Constants you may fix, constants you must not — and the constants that "a principle fixes to an exact value"
In Bonus 5 we counted "of the four constants that build \(\alpha\), you can fix at most three." This time we widen that idea to the constants of physics as a whole. Once you notice that "fixing" comes in two kinds, the whole scene flips over — for humans to fix a unit by declaration is an arbitrary convention and carries no meaning. Whereas for a principle to nail a dimensionless quantity to an exact value is the explanation itself. So a "dimensionless quantity you may fix" is not a quantity that humans should pin down by fiat, but a quantity that a principle fixes on its own. Let's close the constants series with a three-column map.
This is the upshot of the earlier bonuses. It wasn't that \(c\) was special — every constant that carries a unit is, without exception, a bookkeeping convention. The proof is the 2019 SI redefinition: humanity fixed seven constants such as \(c,h,e,k_B,N_A\) all at once, declaring "these are the exact values" (the all-constants version of Bonus 2's fixing of the speed of light). That the values could be set by declaration is the clearest possible proof that they are not physics but unit conventions. \(k_B\) is especially clean — temperature is not a fundamental quantity but energy per degree of freedom. In Episode 9, when \(\tau=\hbar/k_BT\) turned temperature into a period in time, it was because temperature is not an independent quantity.
We saw Column 1 (conventions) and Column 2 (the 26 you measure) in earlier installments. The star of this one is Column 3: dimensionless quantities that are neither pinned down by human fiat nor felt out by measurement, but that a principle rigidly nails to "this value and no other." Here are four.
In the Standard Model, the sum of the electric charges (hypercharges) over one generation must be exactly \(\sum Y=0,\ \sum Y^3=0\). The moment it drifts, quantum effects break gauge symmetry and the theory dies (non-renormalizable, non-unitary). This "fixing to zero" jointly constrains why quark charges come in thirds, why the proton and electron charges cancel exactly (measured to a precision of \(10^{-21}\)), and why there must be three colors. It's the quantum-version fangs of Episode 8's gauge principle.
\(\displaystyle\sum_{\text{one generation}} Y = 0,\qquad \sum_{\text{one generation}} Y^3 = 0\) — unless this holds, the theory cannot exist as a quantum theory.
→ charge quantization, the matching of proton and electron charge, and three colors are fixed by consistency, not by measurement.
Winding numbers, Chern numbers, the quantum Hall conductance \(\sigma=\nu e^2/h\) (where \(\nu\) is an exact integer), Dirac's charge quantization \(eg=2\pi n\). Topology can only emit integers, so these dimensionless quantities have a rigidity that survives even when you soil them. What's fun is that metrology exploits this in reverse — the quantum Hall resistance can serve as a unit standard precisely because its dimensionless part is fixed to an integer and cannot budge. Here Column 1 (a unit fixed arbitrarily) leans on Column 3 (the integer \(\nu\) fixed by a principle), a nesting.
Dimensionless couplings run (Episode 6), but at special values the running stops = a fixed point. QCD's ultraviolet fixed point \(g^*=0\) (asymptotic freedom), the Wilson–Fisher point of critical phenomena, and gravity's dream, "asymptotic safety." The critical exponents around a fixed point are universal dimensionless numbers independent of the details of the matter. This goes one step beyond Episode 6's "how it runs is deeper than the value" — within the running, the "point where it stops" is the deepest of all.
The electron's anomalous magnetic moment \(a_e=(g-2)/2\) is computed by QED to 12 digits and agrees with experiment, given only \(\alpha\). This is not an input (measured) but an output (fixed). A good theory is measured by how many dimensionless numbers it can move from the "measured" column into the "fixed by calculation" column.
Column 3 also has candidates that want in but haven't gotten in yet. Dimensionless quantities we'd like to fix with a principle, but whose principle hasn't been found — this is the current frontier.
| Dimensionless quantity we'd like to fix | Situation |
|---|---|
| \(\theta_{QCD}\) (strong CP) | For some reason almost exactly 0 (\(<10^{-10}\) from the neutron EDM). The mechanism that fixes it to 0 (Peccei–Quinn / axion) is under active search. |
| \(\rho=M_W^2/(M_Z^2\cos^2\theta_W)\) | Custodial symmetry fixes it \(=1\). Any deviation would be a sign of new physics. |
| Cosmological constant \(\Lambda\) | We wish it were fixed near 0, but no principle has been found (the energy-origin story of Bonus 7). |
The "count" of the Standard Model's free parameters changes with how you count (roughly 19–26, depending on whether you include neutrino masses and mixings, etc.). Anomaly cancellation "fixes" charge quantization only given a particular assignment of particles (the representation content); why that assignment itself is what it is remains a Column 2 question. Fixing by topology and fixed points each have their own conditions of applicability.
This three-column map is a rough guide for organizing the constants, not a rigorous taxonomy. Some quantities sit on the boundary (e.g., \(\theta_{QCD}\) shows up in Column 2 and as a Column 3 candidate). Its single purpose is to "split the meaning of fixing into two kinds."
Anything with dimensions is just a "convention" and may be fixed (it's not physics). Most dimensionless numbers are the ~26 "you can only measure by hand," and we don't yet understand them. Of those, only the ones that a principle (anomaly cancellation, topology, a fixed point of the renormalization group, prediction by calculation) nails to an exact value are the "dimensionless quantities you may — and should — fix." Because there, "fixing" is not a convention but an explanation.
So the entire program of physics is to move the free dimensionless numbers (Column 2, measured) into the dimensionless numbers a principle fixes (Column 3). Last time's "reduce the 26" and this time's "fix by principle" were two sides of the same thing. The constants series, which began with Bonus 2's "the absolute value is bookkeeping, only ratios are physics," arrives at this one map — separating the fixing of a convention from the fixing of necessity.
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