Abstract
We give here a mathematical description of Gibbs states for general quantum lattice harmonic oscillators. We determinate the set of Gibbs measures for quadratic interactions on an abstract Hilbert space and then we specialize to the state space adapted to the quantum description. Next, we add physically relevant assumptions on the support of the measures. Under these assumptions, we exhibit a a one-to-one correspondence between the quantum Gibbs states and the Gibbs measures for a classical model of quadratic interactions. Moreover, we give a nice criterium for the existence of uniqueness of a quantum Gibbs state.
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References
Albeverio, S. and Høegh-Krohn, R.: 'Homogeneous random fields and quantum statistical mechanics', J. Funct. Anal. 19 (1975), 242–272.
Albeverio, S., Kondratiev, Y. and Kozitsky, Y.: 'Classical limits of Euclidean Gibbs states for quantum lattice models', Lett. Math. Phys. 48(3) (1999), 221–233.
Albeverio, S., Kondratiev, Y., Röckner, M. and Tsikalenko, T.: 'Uniqueness of Gibbs states for quantum lattice systems', Probab. Theory Related Fields 108 (1997), 193–218.
Albeverio, S., Kondratiev, Y.G. and Tsikalenko, T.V.: 'Stochastic dynamics for quantum lattice systems and stochastic quantization. I. Ergodicity', Random Oper. Stochastic Equations 2(2) (1994), 103–139.
Barbulyak, V. and Kondratiev, Y.: 'Functional integrals and quantum lattice systems: I. Existence of Gibbs states', Reports Nat. Acad. Sci. Ukraine 8 (1991), 31–34.
Barbulyak, V. and Kondratiev, Y.: 'Functional integrals and quantum lattice systems: II. Periodical Gibbs states', Reports Nat. Acad. Sci. Ukraine 9 (1991), 38–40.
Berezansky, Y. and Kondratiev, Y.: Spectral Methods in Infinite-Dimensional Analysis, Kluwer Acad. Publ., Dordrecht, 1995.
Dalecky, Y. and Fomin, S.: Measures and Differential Equations in Infinite-Dimensional Space, Kluwer Acad. Publ., Dordrecht, 1991.
Dobrushin, R.: 'Gaussian random fields-Gibbsian point of view', in R.D. and Ya.G. Sinai (eds), Multicomponent Random Systems, Adv. Probab. Related Topics, Marcel Dekker, New York, 1980.
Faris, W. and Minlos, R.: 'A quantum crystal with multidimensional anharmonic oscillators', J. Statist. Phys. 94(3/4) (1999), 365–387.
Fortet, R.: Vecteurs, fonctions et distributions aléatoires dans les espaces de Hilbert, Hermés, Paris, 1995.
Garet, O.: 'Mesures de Gibbs gaussiennes et dynamiques aléatoires associées sur ℝ;ℤd, Ph.D. Thesis, Université de Lille I, 1998.
Garet, O.: 'Gibbsian fields associated to exponentially decreasing quadratic potentials', Ann. Inst. H. Poincaré, Probabilités 35 (1999), 387–415.
Garet, O.: 'Infinite dimensional dynamics associated to quadratic Hamiltonians', Markov Proc. Rel. Fields 6(2) (2000), 205–237.
Georgii, H.-O.: Gibbs Measures and Phase Transitions, Walter de Gruyter, Berlin, 1988.
Globa, S. and Kondratiev, Y.G.: 'The construction of Gibbs states of quantum lattice systems', Selecta Math. Soviet. 9 (1990), 297–307.
Guyon, X.: Champs aléatoires sur un réseau, Masson, Paris, 1992.
Guyon, X.: Random Fields on a Network, Springer-Verlag, New York, 1995.
Guyon, X. and Künsch, H.R.: 'Asymptotic comparison of estimators in the Ising model', in Stochastic Models, Statistical Methods, and Algorithms in Image Analysis (Rome, 1990), Springer, 1992, pp. 177-198.
Kahane, J.: Séries de Fourier absolument convergentes, Springer, New York, 1970.
Klein, A. and Landau, L.: 'Stochastic processes associated with KMS states', J. Funct. Anal. 42 (1981), 368–428.
Künsch, H.-R.: 'Gaussian Markov random fields', J. Fac. Sci. Univ. of Tokyo 26 (1979), 53–73.
Künsch, H.-R.: 'Reellwertige Zufallsfelder auf einem Gitter: Interpolationsprobleme, Variationsprinzip und statistische Analyse', Ph.D. Thesis, Zürich, 1980.
Kondratiev, Y., Roelly, S. and Zessin, H.: 'Stochastic dynamics for an infinite system of random closed strings', Stochastic Process Appl. 61 (1996), 223–248.
Kondratiev, Y.G. and Sokol, T.: 'Gaussian measures and stochastic quantification', Selecta Math. Soviet. 13(3) (1994), 225–238.
Minlos, R., Verbeure, A. and Zagrebnov, V.: 'A quantum crystal model in the light mass limit: Gibbs states', Rev. Math. Phys. (1999).
Park, Y. and Yoo, H.: 'A characterization of Gibbs states of lattice boson systems'. J. Statist. Phys. 75 (1994), 215–239.
Park, Y. and Yoo, H.: 'Uniqueness and clustering properties of Gibbs states for classical and quantum unbounded spin systems', J. Statist. Phys. 80 (1995), 223–271.
Rosanov, Y.A.: 'On Gaussian fields with given conditional distributions', Theory Probab. Appl. 12 (1967), 433–443.
Simon, B.: Functional Integrals in Quantum Physics, Academic Press, 1986.
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Garet, O. Harmonic Oscillators on an Hilbert Space: A Gibbsian Approach. Potential Analysis 17, 65–88 (2002). https://doi.org/10.1023/A:1015237532453
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DOI: https://doi.org/10.1023/A:1015237532453