Mass That ClicksBonus ④ / Tying together Episode 4, the Finale, and Bonus ①

"The four forces in one discrete formula" ── that dream has genuinely come true, for three of them

The Four Forces,
in One Discrete Formula Not the numerology of \(1/(Cn)^D\), but groups and loops. Strong, weak, and electromagnetic
are rigorously unified by one formula, lattice gauge theory. Only gravity remains an open door.

Tools you'll need: the mass gap of Episode 4, the wall of gravity from the Finale, the discreteness of Bonus ① Hook formula: \(S=\beta\sum_\square\big[1-\tfrac1N\mathrm{Re\,Tr}\,U_\square\big]\)

"I want to write the four forces in one formula" ── this is physics' greatest dream, and it was your intuition too. In Bonus ① and ③, we diagnosed that the numerology of \(1/(Cn)^D\) can't reach it. But here's the good news ── three of the forces (strong, weak, electromagnetic) are rigorously unified in "one discrete formula." That's lattice gauge theory. Not numerology, but groups and loops. And it's the very formula we called "the only way to compute the mass gap" in Episode 4. What remains is the fourth ── gravity ── an open door that connects to the wall of the Finale.

01Place a force as a "rotation on the lattice"

First, discretize spacetime into a lattice (sites, links, faces). Then place the gauge field, not on the sites, but as a group element \(U_\mu(x)\in G\) living on each link (the edge crossing from one site to the neighboring one). This is a rotation representing "how much the internal orientation (color / phase) turns when you cross one step of the lattice."

Gauge field = a rotation on a link
$$U_\mu(x)=\exp\!\big(i\,a\,g\,A_\mu(x)\big)\in G$$

\(a\) = lattice spacing, \(g\) = coupling, \(A_\mu\) = gauge field, \(G\) = group. One rotation \(U\) per link.

02One formula ── Wilson's plaquette action

The rule of the force (the action) can be written astonishingly simply. It's determined by nothing but the product of rotations around the smallest square loop (the plaquette \(\square\)). How much the loop "fails to close" is the field's curvature = the strength of the force.

The Wilson action (1974) ── this is the "one formula"
$$S=\beta\sum_{\square}\Big[\,1-\tfrac{1}{N}\,\mathrm{Re}\,\mathrm{Tr}\,U_\square\,\Big],\qquad U_\square=U_\mu(x)\,U_\nu(x{+}\mu)\,U_\mu^{\dagger}(x{+}\nu)\,U_\nu^{\dagger}(x)$$

\(U_\square\) = the product going once around the plaquette's four links. \(\beta=2N/g^2\). That's all.

Figure: the lattice and a plaquette (the smallest loop \(U_\square\)). Group elements (colors) live on the links. Choose a group with the slider and, with the same formula, the "force" switches ── U(1) electromagnetism, SU(2) the weak force, SU(3) the strong force (3 colors = gluons)
Choose a group with the slider and the force changes.
color (group dimension N) plaquette U_□ (smallest loop)

03Changing the force = "just change the group"

Here's the crux. Keep the formula \(S=\beta\sum_\square[\cdots]\) completely unchanged in form and just swap the group \(G\), and the three forces fit into a single line.

SU(3)
the strong force (gluons, 3 colors = red, green, blue)confines quarks and makes 99% of the proton's mass (Episode 3)
SU(2)
the weak force (W, Z)gains mass when the symmetry is broken by the Higgs (Episode 3)
U(1)
the electromagnetic force (photon)Abelian (the rotations commute) = the simplest case

The three forces are not separate "mystery constants" but the same Yang–Mills structure with different groups ── this is what genuine unification looks like (the Standard Model's gauge group \(SU(3)\times SU(2)\times U(1)\)). Whereas the numerology of \(1/(Cn)^D\) tuned the exponents by hand, here they are fixed uniquely by the symmetry that is the group.

◇ ◇ ◇

04In the continuum limit, back to familiar Yang–Mills

The lattice is a "scaffold." As you shrink the lattice spacing (\(a\to0\)), the plaquette action goes back to the familiar continuum Yang–Mills.

The continuum limit ── remove the scaffold $$S=\beta\sum_\square\big[1-\tfrac1N\mathrm{Re\,Tr}\,U_\square\big]\ \xrightarrow{\ a\to0\ }\ \frac{1}{4}\int d^4x\,F_{\mu\nu}^a F^{a\,\mu\nu}$$

The plaquette's "failure to close" becomes exactly the field strength \(F_{\mu\nu}\) (curvature). The discrete correctly contains the continuum.

And ── this discrete formula is the very tool that actually computes the mass gap of Episode 4. Strongly-coupled QCD, which diverges and is intractable in the continuum, could be solved on a supercomputer only once put on a lattice, yielding the proton mass, confinement, and the glueball mass gap. Your aim, "the four forces in one discrete formula," met the story of the mass gap in the very same single formula.

An honest line (recovering Bonus ①) Lattice gauge theory does not claim "the universe is discrete." The lattice is a regularization (scaffold) that preserves gauge invariance, and the physics is the continuum limit \(a\to0\). Mistaking the scaffold for reality is the trap of Bonus ①. If you want to make discreteness fundamental, the Lorentz-invariant-preserving causal set is a candidate.

