KMS condition

In plain terms

Here is the problem. What makes a state thermal? Temperature talk usually leans on infinite systems, baths, and limits. The Kubo-Martin-Schwinger condition is the sharp answer algebraists settled on: a state is in equilibrium at a given temperature precisely when its correlation functions obey one specific symmetry -- evolve one observable forward in imaginary time by the inverse temperature and the order of the pair swaps.

On the finite field arena this is not an axiom or a limit: the Boltzmann state at the cutoff satisfies the KMS symmetry exactly, by computation. Cool it all the way down and the thermal two-point function converges to the vacuum one -- zero-point contributions cancelling on the way.

Thermal physics, at finite resolution, with no hand-waving about baths.

In CSD

The thermal tier joins the programme's thermodynamics track to the field track: the thermal state IS the Gibbs state of the field Hamiltonian, the partition function factorises across modes, and the propagator's truncation edge is explicit rather than hidden -- the same honesty about the cutoff that the Wick tier maintains.

Labelled breadth, as ever: equilibrium structure on the isolated finite piece, not a claim about continuum thermodynamics.

Mathematically

thermal_kms: for the diagonal field Hamiltonian at cutoff, the thermal two-point function satisfies the exact KMS identity -- complex-time evolution entrywise, the identity reducing to Boltzmann-weight transport and an index shuffle. Companions: closed-form diagonal Boltzmann weights via the continuous functional calculus, mode marginals, the off-diagonal vanishing, and the beta-to-infinity vacuum limit recovering the free propagator with no residue.

The name

Ryogo Kubo (1920-1995), at Tokyo, built the linear-response formalism of statistical mechanics; Paul Martin (1931-2016), at Harvard, and Julian Schwinger (1918-1994) wrote the 1959 paper whose boundary condition carried the same content. It was Haag, Hugenholtz and Winnink in 1967 who recognised the shared condition as the definition of equilibrium for infinite systems and attached the three initials to it.