Computation ①: solve with force ／ ②: unsolvable even with force → ③: confirm, by computation, a force we cannot prove
The close of the "Force and Computation" cluster, and the last of all the bonuses of Force That Clicks. In Episode 11 we saw the confinement of the strong force (a quark cannot be extracted on its own). And its mathematical proof still does not exist ── it remains, as the Yang–Mills mass gap, one of the million-dollar Millennium Prize Problems. So why do physicists believe in confinement? The answer is computation. Carve spacetime itself into a lattice and compute the whole strong force on a supercomputer, and ── confinement, and the masses of hadrons, come out properly. It cannot be proven, but the computation speaks. This is the last face of force and computation.
Electromagnetism has a small coupling \(\alpha\approx1/137\), so it could be computed with high precision even by pencil and paper using approximation (perturbation theory) (like the \(g-2\) of Episode 6). But at low energies the strong force has a coupling \(\alpha_s\sim1\) that is too large, so approximation fails (the running of Episode 8). And confinement (Episode 11) is precisely a phenomenon of this "strongly acting" regime. Neither approximation nor proof works ── the strong force is more than pencil and paper can handle.
So we change the approach. Replace continuous spacetime with a fine lattice (grid). Quarks live on the lattice points, and the gluon field that carries the strong force (the gauge field of Mathematics ③) lives on the links connecting point to point. Then the field theory, which was infinite-dimensional, becomes a finite number of variables that a computer can handle. After that, let a supercomputer compute the enormous summation over all field configurations (the path integral) using the Monte Carlo method (clever dice) ── this is lattice QCD. The very structure we saw in Mathematics ③ as "force = the curvature of a gauge field on lattice links" is put directly onto numbers.
Continuous spacetime → a lattice (finite points and links). The gauge field rides on the links (Mathematics ③).
The infinite path integral → approximated by Monte Carlo (rolling clever dice in bulk).
The strong force, where approximation (perturbation) fails, can be computed whole, numerically, from first principles.
With lattice QCD, compute the energy (potential) \(V(r)\) when a quark and antiquark are separated by a distance \(r\), and ── the farther apart, the more it keeps rising linearly (\(V(r)\approx -a/r+\sigma r\), where \(\sigma\) is the string tension). Because the energy grows without bound, the quarks cannot be pulled apart = confinement is reproduced (the "snapping string" of Episode 11). Furthermore, the masses of the proton and neutron can be computed from first principles using only the properties of quarks as input, and they agree with measurement to within a few percent. In the figure below, watch how the computed \(V(r)\) on the lattice, shaking off the electromagnetic \(-a/r\) (which levels off), climbs linearly.
The conclusion of Force and Computation ③. Even a force that cannot be mathematically proven (confinement) can have its shape confirmed by carving spacetime into a lattice and computing. Lattice QCD reproduced confinement, produced hadron masses from first principles, and made them agree with measurement. The proof (a Millennium Problem) remains unsolved ── yet computation, on the threshold of proof, tells us with overwhelming precision that force indeed behaves that way. To understand (proof) and to confirm (computation) are different. And computation carries us all the way to the edge of what we do not understand.
Lattice QCD is powerful numerical evidence, but it is not a mathematical proof. Confinement (the existence of the Yang–Mills mass gap) remains unsolved as a Millennium Prize Problem. Lattice computation also has its own difficulties ── making time imaginary (Euclideanization), the continuum limit of shrinking the lattice spacing toward zero, finite-volume corrections, and the "sign problem" where the computation breaks down once you put in a density. Even so, the reproduction of confinement and hadron masses is an established result that agrees with experiment.
The figure's \(V(r)=-a/r+\sigma r\) (the Cornell potential) and "computed points" are a schematic showing the qualitative behavior that lattice QCD gives; they are not numerical values from actual lattice data. The lattice and flux tube on the left are also conceptual diagrams.
The confinement of the strong force (Episode 11) has a large coupling where approximation fails, and no mathematical proof (a Millennium Problem). So we carve spacetime into a lattice, ride the gauge field on the links (Mathematics ③), and let a supercomputer compute the enormous integral by Monte Carlo = lattice QCD. As a result, the quark-quark potential climbs linearly (confinement reproduced), and hadron masses too, from first principles, agree with measurement. It cannot be proven, but computation tells the shape of the force.
Force and Computation ① (nature solves with force), ② (unsolvable even with force, prediction has a horizon), ③ (computation lights up the threshold of proof) ── computation was a tool for solving force, a witness to its unsolvability, and a stand-in for proof. To understand (proof) and to confirm (computation) are different, and computation carries us all the way to the edge of what we do not understand. With this we close "Force and Computation," and we close all the bonuses of Force That Clicks.
Print / save as PDF: ⌘+P (Ctrl+P on Windows). On screen, the slider lets you watch the potential on the lattice keep climbing linearly (confinement). "See the answer" opens each solution.