Skip to main content
SpringerLink
Log in

Normal product states for fermions and twisted duality for CCR- and CAR-type algebras with application to the Yukawa2 quantum field model

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

Abstract

We present sufficient conditions that imply duality for the algebras of local observables in all Abelian sectors of all locally normal, irreducible representations of a field algebra if twisted duality obtains in one of these representations. It is verified that the Yukawa2 model satisfies these conditions, yielding the first proof of duality for the observable algebra in all coherent charge sectors in this model. This paper also constitutes the first verification of the assumptions of the axiomatic study of the structure of superselection sectors by Doplicher, Haag and Roberts in an interacting model with nontrivial sectors. The existence of normal product states for the free Fermi field algebra and, thus, the verification of the “funnel property” for the associated net of local algebras are demonstrated.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Araki, H.: von Neumann algebras of local observables for free scalar field. J. Math. Phys.5, 1–13 (1964)

    Google Scholar 

  2. Araki, H.: Types of von Neumann algebras associated with free fields. Prog. Theor. Phys.32, 956–965 (1964)

    Google Scholar 

  3. Araki, H.: On the algebra of all local observables. Prog. Theor. Phys.32, 844–854 (1964)

    Google Scholar 

  4. Balaban, T., Gawedzki, K.: A Low temperature expansion for the pseudoscalar Yukawa model of quantum fields in two space-time dimensions. Ann. Inst. H. Poincaré.

  5. Béllissard, J., Fröhlich, J., Gidas, B.: Soliton mass and surface tension in the (λ|φ|4)2 quantum field model. Commun. Math. Phys.60, 37–72 (1978)

    Google Scholar 

  6. Bisognano, J. J., Wichmann, E. H.: On the duality condition for a hermitian scalar field. J. Math. Phys.16, 985–1007 (1975)

    Google Scholar 

  7. Bisognano, J. J., Wichmann, E. H.: On the duality condition for quantum fields. J. Math. Phys.17, 303–321 (1976)

    Google Scholar 

  8. Buchholz, D.: Product states for local algebras. Commun. Math. Phys.36, 287–304 (1974)

    Google Scholar 

  9. Cochran, J. A.: The analysis of linear integral equations. New York: McGraw-Hill 1972

    Google Scholar 

  10. Cooper, A., Rosen, L.: The weakly coupled Yukawa2 field theory: Cluster expansion and Wightman axioms. Trans. Am. Math. Soc.234, 1–88 (1977)

    Google Scholar 

  11. Dell'Antonio, G. F.: Structure of the algebras of some free systems. Commun. Math. Phys.9, 81–117 (1968)

    Google Scholar 

  12. Doplicher, S., Haag, R., Roberts J.: Fields, observables and gauge transformations, I. Commun. Math. Phys.13, 1–23 (1969)

    Google Scholar 

  13. Doplicher, S., Haag, R., Roberts, J.: Fields, observables and gauge transformations, II. Commun. Math. Phys.15, 173–200 (1969)

    Google Scholar 

  14. Driessler, W.: Duality and absence of locally generated superselection sectors for CCR-type algebras. Commun. Math. Phys.70, 213–220 (1979)

    Google Scholar 

  15. Eckmann, J.-P., Osterwalder, K.: An Application of Tomita's theory of modular Hilbert algebras: duality for free Bose fields. J. Funct. Anal.13, 1–12 (1973)

    Google Scholar 

  16. Foit, G. J.: Ph. D. Thesis, Universität Osnabrück, 1982

  17. Glimm, J., Jaffe, A.: Quantum field theory models. Les Houches Lectures 1970. De Witt, C., Stora, R. (eds.) New York: Gordon and Breach 1970

    Google Scholar 

  18. Glimm, J., Jaffe, A.: Boson quantum field models. In: Mathematics in Contemporary Physics, Streater, R. F. (ed.) London, New York: Academic Press 1972

    Google Scholar 

  19. Glimm, J., Jaffe, A.: Self-Adjointness of the Yukawa2 Hamiltonian. Ann. Phys. 60, 321–383 (1970)

    Google Scholar 

  20. Glimm, J., Jaffe, A.: The λ(φ4)2 quantum field theory without cutoffs, III, The physical vacuum. Acta Math.125, 203–267 (1970)

    Google Scholar 

  21. Glimm, J., Jaffe, A.: The Yukawa2 quantum field theory without cutoffs. J. Funct. Anal.7, 323–357 (1971)

    Google Scholar 

  22. Glimm, J., Jaffe, A., Spencer, T.: The particle structure of the weakly coupledP(φ)2 model and other applications of high temperature expansions. In: Constructive Quantum Field Theory. Velo, G., Wightman, A. (eds.) Lecture Notes in Physics, Vol. 25, New York, Heidelberg: Springer 1973

    Google Scholar 

  23. Glimm, J., Jaffe, A., Spencer, T.: Convergent expansion about mean field theory, I and II. Ann. Phys.101, 610–630 and 631–669 (1976)

    Google Scholar 

  24. Haag, R., Kastler, D.: An Algebraic approach to quantum field theory. J. Math. Phys.5, 848–861 (1964)

    Google Scholar 

  25. Leyland, P., Roberts, J., Testard, D.: Quality for quantum free fields. Marseille CNRS preprint, CPT 78/p. 1016

  26. Magnen, J., Sénéor, R.: The Wightman axioms for the weakly coupled Yukawa model in two dimensions. Commun. Math. Phys.51, 297–313 (1976)

