Source
Baez & Fong (2013), LIT-773, read in full in NOTE-598: Theorem 1 with its proof, the counterexample of Section 1, and Theorems 2–3 of Section 3.
What was actually shown
Take a finite set X, an infinitesimal stochastic H (off-diagonal entries ≥ 0, columns summing to zero) and an observable O : X → ℝ. Theorem 1 proves that four conditions are equivalent:
- [O, H] = 0;
- every polynomial in O has a constant mean along every solution of the master equation;
- O and O² have constant means along every solution;
- O is constant on each connected component of the transition graph, where an edge j → i means Hᵢⱼ ≠ 0.
The key step is that Σᵢ (Oⱼ − Oᵢ)² Hᵢⱼ is a sum of non-negative terms, and it equals a combination of the time derivatives of ⟨O⟩ and ⟨O²⟩ at the point mass on j.
That the mean alone could have sufficed is ruled out by an explicit 3-state example. H moves state 2 to states 1 and 3 at equal rates, and O = (0, 1, 2), so the mean of O is conserved but [O, H] ≠ 0. On a σ-finite measure space the moment form (not the component form) holds for a single stochastic operator (Theorem 3, by Chebyshev’s inequality) and for Markov semigroups (Theorem 2, stated as a consequence).
What this does not say
- Not Noether’s theorem in its original sense. The paper says its version is “somewhat removed” from the Lagrangian original. It generalises the commutator form, in which the observable is both the conserved quantity and the generator of the symmetry.
- Not a theorem about symmetries that permute states. Only multiplication operators are treated. A permutation of X commuting with H, which is what “symmetry” usually means for a Markov chain, is outside it.
- Not that conserved means are uninteresting. An observable with Hᵀ O = 0 has a conserved mean in every state without commuting with H. The paper does not study such observables, and their name and role (harmonic functions of the process) are the reader’s gloss.
- Not the general-space form, as the record’s. Beyond finite X, Section 3 gives only the moment and commutator forms: Theorem 3 for one stochastic operator, by Chebyshev, and Theorem 2 for Markov semigroups, stated as a consequence. Neither is re-derived here, and there is no component form. That is QUESTION-144.
- Not “standard deviation” in the abstract’s sense. Conservation of the mean and of the second moment is what is proved. That is equivalent to a conserved mean and variance, and so to a conserved standard deviation, but the proof works with ⟨O²⟩.
Connections
The commutant described here is a block structure. H can only be commuted with by observables that label regions it never connects. This parallels THEORY-019, where operators commuting with a group respect its isotypic components, and THEORY-042, where superselected observables label sectors no observable connects. Both parallels are the record’s, not the paper’s.
Note of 2026-10-19: Theorem 1, re-derived
The promotion audit’s batch 008 re-derived the finite-state cycle (ADR-047 v2). X is finite, and H is infinitesimal stochastic: off-diagonal Hᵢⱼ ≥ 0, and columns sum to 0. The master equation is ψ′ = Hψ, and O is a function on X. Write d⟨f⟩/dt = Σᵢ f(i)(Hψ)ᵢ = Σⱼ ψⱼ Σᵢ f(i)Hᵢⱼ. This vanishes along every solution iff Σᵢ f(i)Hᵢⱼ = 0 for every j, since the starting point can be any point mass.
- The moments give components. If O and O² have constant means, then for every j, Σᵢ (Oⱼ − Oᵢ)² Hᵢⱼ = Oⱼ² Σᵢ Hᵢⱼ − 2Oⱼ Σᵢ OᵢHᵢⱼ + Σᵢ Oᵢ² Hᵢⱼ = 0. The i = j term is 0, and every other term is non-negative. So Hᵢⱼ ≠ 0 implies Oᵢ = Oⱼ: O is constant on each connected component. It also gives [O, H]ᵢⱼ = (Oᵢ − Oⱼ)Hᵢⱼ = 0.
- Commutation gives every moment. If [O, H] = 0, then (Oᵢ − Oⱼ)Hᵢⱼ = 0, so O is constant on components. Then for every polynomial p, Σᵢ p(Oᵢ)Hᵢⱼ = p(Oⱼ)Σᵢ Hᵢⱼ = 0. Every polynomial in O has a constant mean, which closes the cycle.
- The counterexample. Let H₁₂ = H₃₂ = r and H₂₂ = −2r, with all other entries 0, and O = (0, 1, 2). Then Σᵢ OᵢHᵢ₂ = −2r + 2r = 0, so the mean of O is conserved. But [O, H]₁₂ = −r ≠ 0, and Σᵢ Oᵢ²Hᵢ₂ = 2r, so the second moment is not conserved.