Advertisement

Communications in Mathematical Physics

, Volume 2, Issue 1, pp 271–300 | Cite as

The Schrödinger equation for quantum fields with nonlinear nonlocal scattering

  • James Glimm
Article
  • 82 Downloads

Abstract

This paper considers perturbationsH=H0V of the Hamiltonian operatorH0 of a free scalar Boson field.V is a polynomial in the annihilation creation operators. Terms of any order are allowed inV, but point interactions, such as ∫:0(x)4(x)4:dx, are not considered. Unnormalized solutions for the Schrödinger equation are found. For ε→0, these solutions have a partial asymptotic expansion in powers of ε. The set of all possible pertubation termsV forms a Lie algebra. General properties of this Lie algebra are investigated.

Keywords

Neural Network Statistical Physic Complex System Nonlinear Dynamics Asymptotic Expansion 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Bibliography

  1. [1]
    Friedrichs, K. O.: Mathematical aspects of the quantum theory of fields. New York: Interscience 1953.Google Scholar
  2. [2]
    -- Perturbation of spectra in Hilbert space, Lectures in applied mathematics, Vol. III, American Mathematical Society, 1965.Google Scholar
  3. [3]
    Jacobson, N.: Lie algebras. New York: Interscience 1962.Google Scholar
  4. [4]
    Jost, R.: The general theory of quantized fields, Lectures in applied mathematics, Vol. IV, American Mathematical Society, 1965.Google Scholar
  5. [5]
    Kristensen, P., L. Mejlbo, andE. Poulsen: Tempered distributions in infinitely many dimensions I, Comm. Math. Phys.1, 175–215 (1965).Google Scholar
  6. [6]
    —— —— —— Tempered distributions in infinitely many dimensions II, Math. Scand.14, 129–150 (1964).Google Scholar
  7. [7]
    -- -- -- Tempered distributions in infinitely many dimensions III, to appear.Google Scholar
  8. [8]
    Nelson, E.: Interaction of nonrelativistic particles with a scalar field. J. Math. Phys.5, 1190–1197 (1964).Google Scholar
  9. [9]
    Wightman, A.: Introduction to some aspects of the relativistic dynamics of quantized fields, Institute des Hautes Études Scientifiques, Bures-sur-Yvette (Revised notes for lectures at the French summer school of theoretical physics, Cargèse, Corsica, July 1964).Google Scholar

Copyright information

© Springer-Verlag 1966

Authors and Affiliations

  • James Glimm
    • 1
    • 2
  1. 1.Department of MathematicsMassachusetts Institute of TechnologyUSA
  2. 2.Matematisk InstitutAarhus UniversitetDenmark

Personalised recommendations