Abstract
We consider the infinite volume Dirichlet (or half-Dirichlet)P(φ)2 quantum field theory withP(X)=aX 4+bX 4+bX 2−μX(a>0). If μ≠0 there is a positive mass gap in the energy spectrum. If the gap vanishes as μ → 0, it goes to zero no faster than linearly yielding a bound on a critical exponent.
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References
Fisher, M.E.: Rep. Progr. Phys.30, 615–730 (1967)
Friedrichs, K.: Perturbation of spectra in Hilbert space. A.M.S. 1965
Frohlich, J.: Schwinger functions and their generating functionals. I. Helv. Phys. Acta (to appear)
Glimm, J., Jaffe, A.: Ann. Math.91, 362–401 (1970)
Glimm, J., Jaffe, A.: J. Math. Phys.13, 1568–1584 (1972)
Glimm, J., Jaffe, A.: Phys. Rev. D10, 536 (1974)
Glimm, J., Jaffe, A., Spencer, T.: The Wightman axioms and the particle structure in theP(φ)2 quantum field model. Ann. model. Ann. Math. (to appear)
Glimm, J., Jaffe, A., Spencer, T.: The particle structure of the weakly coupledP(φ)2 model and other applications of high temperature expansions. II. The cluster expansion in [28]
Guerra, F., Rosen, L., Simon, B.: Commun. math. Physics27, 10–22 (1972)
Guerra, F., Rosen, L., Simon, B.: TheP(φ)2 Euclidean quantum field theory as classical statistical mechanics. Ann. Math. (to appear)
Guerra, F., Rosen, L., Simon, B.: Boundary conditions in theP(φ)2 Euclidean quantum field theory (in preparation)
Heins, M.: Selected topics in the classical Theory of functions of a complex variable. New York: Holt 1962
Kato, T.: Perturbation theory for linear operators. Berlin-Heidelberg-New York: Springer 1966
Lebowitz, J.L., Penrose, O.: Phys. Rev. Letters31, 749–752 (1973)
Penrose, O., Lebowitz, J.L.: On the exponential decay of correlation functions. Yeshiva University (preprint)
Penrose, O., McElvey, J.S.N.: J. Phys. A1, 661–664 (1968)
Rado, T.: Subharmonic functions. Berlin: Springer 1937
Reed, M., Simon, B.: Methods of modern mathematical physics. I. Functional analysis. New York: Academic Press 1972; II. Fourier analysis; Self-adjointness. New York: Academic Press 1975; III. Analysis of operators (in preparation)
Ruelle, D.: Statistical mechanics. New York: Benjamin 1969
Simon, B.: Commun. math. Phys.31, 127–136 (1973)
Simon, B.: Correlation inequalities and the mass Gap inP(φ)2. II. Uniqueness of the vacuum for a class of strongly coupled theories. Ann. Math. (to appear)
Simon, B.: J. Funct. Anal.12, 335–339 (1973)
Simon, B.: TheP(φ)2 Euclidean (quantum) field theory. Princeton: University Press 1974
Simon, B., Griffiths, R.: Commun. math. Phys.33, 145–164 (1973)
Spencer, T.: Commun. math. Phys.39, 63–76 (1974)
Stanley, E.: Introduction to phase transitions and critical phenomena. Oxford: Oxford University Press 1971
Titchmarsh, E.C.: The theory of functions. First Edition Oxford: Oxford University Press 1932
Velo, G., Wightman, A.S.: Constructive quantum field theory. Lecture Notes in Physics No. 25. Berlin-Heidelberg-New York: Springer 1973
Whittaker, E.T., Watson, G.N.: A course of modern analysis. Cambridge: Cambridge University Press 1902
Vesentini: Boll. Un. Mat. Ital. (4)1, 427 (1968)
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Communicated by A. S. Wightman
A Sloan Foundation Fellow; partially supported by USNSF under contract No. GP 39048.
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Guerra, F., Rosen, L. & Simon, B. Correlation inequalities and the mass gap inP (φ)2 . Commun.Math. Phys. 41, 19–32 (1975). https://doi.org/10.1007/BF01608544
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DOI: https://doi.org/10.1007/BF01608544