Skip to main content
Log in

On the equilibrium states in quantum statistical mechanics

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Kubo, R.: J. Phys. Soc. Japan12, 570 (1957).

    Google Scholar 

  2. Martin, P. C., andJ. Schwinger: Phys. Rev.115, 1342 (1959).

    Google Scholar 

  3. Araki, H., andE. J. Woods: J. Math. Phys.4, 637 (1963).

    Google Scholar 

  4. Doplicher, S., D. Kastler, andD. W. Robinson: Commun. Math. Phys.3, 1 (1966).

    Google Scholar 

  5. Ruelle, D.: Commun. Math. Phys.3, 133 (1966).

    Google Scholar 

  6. Kastler, D., andD. W. Robinson: Commun. Math. Phys.3, 151 (1966).

    Google Scholar 

  7. Haag, R.: Lecture Notes, Pacific Summer School, Hawaii (1965).

    Google Scholar 

  8. Araki, H., andW. Wyss: Helv. Phys. Acta37, 136 (1964).

    Google Scholar 

  9. Dell Antonio, G. F., S. Doplicher, andRuelle D.: Commun. Math. Phys.2, 223 (1966).

    Google Scholar 

  10. Ruelle, D.: Helv. Phys. Acta36, 789 (1963).

    Google Scholar 

  11. Dyson, F. J., andA. Lenard: Preprint.

  12. Dixmier, J.: Les algèbres d'operateurs dans l'espace Hilbertien (Algèbres de von Neumann). Paris: Gauthier-Villars 1957.

    Google Scholar 

  13. Segal, I.E.: Mat. Fys. Medd. Dan. Vid. Selsk.31, 12 (1959).

    Google Scholar 

  14. Nelson, E.: Ann. Math.70, 572 (1959).

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Haag, R., Hugenholtz, N.M. & Winnink, M. On the equilibrium states in quantum statistical mechanics. Commun.Math. Phys. 5, 215–236 (1967). https://doi.org/10.1007/BF01646342

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01646342

Keywords

Navigation