Communications in Mathematical Physics

, Volume 5, Issue 3, pp 215–236 | Cite as

On the equilibrium states in quantum statistical mechanics

  • R. Haag
  • N. M. Hugenholtz
  • M. Winnink


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.


Boundary Condition Neural Network Statistical Physic Equilibrium State Complex System 
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Copyright information

© Springer-Verlag 1967

Authors and Affiliations

  • R. Haag
    • 1
  • N. M. Hugenholtz
    • 2
  • M. Winnink
    • 2
  1. 1.Department of PhysicsUniversity of IllinoisUrbana
  2. 2.Natuurkundig LaboratoriumRijks-UniversiteitGroningen

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