    Google Scholar 

  27. McBryan, O. A., Park, Y. M.: Lorentz covariance of the Yukawa2 quantum field theory. J. Math. Phys.16, 104–110 (1975)

    Google Scholar 

  28. Osterwalder, K.: Euclidean Green's functions and Wightman distributions. In: Constructive Quantum Field Theory, Velo, G., Wightman, A. (eds.) Lecture Notes in Physics, Vol. 25, Berlin, Heidelberg, New York: Springer 1973

    Google Scholar 

  29. Osterwalder, K.: Duality for free Bose fields. Commun. Math. Phys.29, 1–14 (1973)

    Google Scholar 

  30. Osterwalder, K., Schrader, R.: Euclidean Fermi fields and a Feynman-Kac formula for bosonfermion models. Helv. Phys. Acta.46, 277–302 (1973)

    Google Scholar 

  31. Osterwalder, K., Schrader, R.: Axioms for Euclidean Green's functions, I and II. Commun. Math. Phys.31, 83–112 (1973) and42, 281–305 (1975)

    Google Scholar 

  32. Powers, R. F., Størmer, E.: Free states of the canonical anticommutation relations. Commun. Math. Phys.16, 1–33 (1970)

    Google Scholar 

  33. Roberts, J. E.: The Structure of sectors reached by a field algebra. In: Cargése Lectures in Physics, Vol. 4, Kastler, D. (ed.) New York: Gordon and Breach 1970

    Google Scholar 

  34. Roberts, J. E.: Spontaneously broken gauge symmetries and super-selection rules. In: Proceedings of the International School of Mathematical Physics, Univ. of Camerino, 1974, Gallavotti, E. (ed.) Univ. of Camerino, 1976

  35. Roberts, J. E.: A Survey of local cohomology. In: Proceedings of the Conference on Mathematical Problems in Theoretical Physics, Rome 1977, (eds.) Doplicher, S. Dell'Antonio, G. F., Jona-Lasinio, G. (eds.) Lecture in Physics, Vol. 80, Berlin and New York: Springer 1978

    Google Scholar 

  36. Roberts, J. E.: Net Cohomology and its applications to field theory. In: Quantum Fields-Algebra, Processes, Streit, L. (ed.) Vienna, New York: Springer 1980

    Google Scholar 

  37. Roos, H.: Independence of local algebras in quantum field theory. Commun. Math. Phys.16, 238–246 (1970)

    Google Scholar 

  38. Sakai, S.: C* - and W*-Algebras, Berlin, Heidelberg, New York: Springer 1971

    Google Scholar 

  39. Schrader, R.: A Remark on Yukawa plus Boson selfinteraction in two space-time dimensions. Commun. Math. Phys.21, 164–170 (1971)

    Google Scholar 

  40. Schrader, R.: Yukawa quantum field theory in two space-time dimensions without cutoffs. Ann. Phys.70, 412–457 (1972)

    Google Scholar 

  41. Segal, I. E., Goodman, R. W.: Anti-locality of certain Lorentz-invariant operators. J. Math. Mech.14, 629–638 (1965)

    Google Scholar 

  42. Seiler, E.: Schwinger functions for the Yukawa model in two dimensions with space-time cutoff. Commun. Math. Phys.42, 163–182 (1975)

    Google Scholar 

  43. Seiler, E., Simon, B.: Nelson's symmetry and all that in the Yukawa2 and (φ4)3 field theories. Ann. Phys.97, 470–518 (1976)

    Google Scholar 

  44. Shale, D., Stinespring, W. F.: States on the Clifford algebra. Ann. Math.80, 365–381 (1964)

    Google Scholar 

  45. Schlieder, S.: Einige Bemerkungen über Projektionsoperatoren. Commun. Math. Phys.13, 216–225 (1969)

    Google Scholar 

  46. Streater, R. F., Wightman, A.: PCT, Spin and Statistics, and All That, New York: Benjamin 1964

    Google Scholar 

  47. Kishimoto, A., Takai, H. On the invariant Γ (α) in C*-dynamical systems. Tôhoku Math. J.30, 83–94 (1978)

    Google Scholar 

  48. Glimm, J., Jaffe, A.: The λφ 42 quantum field theory without cutoffs, IV. J. Math. Phys.13, 1568–1584 (1972)

    Google Scholar 

  49. McBryan, O. A.: Convergence of the vacuum energy density, φ-bounds and existence of Wightman functions for the Yukawa2 model. In: Les Méthodes Mathématiques de la Théorie Quantique des Champs, Proceedings of the 1975 Marseille Conference, Editions du CNRS, 1976

  50. Pedersen, G. K.: C*-Algebras and Their Automorphism Groups. London, New York: Academic Press 1979

    Google Scholar 

  51. Rideau, G.: On some representations of the anticommutation relations, Commun. Math. Phys.9, 229–241 (1968)

    Google Scholar 

  52. Heifets, E. P., Osipov, E. P.: The energy-momentum spectrum in the Yukawa2 quantum field theory. Commun. Math. Phys.57, 31–50 (1977)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Fachbereich 4, Universität Osnabrück, D-4500, Osnabrück, Germany

    Stephen J. Summers

Authors
  1. Stephen J. Summers
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by K. Osterwalder

Rights and permissions

Reprints and permissions

About this article

Cite this article

Summers, S.J. Normal product states for fermions and twisted duality for CCR- and CAR-type algebras with application to the Yukawa2 quantum field model. Commun.Math. Phys. 86, 111–141 (1982). https://doi.org/10.1007/BF01205664

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01205664

Keywords

Search

Navigation