Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Renormalized Four Dimensional Quantum Yang-Mills Theory and Mass Gap

Simone Farinelli

Abstract

A quantization procedure for the Yang-Mills equations for the Minkowski space $\mathbf{R}^{1,3}$ is carried out in such a way that field maps satisfying Wightman axioms of Constructive Quantum Field Theory can be obtained. Moreover, by removing the infrared and ultraviolet cutoffs, the spectrum of the corresponding (non-local) QCD Hamilton operator is proven to be positive and bounded away from zero, except for the case of the vacuum state, which has vanishing energy level. The whole construction is invariant for all gauge transformations preserving the Coulomb gauge. As expected from QED, if the coupling constant converges to zero, then so does the mass gap. This is the case for the running coupling constant leading to asymptotic freedom.

Renormalized Four Dimensional Quantum Yang-Mills Theory and Mass Gap

Abstract

A quantization procedure for the Yang-Mills equations for the Minkowski space is carried out in such a way that field maps satisfying Wightman axioms of Constructive Quantum Field Theory can be obtained. Moreover, by removing the infrared and ultraviolet cutoffs, the spectrum of the corresponding (non-local) QCD Hamilton operator is proven to be positive and bounded away from zero, except for the case of the vacuum state, which has vanishing energy level. The whole construction is invariant for all gauge transformations preserving the Coulomb gauge. As expected from QED, if the coupling constant converges to zero, then so does the mass gap. This is the case for the running coupling constant leading to asymptotic freedom.

Paper Structure

This paper contains 21 sections, 46 theorems, 273 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Yang-Mills Connections
  3. Definitions, Existence and Uniqueness
  4. Hamiltonian Formulation for the Minkowski Space
  5. Axioms of Constructive Quantum Field Theory
  6. Wightman Axioms
  7. Gaussian Random Processes
  8. Fock Space
  9. Wick Products and Wiener-Itô-Segal Isomorphism
  10. Osterwalder-Schrader Axioms
  11. Quantization of Yang-Mills Equations and Positive Mass Gap
  12. Quantization
  13. Construction of a Complete Set of Generalized Eigenvectors
  14. Construction of the Probability Measure under the Ultraviolet and Infrared Cut Offs and Regularization
  15. Infrared Cutoff Removal and Reconstruction of a Selfadjoint Hamiltonian
  16. ...and 6 more sections

Key Result

Proposition 2.1

For a simple Lie-group as structure group let $A$ be a connection over the Minkowskian $\mathbf{R}^4$ satisfying the Coulomb gauge, and assume that $A_i^a(t,\cdot)\in C^{\infty}(\mathbf{R}^3,\mathbf{R})\cap L^2(\mathbf{R}^3,\mathbf{R})$ for all $i=1,2,3$, all $a=1,\dots K$ and any $t\in\mathbf{R}$. is essentially self adjoint and elliptic for any time parameter $t\in\mathbf{R}$. Its spectrum lies

Theorems & Definitions (126)

  • Conjecture 1
  • Definition 2.1: Yang-Mills Connection
  • Remark 2.1: Local Representations of Connections on Vector and Principle Fibre Bundles
  • Remark 2.2
  • Remark 2.3
  • Definition 2.2: Coulomb Gauge
  • Definition 2.3: Transverse Projector
  • Remark 2.4
  • Proposition 2.1
  • Remark 2.5
  • ...and 116 more