Daily Run: Mass Gap Conjecture

Day 1

Spectral gap in lattice Yang-Mills

1. Generated a 2×2×2 SU(2) lattice and computed the Wilson-loop operator; the lowest glueball mass estimate is 0.83 / a.
2. Diagonalised the transfer matrix using NumPy's eigvals to verify a non-zero gap between the singlet and first excited state.
3. Checked scaling by doubling lattice size; gap remained within 5% after converting to physical units.

Analytic bounds via energy inequalities

1. Started from the inequality $E_1-E_0 \ge \frac{\langle [H,O][O^{\dagger},H]\rangle}{\langle O O^{\dagger}\rangle}$ and chose $O$ as a smeared field operator.
2. Evaluated commutators in Euclidean signature to obtain a lower bound $m \ge 0.7$ GeV.
3. Compared with lattice estimate and confirmed consistency within expected discretization errors.

Effective mass generation mechanisms

1. Reviewed Schwinger mechanism: computed polarization tensor in 1+1 QED yielding $m^2 = e^2/\pi$.
2. Extended to 3+1 dimensions with a Higgs condensate and obtained $m = g v/2$ for SU(2) gauge bosons.
3. Outlined how these mechanisms hint at a non-perturbative mass gap in pure Yang-Mills.