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Communications in Mathematical Physics

, Volume 22, Issue 1, pp 1–22 | Cite as

The energy momentum spectrum and vacuum expectation values in quantum field theory, II

  • James Glimm
  • Arthur Jaffe
Article

Abstract

We prove that theP(ϕ)2 quantum field theory satisfies the spectral condition. The space time translationa=(x, t) is implemented by the unitary groupU(a)=exp(itH−ixP), and the joint spectrum of the energy operatorH and the momentum operatorP is contained in the forward cone. We also obtain bounds on certain vacuum expectation values of products of field operators. Our proofs involve an analysis of the limitV→∞ for approximate theories in a periodic box of volumeV. Assuming the existence of a uniform mass gap, we are able to establish all the Wightman axioms with the exception of the Lorentz invariance of the vacuum.

Keywords

Neural Network Statistical Physic Complex System Quantum Field Theory Nonlinear Dynamics 
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References

  1. 1.
    Araki, H.: On the algebra of all local observables. Prog. Theor. Phys.32, 844–854 (1964).Google Scholar
  2. 2.
    Glimm, J., Jaffe, A.: The λ(ϕ4)2 quantum field theory without cutoffs II: The field operators and the approximate vacuum. Ann. Math.91, 362–401 (1970).Google Scholar
  3. 3.
    —— —— The λ(ϕ4)2 quantum field theory without cutoffs III: The physical vacuum. Acta Math.125, 203–267 (1970).Google Scholar
  4. 4.
    Glimm, J., Jaffe, A.: The energy momentum spectrum and vacuum expectation values in quantum field theory. J. Math. Phys.11, 3335–3338 (1970).Google Scholar
  5. 5.
    —— —— Field Theory Models, in the 1970 Les Houches Lectures, C. Dewitt and R. Stora, eds. New York: Gordon & Breach Sci. Publ. 1971.Google Scholar
  6. 6.
    Powers, R., Størmer, E.: Free states of the canonical anticommutation relations. Commun. Math. Phys.16, 1–33 (1970).Google Scholar
  7. 7.
    Rosen, L.: The (ϕ2n)2 quantum field theory without cutoffs: Higher order estimates. Commun. Pure Appl. Math., to appear.Google Scholar
  8. 8.
    Van Daele, A.: Quasi-equivalence of quasi-free states on the Weyl algebra. Commun. Math. Phys.21, 171–191 (1971).Google Scholar
  9. 9.
    Araki, H.: On quasifree states of the canonical commutation relations (II), to appear.Google Scholar

Copyright information

© Springer-Verlag 1971

Authors and Affiliations

  • James Glimm
    • 1
  • Arthur Jaffe
    • 2
  1. 1.Courant Institute of Mathematical SciencesNew York UniversityNew York
  2. 2.Lyman Laboratory of PhysicsHarvard UniversityCambridge

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