On the equilibrium states in quantum statistical mechanics

Rudolf Haag, Nicolaas M. Hugenholtz, Marinus Winnink
February 15, 1967
Representations of the $C^*$*-algebra $\frak A$ of observables corresponding to thermal equilibrium of a system at given temperature $T$ and chemical potential $\mu$ are studied. Both for finite and for infinite systems it is shown that the representa- tion is reducible and that there exists a conjugation in the representation space, which maps the von Neumann algebra spanned by the representative of $\frak A$ onto its commutant. This means that there is an equivalent anti-linear representation of $\frak A$ in the commutant. The relation of these properties with the Kubo-Martin-Schwinger boundary condition is discussed.
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KMS condition, modular theory