Abstract
We study a nonrelativistic quantum system coupled, via a quadratic interaction [cf. formula (1.10) below], to a free Boson gas in the Fock state. We prove that, in the low density limit (z 2=fugacity→0), the matrix elements of the wave operator of the system at timet/z 2 in the collective coherent vectors converge to the matrix elements, in suitable coherent vectors of the quantum Brownian motion process, of a unitary Markovian cocycle satisfying a quantum stochastic differential equation driven by some pure number process (i.e. no quantum diffusion part and only the quantum analogue of the purely discontinuous, or jump, processes). This proves that the number (or quantum Poisson) processes, introduced by Hudson and Parthasarathy and studied by Frigerio and Maassen, arise effectively as conjectured by the latter two authors as low density limits of Hamiltonian models.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Accardi, L.: On the quantum Feynman-Kac formula. Rendiconti Mat. Fisico, Milano48, 135–180 (1978)
Accardi, L.: Noise and dissipation in quantum theory, to appear in Rev. Modern Phys. (1991)
Accardi, L., Frigerio, A., Lu, Y.G.: The weak coupling limit as a quantum functional central limit. Commun. Math. Phys.131, 537–570 (1990)
Accardi, L., Frigerio, A., Lu, Y.G.: On the weak coupling limit (II): Langevin equation and finite temperature case, submitted to R.I.M.S.
Accardi, L., Frigerio, A., Lu, Y.G.: The weak coupling for Fermion case. J. Math. Phys.32, 1567–1581 (1991)
Accardi, L., Lu, Y.G.: On the weak coupling limit for nonlinear interactions. Ascona Stochastic Processes, Physics and Geometry, 1–26
Accardi, L., Lu, Y.G.: On the weak coupling limit without rotating wave approximation. Anna. Ins. J. Poincaré-Theo. Phys.54(4), pp. 1–24 (1991)
Accardi, L., Bach, A.: The harmonic oscillator as quantum central limit of Bernoulli processes. Prob. Theoret. Rel. Fields (to appear)
Alicki, R., Frigerio, A.: Quantum Poisson noise and linear Boltzmann equation. Preprint, March 1989
Davies, E.B.: Markovian master equation. Commun. Math. Phys.39, 91–110 (1974)
Dümcke, R.: The low density limit for anN-level system interacting with a free Bose or Fermi gas. Commun. Math. Phys.97, 331–359 (1985)
Fagnola, F.: A martingale characterization of quantum Poisson process. Preprint February, 1989
Frigerio, A.: Quantum Poisson processes: physical motivations and applications. Lecture Notes in Mathematics, vol. 1303, pp. 107–127, Berlin, Heidelberg, New York: Springer 1988
Frigerio, A., Maassen, H.: Quantum Poisson processes and dilations of dynamical semigroups. Prob. Theoret. Rel. Fields83, 489–508 (1989)
Grad, H.: Principles of the kinetic theory of gases. Handbuch der Physik, vol. 12. Berlin, Heidelberg, New York: Springer 1958
Hudson, R.L., Lindsay, M.: Uses of non-Fock quantum Brownian motion and a quantum martingale representation theorem. Lect. Notes in Math., vol. 1136, pp. 276–305. Berlin, Heidelberg, New York: Springer
Hudson, R.L., Parthasarathy, K.R.: Quantum Ito's formula and stochastic evolutions. Commun. Math. Phys.93, 301–323 (1984)
Palmer, P.F.: Ph.D. Thesis, Oxford University
Pulé, J.V.: The Bloch equations. Commun. Math. Phys.38, 241–256 (1974)
Kümmerer, B.: Markov Dilation onW *-Algebras. J. Funct. Anal.63, 139–177 (1985)
Accardi, L., Alicki, R., Frigerio, A., Lu, Y.G.: An invitation to the weak coupling and low density limits, to appear in: Quantum probability and Application VI
Accardi, L., Lu, Y.G.: The low density limit and the quantum Poisson process, to appear in: Proc. 5th Vilnus Conf. Prob. Th.
Author information
Authors and Affiliations
Additional information
Communicated by J.L. Lebowitz
Rights and permissions
About this article
Cite this article
Accardi, L., Lu, Y.G. The number process as low density limit of Hamiltonian models. Commun.Math. Phys. 141, 9–39 (1991). https://doi.org/10.1007/BF02100003
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02100003