Contribution
A characterisation of the observables that commute with the generator of a Markov process, in the commutator form of Noether’s theorem (conserved quantity iff it commutes with the Hamiltonian). In quantum mechanics a constant expectation in every state is enough for commutation. The paper shows that for Markov processes it is not, and that adding a constant second moment (so a constant variance) is exactly enough. For a finite state space, it also identifies what such observables are: functions constant on the connected components of the transition graph.
Key insight
The derivative of ⟨O, ψ(t)⟩ is ⟨1, [O, H]ψ(t)⟩ in the stochastic case. Probability enters linearly, so this can vanish for every ψ without the commutator vanishing. An observable can then have a constant mean while probability still flows between states where it takes different values, as long as the flows balance. Requiring the second moment to stay constant rules that out. Σᵢ (Oⱼ − Oᵢ)² Hᵢⱼ is a sum of non-negative terms, and it vanishes only if no transition ever connects states with different values of O. So a commuting observable can only label regions that the process never leaves.
Assumptions
- Finite case (Sections 1–2): X a finite set; H infinitesimal stochastic, with Hᵢⱼ ≥ 0 for i ≠ j and Σᵢ Hᵢⱼ = 0 (columns sum to zero, acting on distributions). Master equation dψ/dt = Hψ. An observable is a function O : X → ℝ, identified with a diagonal matrix.
- General case (Section 3): X a σ-finite measure space; distributions are in L¹(X); observables are in L^∞(X), acting by multiplication; U(t) is a strongly continuous semigroup of stochastic operators on L¹. Its generator is typically unbounded, so the theorem is stated with [O, U(t)] and not [O, H].
- “For every state.” Each equivalence quantifies over all initial distributions, not one.
Key results
- Theorem 1 (finite X). The following are equivalent: (i) [O, H] = 0; (ii) d/dt ⟨f(O), ψ(t)⟩ = 0 for every polynomial f and every solution; (iii) d/dt ⟨O, ψ(t)⟩ = d/dt ⟨O², ψ(t)⟩ = 0 for every solution; (iv) Oᵢ = Oⱼ whenever i and j lie in the same connected component of the transition graph. The graph has an edge j → i iff Hᵢⱼ ≠ 0, and components are taken ignoring direction.
- The mean is not enough. With H having columns (0, 0, 0), (1, −2, 1), (0, 0, 0) and O = diag(0, 1, 2), d/dt ⟨O, ψ⟩ = 0 at ψ(0) = (0, 1, 0) but [O, H] ≠ 0. The example is stronger than the paper states. Only column 2 of H is non-zero, and Σᵢ Oᵢ Hᵢ₂ = 0 − 2 + 2 = 0, so ⟨O, ψ(t)⟩ is constant for every initial distribution. Meanwhile Σᵢ Oᵢ² Hᵢ₂ = −2 + 4 = 2 ≠ 0, so the second moment moves. (Checked by hand.)
- Theorem 3 (Markov chain, σ-finite X). For a stochastic operator U on L¹(X) and O in L^∞: [O, U] = 0 iff ⟨O, Uψ⟩ = ⟨O, ψ⟩ and ⟨O², Uψ⟩ = ⟨O², ψ⟩ for every distribution ψ. The proof runs through three lemmas. If the moments are preserved, U maps distributions supported in O⁻¹(I) to the same set, for every compact interval I (Lemma 3, by Chebyshev on a fine partition of O’s range). That makes U commute with every χ_I(O) (Lemma 2), and hence with O (Lemma 1).
- Theorem 2 (Markov semigroup). [O, U(t)] = 0 for all t ≥ 0 iff ⟨O, U(t)ψ⟩ and ⟨O², U(t)ψ⟩ are constant in t for every distribution ψ. It is presented as an easy consequence of Theorem 3, with no separate proof.
Claims
| id | claim | strength | support |
|---|---|---|---|
| C1 | In a finite Markov process, commutation with H ⇔ constant mean and second moment in every state ⇔ O constant on transition-graph components | strong | Theorem 1, full proof |
| C2 | Constant mean in every state does not imply commutation | strong | explicit counterexample, Section 1 |
| C3 | The same equivalence (without the component form) holds for Markov chains and semigroups on σ-finite measure spaces | strong for Theorem 3; Theorem 2 asserted as a corollary | Section 3 |
| C4 | Quantum Noether in commutator form: [O, H] = 0 iff ⟨ψ, Oψ⟩ is constant for every solution | strong (standard; bounded operators only) | Section 1, via polarisation |
Concepts
- stochastic operator: linear, positivity-preserving, and integral-preserving (⟨1, Uψ⟩ = ⟨1, ψ⟩).
- infinitesimal stochastic operator: the generator H of a Markov semigroup: non-negative off-diagonal entries, columns summing to zero.
- Markov semigroup: U(t) = exp(tH), stochastic for t ≥ 0, continuous in t, with U(s + t) = U(s)U(t) and U(0) = I.
- transition graph: vertices X, an edge j → i iff Hᵢⱼ ≠ 0; connected components are taken ignoring direction.
- “stochastic mechanics”: Baez’s name for the analogy in which probabilities replace amplitudes and the master equation replaces Schrödinger’s.
Connections
- Baez’s Network theory notes, the source of the stochastic-mechanics analogy, cited as [1] and not held by the record.
- Noether’s theorem. The paper is explicit that its version is “somewhat removed from the original form”: it generalises the Poisson-bracket and commutator form, not the Lagrangian one.
- THEORY-019 (Proposed) says an operator commuting with a group has eigenspaces that are the group’s isotypic components. In the stochastic case of Theorem 1, the commutant of H among diagonal observables is exactly the functions constant on the components of the transition graph. That is the block structure H is forced to respect. The parallel is mine.
- THEORY-042 (Active), the superselection account. A commuting stochastic observable labels sectors that no transition connects, as a superselected charge labels sectors that no observable connects. The parallel is mine, and the two settings differ: one concerns dynamics, the other observables.
Bearing on the record
- New THEORY filed: THEORY-158, on what the commuting conserved quantities of a Markov process are, and on the gap between a conserved mean and commutation.
- No existing THEORY is supported or contradicted directly; THEORY-019 and THEORY-042 are parallels, noted above.
- Anthology. No instruction for machine-learning practice.
Limitations
- Only diagonal observables. O is a function on states, a multiplication operator. Commutation with non-diagonal operators (symmetries that permute states) is not treated, so the link to symmetries is thinner than the title suggests. The “symmetry” side of this Noether theorem is O itself as a generator, and the paper does not discuss what exp(sO) means for a Markov process.
- Theorem 2 is not proved separately from Theorem 3, and the step from a semigroup to every U(t) is left to the reader.
- The component characterisation (iv) is given only for finite X.
- Wording slips. The text says “the variance is the standard deviation of O”, meaning its square. In the Chebyshev step, after defining ((M/n)/d)², it sets c = (Md)², where (M/d)² is meant. Neither affects the result.
- The quantum version is stated for bounded operators only.
Open questions
- Is there a stochastic Noether theorem for non-diagonal symmetries, for example permutations of X commuting with H, linking them to conserved quantities? The paper does not attempt one.
- What do observables with a constant mean but not a constant variance mean in the dynamics, like the counterexample’s O? They satisfy Hᵀ O = 0, a harmonic or martingale condition, but the paper does not name or study them.
- Does (iv) extend to general state spaces, with “connected component” replaced by a suitable notion of invariant set?