Topics on euclidean classical field equations with unique vacuua

  • Jeffrey Rauch
  • David N. Williams
Session III
Part of the Lecture Notes in Physics book series (LNP, volume 106)


We discuss the real classical field equation (-δ+μ2)φ+λF(φ) = f, where μ2>o, λ≥o, F ∈ c(R), and aF (a)≥0 for all a ∈ R, and where the source function f belongs to various function spaces contained in the Sobolev space H−1 (Rd). We review a number of results whose proofs are to appear elsewhere, on existence of a solution φ ∈ Hl(Rd), the correspondences between the function spaces of f and φ (regularity properties), contractivity properties and uniqueness of φ, and analytic dependence on λ and functional differentiability in f. We mention some of the ideas from the proofs, including a few alternate methods we have not discussed elsewhere. We prove a new result on positivity preservingness: if f is a nonnegative measure in H−1(Rd), then 0 ≤ φ ∈ H1(Rd), a result that is also valid for μ2 = 0. Finally, we give an intuitive interpretation of some of the results for the case of d = 1 dimension.


Point Particle Intuitive Interpretation Contraction Mapping Principle Euclidean Path Unique Vacuum 
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.
    B. Simon, The P(ø) 2 Euclidean (Quantum) Field Theory, Princeton University Press, Princeton, 1974.Google Scholar
  2. 2.
    D. N. Williams, The Euclidean loop expansion for massive λ:Φ44: Through one loop. Commun. math.Phys. 54, 193–218 (1977).Google Scholar
  3. 3.
    J. Rauch and D. N. Williams, Euclidean Nonlinear Classical Field Equations with Unique Vacuum. Commun. math Phys., to appear (1978).Google Scholar
  4. 4.
    I. V. Volovich, Classical Equations of Euclidean Field Theory. Theoretical and Mathematical Physics (translated from Russian) 34, 9–14 (1978).Google Scholar

Copyright information

© Springer-Verlag 1979

Authors and Affiliations

  • Jeffrey Rauch
    • 1
  • David N. Williams
    • 2
  1. 1.Department of MathematicsUniversity of MichiganMichigan
  2. 2.The Harrison M. Randall Laboratory of PhysicsUniversity of MichiganMichigan

Personalised recommendations