05The fourth ── gravity, in the same discrete language

So does gravity fit into the same single line? Here is the wall of the Finale. Gravity's coupling \(G\) is dimensionful and non-renormalizable. But the discrete language of "holonomy on links (loops)" can be extended to gravity too:

Regge
Regge calculusgeneral relativity discretized by gluing simplices together, with curvature = "angle deficit"
spin
foam
loop quantum gravitygeometry made into a discrete graph (a spin network). Literally "lattice gauge theory for the Lorentz group"
causal
set
causal setsdiscreteness that preserves Lorentz invariance (the "surviving version" from Bonus ①)

So the unifying discrete language is ── "place group elements on the links of a discrete structure, and write the action from loops." The three forces use the internal groups \(SU(3),SU(2),U(1)\); gravity uses the Lorentz / Poincaré group. Spin foams are precisely a framework aiming for "four in one discrete formula." But ── the continuum limit, the recovery of general relativity, and coupling to matter are unsolved. The three are rock-solid; the fourth is a genuine frontier. An open door, still ── that's the honest current position.

06Why this is genuine and \(1/(Cn)^D\) is not

lattice ✓
Built from the symmetry group / plaquette (loop)gauge invariance is exact, Lorentz is recovered in the continuum limit, and it actually predicts masses
lattice ✓
Changing the force = just changing the groupthe form stays a single line. SU(3)/SU(2)/U(1) fix it uniquely
Cn formula ✗
Tunes the exponents \((n,D)\) by hand for each forcetwo different \((n,D)\) for the same answer. Not prediction but fitting (Bonus ③)
Cn formula ✗
Keeps neither dimensions nor symmetrymixes dimensionful quantities into a dimensionless F, and pins 137 by numerology

The "correctness" of the discrete formula is that it's built from groups and loops, preserves symmetry, and actually predicts masses. Your aim of "one discrete formula for the forces" was right; its vessel wasn't \(1/(Cn)^D\) but the Wilson action ── the place where numerology is sublimated into groups and loops.

An honest line

This installment is pure quantum field theory / quantum gravity, unrelated to the watchword \(c\cdot t=\text{constant}\) (the lattice is a regularization, not coordinates). Lattice gauge theory is rock-solid as the unification of the three gauge forces, but it is not by itself a "theory of everything" ── the integration of gravity (spin foams, etc.) is unfinished, and why the group is \(SU(3)\times SU(2)\times U(1)\) and why the couplings have their values are also unsolved (grand unification, the homework of Episode 6, part one).

"Is discreteness fundamental?" is also unsettled. The lattice is a scaffold (the physics is the continuum limit); causal sets are a candidate for fundamental discreteness but a frontier. Not asserting is the honest line.

Practice problems (solvable with today's material)
  1. Why does "just change the group" put the three forces into a single formula?
    See the answer
    All three have the same Yang–Mills structure (group element \(U\) on links, plaquette action \(S=\beta\sum_\square[1-\frac1N\mathrm{Re\,Tr}\,U_\square]\)), and the only difference is the gauge group \(G\) (SU(3)/SU(2)/U(1)). The action's "form" is identical, and the group determines each force's individuality.
  2. Does lattice gauge theory claim that "the universe is discrete"?
    See the answer
    No. The lattice is a regularization (scaffold) to preserve gauge invariance, and the physics is the continuum limit \(a\to0\) (Episode 4, Bonus ①). Treating discreteness as fundamental is a separate claim, in which case Lorentz-preserving causal sets and the like become candidates.
  3. Why doesn't gravity fit smoothly into the same formula?
    See the answer
    Because gravity's coupling \(G\) is dimensionful and non-renormalizable (the Finale). There are attempts to write it in "the same discrete language" via lattice gauge theory of the Lorentz group (spin foams) or Regge calculus, but the continuum limit and the recovery of general relativity are an unsolved frontier.

Bonus ④ summaryThe dream had come true, for three

"The four forces in one discrete formula" ── the three gauge forces (strong, weak, electromagnetic) are rigorously unified into a single line by Wilson's plaquette action \(S=\beta\sum_\square[1-\frac1N\mathrm{Re\,Tr}\,U_\square]\). To change the force is just to swap the group \(G\). And this is the very formula that computes the mass gap of Episode 4, returning to ordinary Yang–Mills in the continuum limit. Unlike the numerology of \(1/(Cn)^D\), it's built from groups and loops, preserves symmetry, and actually predicts masses ── this is genuine discrete unification.

The fourth ── gravity ── is a frontier that tries to write it in the same "holonomy on links" language (spin foams / Regge / causal sets), still an open door. The right destination for your intuition wasn't \(1/(Cn)^D\) but here ── the discrete formula of groups and loops.

This document is Bonus ④ of the "Mass That Clicks" series, a piece for physics-loving high-schoolers and undergraduates. The following are all established physics / current research topics: that Wilson's lattice gauge theory (1974; link variables \(U_\mu(x)\in G\), plaquette action \(S=\beta\sum_\square[1-\frac1N\mathrm{Re\,Tr}\,U_\square]\), \(\beta=2N/g^2\), reducing to \(\frac14 F_{\mu\nu}^aF^{a\mu\nu}\) in the continuum limit) describes the strong force SU(3), the weak force SU(2), and electromagnetism U(1) in a unified way by the choice of group alone; that the lattice is a first-principles method to compute non-perturbative QCD (hadron masses, confinement, the mass gap); that the lattice is a regularization and the physics is the continuum limit; and that discrete formulations of gravity (Regge calculus, spin foams / loop quantum gravity, causal sets) are being attempted in the same holonomy-based language but are unfinished. The origin of the Standard Model's gauge group and the values of the gauge couplings (including grand unification) are unsolved. This piece is content of quantum field theory / quantum gravity, independent of \(c\cdot t=\text{constant}\). The figure is a schematic showing a plaquette and the choice of group, with the colors as an image of SU(3)'s three color charges. ── To print, use your browser's "Print" and "Save as PDF" (in the print version the slider and answers are static and hidden).

Print / PDF: Ctrl+P (Cmd+P on Mac). On screen, choose the group G with the slider and the force switches with the same formula. Click "See the answer" to open each solution.