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On the equilibrium states in quantum statistical mechanics

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Representations of theC*-algebra\(\mathfrak{A}\) of observables corresponding to thermal equilibrium of a system at given temperatureT and chemical potential μ are studied. Both for finite and for infinite systems it is shown that the representation is reducible and that there exists a conjugation in the representation space, which maps the von Neumann algebra spanned by the representative of\(\mathfrak{A}\) onto its commutant. This means that there is an equivalent anti-linear representation of\(\mathfrak{A}\) in the commutant. The relation of these properties with the Kubo-Martin-Schwinger boundary condition is discussed.

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Haag, R., Hugenholtz, N.M. & Winnink, M. On the equilibrium states in quantum statistical mechanics. Commun.Math. Phys. 5, 215–236 (1967).

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