The Yang–Mills Existence and Mass Gap problem lies at the intersection of quantum field theory and mathematics. It seeks to establish a rigorous foundation for Yang–Mills theory, which underpins the Standard Model of particle physics.
The challenge is to show that for any compact simple gauge group, a non-trivial quantum Yang–Mills theory exists on ℝ⁴ and exhibits a mass gap — meaning that the lowest possible energy state above the vacuum has strictly positive energy.
Resolving this would bridge a fundamental gap between physics and pure mathematics, confirming key aspects of quantum behavior through mathematical rigor.
Honest status note:
What follows is a research hypothesis, not a completed proof. I'm publishing it here — what's
established and what's still open — because MillenniumChecked exists to test the
relational-mathematics paradigm against real, hard problems, not to showcase only what already
works.
I build the Yang–Mills Hamiltonian in the standard way — configuration space of gauge fields
modulo gauge transformations, Hilbert space L²(𝒞, μ), a self-adjoint Hamiltonian bounded below —
and analyze its spectrum. For the free theory (coupling g = 0) this gives a clean mass gap from
the oscillator frequencies, and for small g the gap survives under perturbation. I then work
toward extending this to arbitrary finite coupling.
Where things currently stand:
The free-field case (g = 0) and the small-coupling perturbative case are established using
standard, uncontested arguments — the mass gap here is genuinely there.
The gauge-invariant measure μ on the configuration space is sketched via two routes (lattice
limit, functional/Gaussian construction) and I invoke Kolmogorov's consistency theorem and
Prokhorov's tightness theorem for the lattice-to-continuum limit. What I haven't yet done is
verify their hypotheses for this specific setting — in particular, showing the local energy
contributions stay uniformly bounded as the lattice limit is taken, which is exactly the
continuum-limit question at the center of the Millennium Problem.
Self-adjointness of the Hamiltonian is currently argued at the level of "sum of non-negative
terms," not through an actual domain and closure analysis on the infinite-dimensional space.
For finite coupling g, I state a "Functional Lower Bound" — a Poincaré-type inequality
⟨ψ,Hψ⟩ ≥ λ‖ψ‖² for some λ > 0 — as the mechanism that would extend the gap beyond the
perturbative regime, citing the Friedrichs inequality and Friedrichs extension. I haven't yet
derived a value for λ or verified that these classical results, usually stated for bounded or
compact settings, transfer to this infinite-dimensional nonlinear one. This is the crux of the
problem, not a peripheral detail.
A small GUI/lattice-simulation prototype exists but currently returns placeholder values rather
than running an actual lattice computation.
In short: this page documents a direction I'm exploring, with solid results at the free and
weak-coupling end and the actual hard part of the problem — the continuum limit and a
non-perturbative spectral gap at arbitrary coupling — still open.
📌 Status
Free-field and weak-coupling mass gap: established, standard argument