Yang–Mills existence and mass gap

Let $G$ be a compact simple Lie group with Lie algebra $\mathfrak{g}$ equipped with an invariant inner product $\langle\cdot,\cdot\rangle$, let $g>0$ be a coupling constant, and consider on $\mathbb{R}^{4}$ the classical Yang\u2013Mills action

$$S(A)=\frac{1}{4g^{2}}\int_{\mathbb{R}^{4}}\left\langle F_{\mu\nu}(A),\,F^{\mu\nu}(A)\right\rangle\,d^{4}x,\qquad F_{\mu\nu}(A)=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}+[A_{\mu},A_{\nu}],$$

for $\mathfrak{g}$-valued gauge fields $A=(A_{\mu})_{\mu=0}^{3}$.

By a quantum field theory satisfying the Wightman axioms is meant the following data. A separable complex Hilbert space $\mathcal{H}$ carries a continuous unitary representation $U(a,L)$ of the (covering group of the) Poincar\u00e9 group; the self-adjoint generators $P_{0},P_{1},P_{2},P_{3}$ of the translation subgroup satisfy the spectral condition

$$P_{0}\ge 0,\qquad P_{0}^{2}-P_{j}P_{j}\ge 0,$$

and there is a vector $\Omega\in\mathcal{H}$, unique up to phase, invariant under all $U(a,L)$. For every Schwartz test function $f$ on $\mathbb{R}^{4}$ there are operators $A_{1}(f),\dots,A_{n}(f)$ which, together with their adjoints, are defined on a common dense domain $D\subseteq\mathcal{H}$ containing $\Omega$; the fields are operator-valued tempered distributions, and $\mathcal{H}$ is spanned by polynomials in the $A_{i}(f)$ applied to $\Omega$. The fields transform covariantly,

$$U(a,L)^{\dagger}A(x)U(a,L)=S(L)\,A\!\left(L^{-1}(x-a)\right),$$

for a finite-dimensional representation $S$ of the Lorentz group (or of $\mathrm{SL}(2,\mathbb{C})$ for half-integer spin), and local commutativity holds: if the supports of two smeared fields are space-like separated, the corresponding operators either commute or anticommute.

Such a theory is non-trivial if its Wightman functions are not those of a free field. Writing $H=P_{0}$ for the Hamiltonian, the theory has a mass gap $\Delta>0$ if

$$\operatorname{spec}(H)\subseteq\{0\}\cup[\Delta,\infty),$$

with $0$ a simple eigenvalue whose eigenvector is $\Omega$; equivalently, for a field $\phi$ of the theory the Euclidean two-point function decays as

$$\langle\phi(0,t)\phi(0,0)\rangle\sim\sum_{n}A_{n}\exp(-\Delta_{n}t)$$

with $\Delta_{0}=\Delta>0$ the smallest exponent.

For every compact simple gauge group $G$ there exists a non-trivial quantum Yang\u2013Mills theory on $\mathbb{R}^{4}$ with gauge group $G$, quantizing the action $S(A)$ above, which satisfies the Wightman axioms (or axiomatic properties at least as strong, such as the Osterwalder\u2013Schrader axioms for its Euclidean Green's functions), and which has a mass gap $\Delta>0$.