Existence and Mass Gap for Yang–Mills Theory
Field
Mathematical physics
Problem overview
The Yang–Mills Existence and Mass Gap problem asks whether non-abelian gauge theories in four dimensions admit:
- global mathematical existence in a rigorous quantum-field framework
- a strictly positive mass gap separating the vacuum from the lowest nontrivial excitation
Yang–Mills theory lies at the core of modern particle physics, yet the mathematical origin of confinement-scale mass generation remains unresolved. A central challenge is to understand how nonlinear gauge interactions can generate stable positive spectral scales without inserting them by hand.
Why the problem is difficult
The problem sits at the intersection of nonlinear partial differential equations, spectral theory, and quantum field theory.
The main difficulties include:
- strong nonlinear self-interaction of gauge fields
- gauge redundancy, which complicates the choice of physical variables
- controlling behavior across many spatial and energy scales
- passing rigorously from finite approximations to continuum theories
- proving that a positive mass scale emerges dynamically rather than by assumption
Related physical theories strongly suggest that such a gap should exist. One does not observe freely propagating massless strong-force gauge carriers at low energies, and lattice gauge calculations indicate a positive nonzero excitation scale. The open problem is to establish this rigorously in four-dimensional Yang–Mills mathematics.
Coherence-geometric approach
In the Coherence Geometry (CG) framework, gauge-type systems are modeled as interacting coherence channels governed by a variational stability structure.
Rather than beginning with fields as primitive objects, the approach studies how collective phase-locking, compatibility constraints, and local curvature can generate effective excitation structure.
From this viewpoint, a mass gap is interpreted as the emergence of robust quadratic curvature after internal modes organize into stable locked sectors.
Key organizing ideas
- local coupling alone need not generate a persistent positive gap
- relative-channel locking can create stable gapped internal modes
- remaining soft collective modes determine whether the full system remains gapless or becomes massive
- spectral scales arise from structural curvature of the stabilized configuration space
Author’s Intuition
Many systems remain dynamically active even after reaching stability. They may continue to fluctuate or rebalance internally at low cost, while still resisting larger changes unless additional energy is supplied.
A useful analogy appears in protein folding: a folded protein does not become motionless, but continues to move within a stable basin. Small adjustments remain possible, while larger rearrangements require overcoming stronger barriers. Another familiar analogy is the photoelectric effect, where internal activity may persist while a definite energy input is required to enter a qualitatively different state.
The Yang–Mills mass-gap question asks whether the gauge field behaves similarly: local motion may remain possible, yet any genuinely new excitation must cross a positive minimum energy threshold.
The Local Hessians paper studies whether such thresholds can arise directly from interaction geometry. It shows that simply coupling neighboring parts of a system is usually not enough. But when multiple internal components lock together strongly enough to eliminate freely drifting collective modes, a true gap can emerge.
This suggests that mass gaps are not a generic consequence of locality alone, but of structured locking that removes cost-free large-scale motion.
Projection into standard Yang–Mills language
When translated into conventional operator and lattice terminology, this perspective leads to a structural question:
Which local Hessian configurations admit uniform positive spectral gaps in large-volume limits?
The projected analysis shows a sharp dichotomy:
- nearest-neighbor gradient locality produces Laplacian-type spectra whose lowest nonzero eigenvalue typically scales like \(L^{-2}\), becoming gapless in infinite volume;
- multi-component locking can gap relative internal sectors uniformly;
- a full uniform gap requires positive quadratic curvature in the final remaining soft sector.
This identifies a precise structural mechanism relevant to mass-gap phenomena: locality alone is insufficient, while curvature in the surviving low-energy sector is decisive.
Status
Exploratory study available
Technical Review
These documents are shared for technical evaluation. Corrections, counterexamples, simplifications, literature pointers, and independent verification are all welcome.
Available document
- Title: Local Coherence Hessians: A Structural Classification of Spectral Gaps
- Identifier: CGI-RSR-000003
- Author: Barry L. Petersen
- Length: 15 pages
- Version: v1.0
- Repository: Zenodo (DOI link)
- Citation: Petersen, B. L. (2026). Local Coherence Hessians: A Structural Classification of Spectral Gaps. Zenodo. https://doi.org/10.5281/zenodo.19969141
The Zenodo-hosted PDF is the authoritative technical record. This page is descriptive only.
Relationship to the Clay problem
This work engages the Yang–Mills problem in the spirit for which it was established: identifying whether deeper structural principles clarify the existence and mass gap questions that resist resolution under traditional formulations.
The financial prize associated with the Clay Millennium Problems is not a motivating factor. The motivating factor is determining whether coherence-regulated structure provides a natural resolution of one of the central open problems in mathematical physics.
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