Area law for Wilson loops in 4D lattice gauge theory at weak coupling
In plain words
Quark confinement shows up mathematically as a loop observable that decays with the area enclosed by the loop. This is proven only when the coupling is strong, far from the regime that describes nature.
Precise statement
For 4D lattice $\mathrm{SU}(2)$ or $\mathrm{SU}(3)$ gauge theory with Wilson action at large $\beta = 2N/g^{2}\ (N = 2\ \mathrm{or}\ 3)$, prove $\langle W(C)\rangle \le \exp(-\sigma(\beta)\operatorname{Area}(C))$ with string tension $\sigma(\beta) > 0$ for all rectangular loops $C$, uniformly in the infinite-volume limit. Answer: a proof, ideally with $\sigma(\beta) \sim \exp(-c \beta)$ as predicted by asymptotic freedom.
What would settle it
A proof of a positive string tension for all $\beta$ in four dimensions.
Status in the literature
The area law is proven only at strong coupling (small $\beta$), going back to Osterwalder and Seiler (1978).
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