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Discrete Dyson-Schwinger equations

Marco Frasca

Abstract

We develop the discrete set of Dyson-Schwinger equations for scalar fields. We show that their solution is Gaussian as expected from the theorems of Aizenman and of Aizenman and Duminil-Copin for $d\ge 4$. Extension to lower dimensionality fails, as it should, by observing the the triviality theorems used in our proof are not applicable in such cases.

Discrete Dyson-Schwinger equations

Abstract

We develop the discrete set of Dyson-Schwinger equations for scalar fields. We show that their solution is Gaussian as expected from the theorems of Aizenman and of Aizenman and Duminil-Copin for . Extension to lower dimensionality fails, as it should, by observing the the triviality theorems used in our proof are not applicable in such cases.
Paper Structure (12 sections, 95 equations)

This paper contains 12 sections, 95 equations.

Table of Contents

  1. Introduction
  2. Discrete Scalar Field Theory
  3. Classical fields
  4. Solution of the equation of motion
  5. Higher Functional Derivatives
  6. Quantum fields
  7. Technique
  8. Gaussianity
  9. Constant background
  10. Non-trivial background
  11. Conclusions
  12. Solution of the homogeneous Lamé equation on the lattice