Communications in Mathematical Physics

, Volume 353, Issue 2, pp 691–716 | Cite as

The Resolvent Algebra for Oscillating Lattice Systems: Dynamics, Ground and Equilibrium States

  • Detlev BuchholzEmail author


Within the C*-algebraic framework of the resolvent algebra for canonical quantum systems, the structure of oscillating lattice systems with bounded nearest neighbor interactions is studied in any number of dimensions. The global dynamics of such systems acts on the resolvent algebra by automorphisms and there exists a (in any regular representation) weakly dense subalgebra on which this action is pointwise norm continuous. Based on this observation, equilibrium (KMS) states as well as ground states are constructed, which are shown to be regular. It is also indicated how to deal with singular interactions and non-harmonic oscillations.


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  1. 1.
    Albeverio, S., Kondratiev, Y., Kozitsky, Y., Röckner, M.: The Statistical Mechanics of Quantum Lattice Systems: A Path Integral Approach. EMS Tracts Math 8 (2009)Google Scholar
  2. 2.
    Araki H.: Relative Hamiltonian for faithful normal states of a von Neumann algebra. Pub. RIMS Kyoto Univ. 9, 165–209 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bakhrakh V.L., Vetchinkin S.I., Khristenko S.V.: Green’s function of a multidimensional isotropic harmonic oscillator. Theor. Math. Phys. 12, 776–778 (1972)CrossRefGoogle Scholar
  4. 4.
    Bratteli O., Robinson D.W.: Operator Algebras and Quantum Statistical Mechanics II. Springer, Berlin (1981)CrossRefzbMATHGoogle Scholar
  5. 5.
    Buchholz D.: The resolvent algebra: Ideals and dimension. J. Funct. Anal. 266, 3286–3302 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Buchholz D., Grundling H.: Algebraic supersymmetry: a case study. Commun. Math. Phys. 272, 699–750 (2007)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Buchholz D., Grundling H.: The resolvent algebra: A new approach to canonical quantum systems. J. Funct. Anal. 254, 2725–2779 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Buchholz, D., Grundling, H.: Quantum systems and resolvent algebras. In: Blanchard, P., Fröhlich, J., (eds.) The Message of Quantum Science: Attempts Towards a Synthesis. Lect. Notes Phys. 899, pp. 33–45. Springer, Berlin (2015)Google Scholar
  9. 9.
    Dereziński J., Jaksić V., Pillet C.-A.: Perturbation theory of W-*- dynamics: Liouvilleans and KMS-states. Rev. Math. Phys. 15, 447–489 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Kanda, T., Matsui, T.: KMS states of weakly coupled anharmonic crystals and the resolvent CCR algebra. e-print arXiv:1601.04809
  11. 11.
    Minlos R.A., Verbeure A., Zagrebnov V.A.: A quantum crystal model in the light-mass limit: Gibbs states. Rev. Math. Phys. 12, 981–1032 (2000)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Nachtergaele B., Raz H., Schlein B., Sims R.: Lieb–Robinson bounds for harmonic and anharmonic lattice systems. Commun. Math. Phys. 286, 1073–1098 (2009)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Nachtergaele, B., Schlein, B., Sims, R., Starr, S., Zagrebnov, V.A.: On the existence of the dynamics for anharmonic quantum oscillator systems. Rev. Math. Phys. 22, 207–231 (2010)Google Scholar
  14. 14.
    Nachtergaele, B., Sims, R.: On the dynamics of lattice systems with unbounded on-site terms in the Hamiltonian. e-print arXiv:1410.8174v1
  15. 15.
    Reed M., Simon B.: Methods of Modern Mathematical Physics IV: Analysis of Operators. Acacemic Press, New York (1978)zbMATHGoogle Scholar
  16. 16.
    Ruelle D.: Natural nonequilibrium states in quantum statistical mechanics. J. Stat. Mech. 98, 57–75 (2000)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Møller J.S.: Fully coupled Pauli-Fierz systems at zero and positive temperature. J. Math. Phys. 55, 075203 (2014)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Takesaki M.: Disjointness of the KMS-states of different temperatures. Commun. Math. Phys. 17, 33–41 (1970)ADSMathSciNetCrossRefGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Institut für Theoretische PhysikUniversität GöttingenGöttingenGermany

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