Communications in Mathematical Physics

, Volume 6, Issue 3, pp 189–193 | Cite as

On the factor type of equilibrium states in quantum Statistical Mechanics

  • N. M. Hugenholtz


A theorem is derived giving sufficient conditions for a factor to be either finite or purely infinite. These conditions are:
  1. i.

    In the Hilbert space\(\mathfrak{H}\) exists a conjugation operatorJ transforming the factor ℜ into its commutant ℜ′.

  2. ii.

    There exists a one parameter abelian group of automorphisms of ℜ implemented by unitary operatorsU t weakly continuous int and commuting withJ.

  3. iii.

    There is a cyclic and separating vector Ω, which is invariant forJ and which is the only vector in\(\mathfrak{H}\) invariant forU t .


This theorem is of interest for Statistical Mechanics since representations of thermal equilibrium states satisfy these conditions [1]. One finds that the representations of equilibrium states corresponding to one phase are factors of type III.


Neural Network Statistical Physic Hilbert Space Equilibrium State Complex System 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Haag, R., N. M. Hugenholtz, andM. Winnink: On the equilibrium states in quantum statistical mechanics. To appear in Commun. Math. Phys.Google Scholar
  2. 2.
    Doplicher, S., D. Kastler, andD. W. Robinson: Commun. Math. Phys.3, 1 (1966).Google Scholar
  3. 3.
    Ruelle, D.: Commun. Math. Phys.3, 133 (1966).Google Scholar
  4. 4.
    Dixmier, J.: Les algèbres d'opérateurs dans l'espace hilbertien (algèbres de vonNeumann). Chapter I, § 6, theorem 2. Paris: Gauthier-Villars 1957.Google Scholar
  5. 5.
    Ref. 4, chapter III, § 1, theorem 6.Google Scholar
  6. 6.
    Kastler, D., andD. W. Robinson: Commun. Math. Phys.3, 151 (1966).Google Scholar
  7. 7.
    Araki, H., andE. J. Woods: J. Math. Phys.4, 637 (1963).Google Scholar
  8. 8.
    —— Progr. Theor. Phys.32, 956 (1964).Google Scholar

Copyright information

© Springer-Verlag 1967

Authors and Affiliations

  • N. M. Hugenholtz
    • 1
  1. 1.Natuurkundig Laboratorium der RijksuniversiteitGroningen

Personalised recommendations