Stability of Matter Interacting with Classical Non-Abelian Gauge Fields

Abstract

Stability of matter is the property that the ground state energy of $N$ charged quantum particles is bounded below by $-CN$, with $C$ independent of $N$. We ask whether this holds for colour-charged fermions interacting with a classical non-abelian gauge field. The field is a dynamical variable on the same footing as the matter, and the energy is minimized over both. The charges are matrices, and every pair of particles has an attractive colour channel. Non-relativistic spinless fermions are stable for every value of the coupling constant $α$. Relativistic spinless fermions and non-relativistic spin-$\frac12$ fermions are stable for small $α$ and unstable for large $α$. For Dirac fermions the choice of positive-energy subspace decides: with that of the full Dirac operator they are stable for small $α$, with the free one unstable for every $α$. Two species of bosons are unstable. In every case the outcome is the same as for electric charges, although essential steps of the abelian proofs fail. The key tool is an electrostatic inequality for matrix-valued charges, uniform in the field, which replaces the uniformly charged sphere of Newton's theorem by the surface charge of a covariant harmonic extension. The stability results hold for every compact gauge group, and most of them for non-smooth fields of finite energy.