[Paper Review] Exact solutions for classical Yang-Mills fields

Marco Frasca|arXiv (Cornell University)|Sep 8, 2014

Particle physics theoretical and experimental studies6 references3 citations

TL;DR

This paper presents exact classical solutions to the Yang-Mills equations that exhibit a massive dispersion relation despite the underlying theory being massless. Using a gauge-invariant ansatz involving Jacobi elliptic functions, the solutions are shown to be exact in any gauge, with degeneracy among field components restored only in the Lorenz (Landau) gauge, providing a classical foundation for the mass gap in quantum Yang-Mills theory.

ABSTRACT

We provide a set of exact solutions of the classical Yang-Mills equations. They have the property to satisfy a massive dispersion relation and hold in all gauges. These solutions can be used to describe the vacuum of the quantum Yang-Mills theory and so, they provide a general framework to build a quantum field theory. The components of the field become separated on a generic gauge but are all equal just in the Lorenz (Landau) gauge.

Motivation & Objective

• To derive exact classical solutions for the Yang-Mills equations that are valid in all gauges.
• To establish a connection between classical Yang-Mills solutions and the vacuum structure of quantum Yang-Mills theory.
• To demonstrate that these solutions satisfy a massive dispersion relation, implying a mass gap in the quantum theory.
• To resolve prior criticisms of a scalar-to-Yang-Mills mapping by proving its exactness under proper gauge conditions.
• To generalize Smilga's solutions to arbitrary gauges while preserving exactness and physical consistency.

Proposed method

• An ansatz is introduced where diagonal components of the gauge field are proportional to the Jacobi elliptic function $\text{sn}(p\cdot x, -1)$, with coefficients $X$, $Y$, $Z$.
• The Yang-Mills equations are reduced to a system of algebraic equations by substituting the ansatz into the field equations in a general gauge.
• The dispersion relation $p^2 = \mu^2 g$ is imposed, linking momentum to coupling and an integration constant $\mu$.
• The resulting algebraic system is solved exactly, showing consistency across all components for any gauge parameter $\alpha$.
• The solutions are verified to satisfy the full Yang-Mills equations in all gauges, with exactness maintained via proper functional form and gauge choice.
• The Lorenz gauge ($\alpha = 1$) restores degeneracy among components, recovering the symmetric solution from prior works.

Experimental results

Research questions

• RQ1 Can exact classical solutions of the Yang-Mills equations be constructed that satisfy a massive dispersion relation in all gauges?
• RQ2 How does the mapping from scalar field solutions to Yang-Mills solutions behave under general gauge fixing?
• RQ3 What is the role of the gauge parameter $\alpha$ in preserving or breaking component degeneracy in the solutions?
• RQ4 Can the classical solutions derived here describe the vacuum expectation value in quantum Yang-Mills theory?
• RQ5 Does the nonlinear structure of Yang-Mills theory naturally lead to a mass gap at the classical level?

Key findings

• Exact classical solutions of the Yang-Mills equations are constructed for all gauges, with the field components expressed as $A^{a}_{a} = f_a \cdot \text{sn}(p\cdot x, -1)$.
• The solutions satisfy a massive dispersion relation $p^2 = \mu^2 g$, even though the original theory is massless, indicating spontaneous mass generation at the classical level.
• In the Lorenz gauge ($\alpha = 1$), the components become degenerate: $A^1_1 = A^2_2 = A^3_3 = \frac{\mu}{(2g^2)^{1/4}} \cdot \text{sn}(p\cdot x, -1)$.
• For $\alpha \neq 1$, the components are no longer degenerate, but the solutions remain exact and depend on the gauge parameter through the algebraic constraints.
• The mapping from scalar $\phi^4$ solutions to Yang-Mills solutions is shown to be exact when the correct gauge-dependent ansatz is used, resolving prior criticisms.
• The solutions provide a classical foundation for the mass gap in quantum Yang-Mills theory, as they describe a nontrivial vacuum expectation value with massive dispersion.