Communications in Mathematical Physics

, Volume 68, Issue 1, pp 45–68 | Cite as

The classical field limit of scattering theory for non-relativistic many-boson systems. II

  • J. Ginibre
  • G. Velo


We study the classical field limit of non relativistic many-boson theories in space dimensionn≧3, extending the results of a previous paper to more singular interactions. We prove the expected results: when ħ tends to zero, the quantum theory tends in a suitable sense to the corresponding classical field theory, and the quantum fluctuations are governed by the equation obtained by linearizing the quantum evolution equation around the classical solution. These results hold uniformly in time and therefore apply to scattering theory. The interactions considered here are so singular as to require a change of domain in order to define the generator of the evolution of the fluctuations, but sufficiently regular so that no energy renormalization is needed.


Neural Network Statistical Physic Field Theory 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.
    Friedrichs, K.O.: Perturbation of spectra in Hilbert space. Providence, RI: Am. Math. Soc. 1965Google Scholar
  2. 2.
    Ginibre, J., Velo, G.: The classical field limit of scattering theory for non-relativistic many-boson systems. I. Commun. Math. Phys.66, 37–76 (1979)Google Scholar
  3. 3.
    Ginibre, J., Velo, G.: On a class of non linear Schrödinger equations with non local interactions. Preprint, 1979Google Scholar
  4. 4.
    Glimm, J., Jaffe, A.: Quantum Field models. In: Statistical mechanics and field theory, pp. 1–108. Les Houches Lectures 1970. De Witt, C., Stora, R. (eds.) London: Gordon and Breach 1971Google Scholar
  5. 5.
    Hepp, K.: Commun. Math. Phys.35, 265–277 (1974)Google Scholar
  6. 6.
    Kato, T.: J. Fac. Sci. Univ. Tokyo, Sec. I,17, 241–258 (1970)Google Scholar
  7. 7.
    Nelson, E.: J. Funct. Anal.11, 211–219 (1972)Google Scholar
  8. 8.
    Ruelle, D.: Statistical mechanics. New York: Benjamin 1969Google Scholar

Copyright information

© Springer-Verlag 1979

Authors and Affiliations

  • J. Ginibre
    • 1
  • G. Velo
    • 2
  1. 1.Laboratoire de Physique Théorique et Hautes EnergiesUniversité de Paris-SudOrsayFrance
  2. 2.Istituto di Fisica A. RighiUniversità di Bologna, and INFN, Sezione di BolognaI-BolognaItaly

Personalised recommendations