Math 1 action / 2 symmetry → 3 survey force in the language of geometry and groups
This is the closing of "Force and Mathematics." In Math 1 we gave the leading role to the action, and in Math 2 to symmetry. Finally, we survey force itself in the language of geometry. In Episode 9 we saw "force = the 'connector' that links the freedom at each point," and in Episode 12 "the leftover you can't erase = curvature." This "connector" and "curvature" have proper mathematical names ── connection and curvature. And, astonishingly, gravity, electromagnetism, and the strong force can all be written in the same form, as "the bending (curvature) of some space." What decides "which force" is the mathematics called a group. We'll fold the true nature of force onto a single page with geometry and groups.
The most elementary face of curvature shows up in parallel transport. Carry an arrow "keeping its direction fixed" once around a closed path. On a flat surface (zero curvature), the returning arrow points in the same direction as it started. But on a curved surface (a sphere, say), even when you carry it the same way, when it returns, the direction is off. This offset (holonomy) is exactly curvature ── and it is the true nature of force. It is that very wrap-around we spoke of in Episode 12, when we said "even if you can erase force at a single point, you can't erase the wrap-around over a full loop (the curvature)."
Here Episode 9 and Waves 3 connect. Episode 9's "connector \(A\)" is the rule for how you rotate an arrow as you carry it to the neighbor = the connection. That the electron wave's phase shifted after going around once, in the Aharonov–Bohm effect of Waves 3, is precisely this holonomy. Force (field strength) = the curvature of the connection = the offset over a loop. The distinction (Episode 12) between the "ledger (gauge)" you can erase locally and the "curvature (the body of force itself)" you cannot erase becomes crisp in the language of geometry.
For each force, the only difference is "what is bending."
| Force | Bending (curvature) of what |
|---|---|
| Gravity | Bending of spacetime itself (Riemann curvature). Episode 5's "gravity = geometry." |
| Electromagnetism | Bending of the phase (internal space) at each point. Field strength \(F_{\mu\nu}\) = the curvature of the gauge field \(A\). |
| Weak force / strong force | Bending of a larger internal space (the curvature of non-abelian gauge fields). |
Gravity is the bending of a visible space, "spacetime"; electromagnetism is the bending of an internal space, "the phase attached at each point." The stages differ, but the mathematical form force = the curvature of a connection is exactly the same. Episode 5's "gravity = geometry" was not a special case; every force was geometry (curvature) ── this is the unified picture of force as seen from mathematics.
So where does the difference between electromagnetism, weak, and strong come from? The group. It's what we called in Episode 9 "the difference in which local symmetry you admit" ── turning a single phase \(U(1)\) (electromagnetism), \(SU(2)\) (weak force), turning three colors \(SU(3)\) (strong force). These are Lie groups, collections of continuous symmetry transformations. Which group's "bending" it is decides the kind of force. Furthermore, the representation theory of the group classifies "which particles feel that force," and topology governs the quantization of magnetic flux and the Aharonov–Bohm effect (Waves 3). The world of forces can be written out in full, in the language of geometry and group theory.
· The body of force = the curvature (the leftover you can't erase, Episode 12) of the connection (the connector, Episode 9) = the offset over a loop.
· Which force = the bending of which group (\(U(1)/SU(2)/SU(3)\); for gravity, the symmetry of spacetime).
· Who feels it = the representation of the group; how it acts globally = topology (Waves 3).
The conclusion of "Force and Mathematics." Force is the curvature of a connection. Its kind is decided by the group. Episode 5 (gravity = geometry), Episode 9 (force = the gauge connector), Episode 12 (curvature you can't erase), Waves 3 (the holonomy of phase) ── the vistas we saw separately in the main series and the waves cluster all fit into a single piece of mathematics: geometry (connection and curvature) and groups. Combined with Math 1's action and Math 2's symmetry (Noether) ── nature makes the action stationary, and its symmetry gives birth to force (curvature) and conservation laws. This is the mathematical whole of force.
The mathematics of gauge theory is, strictly, formulated with a connection on a fiber bundle and its curvature (the field strength \(F=dA+A\wedge A\)), and gravity is written with the (Levi–Civita) connection of spacetime and the Riemann curvature. The "offset over a loop = force" here is a plain-language picture of its essence (holonomy = curvature). The groups \(U(1)/SU(2)/SU(3)\) are Lie groups; particle classification corresponds to representation theory, and flux quantization and the AB effect to topology (characteristic classes, holonomy). Quantizing gravity within the same framework as the other forces remains unfinished (the same homework as the main-series finale and the sister series).
The figure schematizes the key point of holonomy ── "go around once by parallel transport and the direction is off by the amount of curvature" ── as a rotation in the plane (it is not a rigorous depiction of actual parallel transport on a sphere).
The body of force is the curvature (Episode 12's leftover you can't erase) of a connection (Episode 9's connector) = the offset in the direction of an arrow after taking it once around a loop (holonomy, the AB effect of Waves 3). Gravity is the bending of spacetime (Episode 5), electromagnetism the bending of the phase (internal space), weak and strong the bending of a larger internal space ── the stages differ, but the form "force = the curvature of a connection" is common. And "which force" is decided by the group \(U(1)/SU(2)/SU(3)\), "who feels it" by representation theory, "how it acts globally" by topology.
With Math 1 (action), 2 (symmetry → conservation law), and 3 (force = curvature, groups), we close "Force and Mathematics." Nature makes the action stationary, symmetry gives birth to force (curvature) and conservation laws, and groups classify the forces. What we peeled back in the main series as "force is relation" had, to the eye of mathematics, the clearest possible form imaginable: "the curvature of a connection." Next is the final cluster, "Force and Computation" ── the story that nature computes with that force.
Print / save as PDF: ⌘+P (on Windows, Ctrl+P). On screen, the slider lets you see how raising the curvature widens the offset of the arrow after one loop (= force). Click "See the answer" to open a solution.