Yang–Mills Mass Gap

Why non-abelian gauge fields have particle-like excitations with mass.

Last updated: 2025-10-25

Quantum Yang–Mills theory underlies the strong nuclear force. The mass gap problem asks for a rigorous construction in 4D with a positive lower bound on the energy spectrum (a mass gap).

It bridges abstract gauge symmetry and the observed spectrum of particles.

The model

With compact gauge group G, the classical Lagrangian density is

Construct the corresponding quantum theory nonperturbatively in 4D.
Show \(\exists\, m>0\) such that the spectrum obeys \(E\ge m\) for one-particle states (mass gap).

\mathcal{L} = -\tfrac{1}{4} F_{\mu\nu}^a F^{a\,\mu\nu},\quad F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu + [A_\mu, A_\nu]

Confinement sketch: flux tube between color charges.

Why it matters

A proof would ground QCD-like phenomena (confinement, hadron masses) in mathematics, linking lattice evidence to a continuum construction.

The model

Why it matters

Further reading