Abstract
LetE(λg) be the vacuum energy for theP(ø)2 Hamiltonian with space cutoffg(x)≧0 and coupling constant λ≧0. For suitable families of cutoffsg→1, the vacuum energy per unit volume converges; i.e., −E(λg)/∫g(x)dx→α ∞(λ). We obtain bounds on the λ dependence ofα ∞(λ) for large and small λ. These lead to estimates forE(λg) as a functional ofg that permit a weakening of the standard regularity conditions forg. Typical of such estimates is the “linear lower bound”, −E(g)≦const ∫g(x)2 dx, valid for allg≧0 provided thatP is normalized so thatP(0)=0. Finally we show that the perturbation series forα ∞(λ) is asymptotic to second order.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Glimm, J.: Boson fields with nonlinear self-interaction in two dimensions. Commun. math. Phys.8, 12 (1968).
Glimm, J., Jaffe, A.: A λø4 quantum field theory without cutoffs. I. Phys. Rev.176, 1945 (1968).
Glimm, J., Jaffe, A.: The λ(ø4)2 quantum field theory without cutoffs. III., The Physical Vacuum. Acta Math.125, 203 (1970).
Glimm, J., Jaffe, A.: The λ(φ4)2 quantum field theory without cutoffs. IV., Perturbations of the Hamiltonian (to appear).
Glimm, J., Jaffe, A.: Quantum field theory models. In: 1970 Les Houches Lectures, De Witt, C., Stora, R. (Ed.), New York: Gordon and Breach, 1972.
Guerra, F.: Uniqueness of the vacuum energy density and Van Hove phenomenon in the infinite volume limit for two-dimensional self-coupled bose fields. Phys. Rev. Lett.28, 1213 (1972).
Guerra, F., Rosen, L., Simon, B.: Nelson's Symmetry and the infinite volume behaviour of the vacuum inP(φ)2. Commun. math. Phys.27, 10 (1972).
Nelson, E.: A quartic interaction in two dimensions. In: Mathematical Theory of Elementary Particles, R. Goodman and I. Segal, Eds., Cambridge: M.I.T. Press 1966.
Nelson, E.: Quantum fields and Markoff fields. Proc. 1971 Amer. Math. Soc. Summer Conference (Berkeley).
Nelson, E.: The free Markoff field. Princeton Univ. preprint.
Osterwalder, K., Schrader, R.: On the uniqueness of the energy density in the infinite volume limit for quantum field models. Helv. Phys. Acta, to appear.
Rosen, L.: A λφ2n field theory without cutoffs. Commun. math. Phys.16, 157 (1970).
Rosen, L.: The (φ2n)2 quantum field theory: higher order estimates. Comm. Pure Appl. Math.24, 417 (1971).
Rosen, L.: Renormalization of the Hilbert space in the mass shift model. J. Math. Phys.13, 918 (1972).
Rosen, L., Simon, B.: The (φ2n)2 field Hamiltonian for complex coupling constant. Trans. Amer. Math. Soc.165, 365 (1972).
Segal, I.: Construction of nonlinear local quantum processes. I. Ann. Math.92, 462 (1970).
Simon, B.: Essential self-adjointness of Schrödinger operators with positive potentials. Math. Ann. (to appear).
Simon, B., Hoegh-Krohn, R.: Hypercontractive semigroups and two dimensional self-coupled bose fields. J. Funct. Anal.9, 121 (1972).
Stein, E., Weiss, G.: Introduction to Fourier analysis on Euclidean spaces. Princeton: Univ. Press 1971.
Author information
Authors and Affiliations
Additional information
Research partially supported by AFOSR under Contract No. F 44620-71-C-0108.
Postal address after September 30, 1972: via A. Falcone 70, 80127, Napoli, Italy.
Rights and permissions
About this article
Cite this article
Guerra, F., Rosen, L. & Simon, B. The vacuum energy forP(ø)2: Infinite volume limit and coupling constant dependence. Commun.Math. Phys. 29, 233–247 (1973). https://doi.org/10.1007/BF01645249
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01645249