Communications in Mathematical Physics

, Volume 5, Issue 5, pp 343–386 | Cite as

Yukawa coupling of quantum fields in two dimensions. I

  • James Glimm


A renormalization procedure is proposed. It gives rigorous mathematical meaning to the infinite cancellations in this model. A space cutoff is introduced in the interaction termV and soV has the form\(\int\limits_{\left| x \right| \leqq K} {V\left( x \right)} dx\), but there are no momentum cutoffs inV. There is an infinite constant and an infinite boson mass renormalization in this model. The main result is that the renormalized Hamiltonian is rigorously defined as a bilinear form in the Fock Hilbert space.


Neural Network Statistical Physic Hilbert Space Complex System Nonlinear Dynamics 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Berezin, F.: On the Lee model. Mat. Sbornik60, 425–446 (1963).Google Scholar
  2. 2.
    Friedrichs, K.: Perturbation of spectra in Hilbert space. Am. Math. Soc. Providence, 1965.Google Scholar
  3. 3.
    Glimm, J.: The Schrödinger equation for quatum fields with nonlinear nonlocal scattering. Commun. Math. Phys.2, 271–300 (1963).Google Scholar
  4. 4.
    Kastler, D.: Introduction a l'electrodynamique Quantique. Paris: Dunod 1961.Google Scholar
  5. 5.
    Kristensen, P., L. Mejlbo, andE. Poulsen: Tempered distributions in infinitely many dimensions II. Math. Scand.14, 129–150 (1964).Google Scholar
  6. 6.
    Lanford, O.: Construction of quantum fields interacting by a cutoff Yukawa coupling. Princeton University Thesis 1966.Google Scholar
  7. 7.
    Nelson, E.: A quartic interaction in two dimensions. In: Mathematical theory of elementary particles, ed. byR. Goodman andI. Segal, pp. 69–73. M.I.T. Press 1965.Google Scholar
  8. 8.
    Streater, R., andA. Wightman: PCT, spin and statistics and all that. New York: Benjamin Inc. 1964.Google Scholar
  9. 9.
    Wightman, A.: Introduction to some aspects of the relativistic dynamics of quantized fields. Institute des Hautes Etudes Scientifiques, Bures-sur-Yvette (Revised notes for lectures at the French summer school of theoretical physics, Cargese, Corsica, July 1964).Google Scholar

Copyright information

© Springer-Verlag 1967

Authors and Affiliations

  • James Glimm
    • 1
  1. 1.Mass. Institute of TechnologyCambridgeUSA

Personalised recommendations