Figure 1: Non-Abelian thermal state.

We derive the form of the thermal state of a system that has charges that might not commute with each other. Example charges include the components J_i of the spin J. We derive the thermal state's form by introducing an approximate microcanonical state. An ordinary microcanonical ensemble could lead to the thermal state's form if the charges commuted: suppose, for example, that the charges were a Hamiltonian H and a particle number N that satisfied [H, N]=0. Consider many copies of the system. The composite system could have a well-defined energy E_tot and particle number N_tot simultaneously. E_tot and N_tot would correspond to some eigensubspace shared by the total Hamiltonian and the total-particle-number operator. The (normalized) projector onto would represent the composite system's microcanonical state. Tracing out the bath would yield the system's thermal state. But the charges J_i under consideration might not commute. The charges might share no eigensubspace. Quantum noncommutation demands a modification of the ordinary microcanonical argument. We define an approximate microcanonical subspace . Each state in simultaneously has almost-well-defined values of noncommuting whole-system charges: measuring any such whole-system charge has a high probability of outputting a value close to an ‘expected’ value analogous to E_tot and N_tot. We derive conditions under which the approximate microcanonical subspace exists. The (normalized) projector onto represents the whole-system state. Tracing out most of the composite system yields the reduced state of the system of interest. We show that the reduced state is, on average, close to the NATS. This microcanonical derivation of the NATS’s form strengthens the link between Jaynes's information-theoretic derivation and physics.