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LIT-773: A Noether theorem for Markov processes

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Properties5
StatusActive — read 2026-10-09 (NOTE-598); worth reading as the stochastic counterpart of the commutator form of Noether's theorem: for a Markov process, an observable commutes with the generator iff its mean and second moment are conserved in every state. On a finite state space that holds iff it is constant on each connected component of the transition graph. A conserved mean alone is not enough, unlike quantum mechanics, as an explicit 3-state example shows. Only diagonal observables are treated, not symmetries that permute states.
Tagsmathematics, probabilistic-modeling
SupportsCLAIM-094 — A harmonic observable conserves only its mean, and Baez and Fong show that mean conservation for every initial state does not make an observable a symmetry
Cited as “paper” byNOTE-598 — Baez & Fong — A Noether theorem for Markov processes
Cited as “source” byTHEORY-158 — For a Markov process on a finite state space, an observable commutes with the generator exactly when its mean and variance are both conserved in every state, which is exactly when it is constant on each connected component of the transition graph, so a conserved mean alone does not give a symmetry

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John C. Baez and Brendan Fong (2013), Journal of Mathematical Physics 54(1):013301 — ARXIV-1203.2035

Key takeaways

  • Theorem 1 (finite state space, generator H with non-negative off-diagonal entries and zero column sums, diagonal observable O). Four conditions are equivalent: [O, H] = 0; every polynomial in O has a conserved mean; O and O² have conserved means, in every state; O is constant on each connected component of the transition graph (an edge wherever Hᵢⱼ ≠ 0).
  • The mean is not enough. A 3-state process in which the middle state leaks equally to its neighbours, with O = (0, 1, 2), conserves the mean of O in every state but does not commute with H. In quantum mechanics the conserved mean alone gives commutation, by polarisation.
  • General state spaces (Section 3). On a σ-finite measure space the moment form holds for a single stochastic operator (Theorem 3, through Chebyshev’s inequality) and for Markov semigroups, stated with [O, exp(tH)] to avoid unbounded generators (Theorem 2). The connected-component form is not extended.
  • Scope. Only multiplication operators. The “symmetry” is O itself as a generator; permutations of states that commute with H are not treated.

Standing in the record

Filed on 2026-10-09 at the owner’s request, as one of the works in the reference list of the owner’s working manuscript (October 2026) that the record did not yet hold. See the curation entry of that day.

Read in full the same day (NOTE-598). Its theorem is filed as THEORY-158. The commutant it describes, observables that label regions the dynamics never connects, parallels the record’s accounts of commutants and isotypic blocks (THEORY-019) and of superselection sectors (THEORY-042). Both parallels are mine, not the paper’s. Fritz’s Markov categories (LIT-772), filed with it, treat the same kernels compositionally, and neither paper cites the other.

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