AI 中文总结
本文研究经典非阿贝尔规范场与色荷费米子相互作用时物质的稳定性，发现稳定性性质与电荷情形一致，关键工具是对矩阵值电荷的静电力不等式。
AI 中文摘要
物质的稳定性是指$N$个带电量子粒子的基态能量具有下界$-CN$，其中$C$与$N$无关。我们探讨这一性质是否适用于与经典非阿贝尔规范场相互作用的色荷费米子。该场是与物质同等地位的动力变量，能量同时对两者取极小。电荷是矩阵，且每对粒子之间存在吸引性的色通道。非相对论无自旋费米子对耦合常数$\alpha$的任意值都是稳定的。相对论无自旋费米子和非相对论自旋-$\frac12$费米子在$\alpha$较小时稳定，在$\alpha$较大时不稳定。对于狄拉克费米子，正能子空间的选择起决定作用：若采用全狄拉克算子的正能子空间，则它们在$\alpha$较小时稳定；若采用自由正能子空间，则对任意$\alpha$都不稳定。两类玻色子则不稳定。在每种情形下，结果与电荷情形相同，尽管阿贝尔证明的关键步骤失效。关键工具是对矩阵值电荷的静电力不等式，该不等式对场一致成立，用协变调和延拓的表面电荷取代牛顿定理中均匀带电球体的电荷分布。稳定性结果对所有紧致规范群成立，且大部分结果对具有有限能量的非光滑场也成立。
英文摘要
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.
Comments49 pages
