between quantum and classical dynamics. J. Math. Phys.4, 1055–1057 and 1058–1073 (1963)

    Google Scholar 

  33. Magnen, J., Sénéor, R.: The infinite volume limit of the φ 43 model. Ann. Inst. H. Poincaré24, 95–159 (1976)

    Google Scholar 

  34. Moyal, J.E.: Quantum mechanics as a statistical theory. Proc. Camb. Philos. Soc.45, 99–124 (1949)

    Google Scholar 

  35. von Neumann, J.: In: Collected Works, Vol. 3, Taub, A. (ed.). New York: Pergamon Press 1963

    Google Scholar 

  36. Polley, L., Reents, G., Streater, R.F.: Some covariant representations of massless Boson field. Preprint Darmstadt (1980)

  37. Ruelle, D.: Statistical mechanics. New York: Benjamin 1969

    Google Scholar 

  38. Segal, I.: Distributions in Hilbert space and canonical systems of operators. Trans. Am. Math. Soc.88, 12–41 (1958); and Foundations of the theory of dynamical systems of infinitely many degrees of freedom. I. Kgl. Danske Videnskab Selskab. Mat. Phys. Medd.31, No. 12 (1959)

    Google Scholar 

  39. Segal, I.E.: Explict formal construction of nonlinear quantum fields. J. Math. Phys.5, 269–282 (1964)

    Google Scholar 

  40. Segal, I.E.: In: Proceedings of the conference on theory and applications of analysis on function space, Martin, W.Ted., Segal, I.E. (eds.). Cambridge, MA: MIT Press 1972

    Google Scholar 

  41. Segal, I.E.: In: Differential geometric methods in mathematics and physics, Marsden, J. (ed.). In: Lecture Notes in Mathematics Berlin, Heidelberg, New York: Springer 1979

    Google Scholar 

  42. Simon, B.: TheP2) Euclidean (quantum) field theory. Princeton, NJ: Princeton University Press 1974

    Google Scholar 

  43. Streater, R.F.: Canonical quantization. Commun. Math. Phys.2, 354–374 (1966)

    Google Scholar 

  44. Streit, L.: A generalization of Haag's theorem. Nuovo Cimento A10, 673–680 (1969)

    Google Scholar 

  45. Symanzik, K.: Euclidean quantum field theory. I. Equations for a scalar model. J. Math. Phys.7, 510–525 (1966)

    Google Scholar 

  46. Umemura, Y.: Carriers of continuous measures in a Hilbertian norm. Publ. R.I.M.S. (Kyoto) A1, 1–47 and 49–54 (1965)

    Google Scholar 

  47. Wentzel, G.: Quantum theory of fields. New York: Interscience 1949

    Google Scholar 

  48. Weyl, H.: The theory of groups and quantum mechanics. 2nd edn. London: Methuen 1931

    Google Scholar 

  49. Xia, Dao-Xing: Measure and integration on infinite dimensional spaces. New York: Academic Press 1972

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Additional information

Communicated by H. Araki

Rights and permissions

Reprints and permissions

About this article

Cite this article

Albeverio, S., Blanchard, P., Combe, P. et al. Local relativistic invariant flows for quantum fields. Commun.Math. Phys. 90, 329–351 (1983). https://doi.org/10.1007/BF01206886

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01206886

Keywords

Navigation