Abstract
Following 't Hooft we extend the Borel sum and the Watson-Nevanlinna criterion by allowing distributional transforms. This enables us to prove that the characteristic function of the measure of anyg −2 φ 4 finite lattice field is the sum of a power series expansion obtained by fixing exponentially small terms in the coefficients. The same result is obtained for the trace of the double well semigroup approximated by then th order Trotter formula.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Beltrami, E.J., Wohlers, M.R.: Distributions and the boundary values of analytic functions. New York: London: Academic Press 1966
Boas, R., Jr.: Entire functions. New York: Academic Press 1954
Brézin, E., Parisi, G., Zinn-Justin, J.: Perturbation theory at large orders for a potential with degenerate minima. Phys. Rev.16 D, 408 (1977)
Colombeau, J.F.: New generalized functions and multiplication of distributions. Amsterdam: North-Holland 1984
Crutchfield, W.Y. II.: No horn of singularities for the double well anharmonic oscillator. Phys. Lett.77 B, 109–113 (1978)
Crutchfield, W.Y. II.: Method for Borel-summing instanton singularities. Introduction. Phys. Rev.19 D, 2370–2384 (1979)
Eckmann, J.-P., Wittwer, P.: Computer methods and Borel summability applied to Feigenbaum equation. Berlin, Heidelberg, New York: Springer 1985
Graffi, S., Grecchi, V.: The Borel sum of the double-well perturbation series and the Zinn-Justin conjecture. Phys. Lett.121 B, 410–414 (1983)
Hardy, G.H.: Divergent series. Oxford: Clarendon Press 1949
't Hooft, G.: The ways of subnuclear physics. Proceedings of the international school of subnuclear physics, Erice (1979), Zichichi, A. (ed.), pp. 943–971. New York: Plenum 1979
't Hooft, G.: Private communication via B. Simon
Khuri, N.N.: Zeros of the Gell-Mann-Low function and Borel summations in renormalizable theories. Phys. Lett.82 B, 83–88 (1979)
Nevanlinna, F.: Ann. Acad. Sci. Fenn.12 A, No. 3 (1918–1919)
Reed, M., Simon, B.: Methods of modern mathematical physics. Vol. IV. New York: Academic Press 1978
Rosen, J.: Proc. Am. Math. Soc.66, 114–118 (1977)
Simon, B.: Functional integration and quantum physics. New York: Academic Press 1979
Simon, B.: Int. J. Quantum Chem.21, 3–25 (1982)
Sokal, A.: An improvement of Watson's theorem on Borel summability. J. Math. Phys.21, 261–263 (1980)
Author information
Authors and Affiliations
Additional information
Communicated by G. Parisi
Rights and permissions
About this article
Cite this article
Caliceti, E., Grecchi, V. & Maioli, M. The distributional Borel summability and the large coupling φ4 lattice fields. Commun.Math. Phys. 104, 163–174 (1986). https://doi.org/10.1007/BF01210798
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01210798