In the statistical mechanics of quantum mechanical systems and quantum field theory, the properties of a system in thermal equilibrium can be described by a mathematical object called a Kubo–Martin–Schwinger (KMS) state: a state satisfying the KMS condition. Ryogo Kubo introduced the condition in 1957, Paul C. Martin and Julian Schwinger used it in 1959 to define thermodynamic Green's functions, and Rudolf Haag, Marinus Winnink and Nico Hugenholtz used the condition in 1967 to define equilibrium states and called it the KMS condition.
Overview The simplest case to study is that of a finite-dimensional Hilbert space, in which one does not encounter complications like phase transitions or spontaneous symmetry breaking. The density matrix of a thermal state is given by
ρ β , μ = e − β ( H − μ N ) T r [ e − β ( H − μ N ) ] = e − β ( H − μ N ) Z ( β , μ ) {\displaystyle \rho _{\beta ,\mu }={\frac {\mathrm {e} ^{-\beta \left(H-\mu N\right)}}{\mathrm {Tr} \left[\mathrm {e} ^{-\beta \left(H-\mu N\right)}\right]}}={\frac {\mathrm {e} ^{-\beta \left(H-\mu N\right)}}{Z(\beta ,\mu )}}}
where H is the Hamiltonian operator and N is the particle number operator (or charge operator, if we wish to be more general) and
Z ( β , μ ) = d e f T r [ e − β ( H − μ N ) ] {\displaystyle Z(\beta ,\mu )\ {\stackrel {\mathrm {def} }{=}}\ \mathrm {Tr} \left[\mathrm {e} ^{-\beta \left(H-\mu N\right)}\right]}
is the partition function. We assume that N commutes with H, or in other words, that particle number is conserved. In the Heisenberg picture, the density matrix does not change with time, but the operators are time-dependent. In particular, translating an operator A by τ into the future gives the operator
α τ ( A ) = d e f e i H τ A e − i H τ {\displaystyle \alpha _{\tau }(A)\ {\stackrel {\mathrm {def} }{=}}\ \mathrm {e} ^{iH\tau }A\mathrm {e} ^{-iH\tau }} . A combination of time translation with an internal symmetry "rotation" gives the more general
α τ μ ( A ) = d e f e i ( H − μ N ) τ A e − i ( H − μ N ) τ {\displaystyle \alpha _{\tau }^{\mu }(A)\ {\stackrel {\mathrm {def} }{=}}\ \mathrm {e} ^{i\left(H-\mu N\right)\tau }A\mathrm {e} ^{-i\left(H-\mu N\right)\tau }}
A bit of algebraic manipulation shows that the expected values
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