Abstract
We demonstrate, under circumstances that allow the construction of a “thermodynamic” hamiltonian, that Gibbs equilibrium states ω are modular states in the Tomita-Takesaki sense. The thermodynamic Greens functionsG are connected to these modular states, and the associated group of modular automorphisms σ, by the identification
(A andB are observables) whenever the thermodynamic Hamiltonian is self-adjoint and defines a derivation of the algebra of observables in a certain sense. Our results apply to a class of interacting quantum gases at small fugacity and Bose gases with repulsive interactions at all fugacitiesz<1.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Robinson, D. W.: Commun. math. Phys.7, 337 (1968)
Streater, R.: Commun. math. Phys.7, 93 (1968)
Ruelle, D.: Statistical Mechanics. New York: Benjamin 1969
Takesaki, M.: Lecture Notes in Mathematics, No. 128. Berlin-Heidelberg-New York: Springer 1970
Ruskai, M. B.: Commun. math. Phys.20, 195 (1971)
Ruelle, D.: Helv. Phys. Acta45, 215 (1972)
Sirigue, M., Winnink, M.: Marseille preprint (1970) unpublished. See Winnink, M.: Statistical Mechanics and Field Theory (eds. R. Sen, C. Weil). Jerusalem: Keter Publishing House 1971
Pulvirenti, M., Tirozzi, B.: Commun. math. Phys.30, 83 (1973)
Duneau, M., Souillard, B.: Commun. math. Phys.31, 113 (1973)
Haag, R., Hugenholtz, N., Winnink, M.: Commun. math. Phys.5, 215 (1967)
Riesz, F., Nagy, R.: Lecon d'Analyse Fonctionelle. Paris: Gauthiers Villars (1965)
Wigner, E. P., Yanase, M.-M.: Proc. Nat. Acad. Sci. USA49, 910 (1963)
Bratteli, O.: Comm. Pure Appl. Math.25, 759 (1972)
Ginibre, J.: Les Houches Summer School (1970). New York: Gordon and Breach (1972)
Robinson, D. W.: Commun. math. Phys.19, 219 (1970)
Ruelle, D.: Commun. math. Phys.18, 127 (1970)
Suhov, Ju. M.: Unpublished manuscript
Courbage, M., Miracle-Sole, S., Robinson, D. W.: Ann. Inst. Henri. Poincaré16, 71 (1971)
Novikov, I. D.: Funct. Anal. Appl.4, ⋆⋆⋆ (1970)
Suhov, Ju. M.: Russ. Math. Rev.83, 491 (1970)
Author information
Authors and Affiliations
Additional information
Communicated by J. L. Lebowitz
Rights and permissions
About this article
Cite this article
Bratteli, O., Robinson, D.W. Greens functions, Hamiltonians and modular automorphisms. Commun.Math. Phys. 50, 133–156 (1976). https://doi.org/10.1007/BF01617992
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01617992