SciELO - Scientific Electronic Library Online

vol.14 número2An Immediate Derivation of Maximum Principle in Banach spaces, Assuming Reflexive Input and State SpacesOn the global behavior of índice de autoresíndice de materiabúsqueda de artículos
Home Pagelista alfabética de revistas  

Servicios Personalizados




Links relacionados


Cubo (Temuco)

versión On-line ISSN 0719-0646

Cubo vol.14 no.2 Temuco  2012 

CUBO A Mathematical Journal Vol.14, No 02, (91-109). June 2012

Texto completo disponíble en formato PDF

Higher order terms for the quantum evolution of a Wick observable within the Hepp method


Sébastien Breteaux

IRMAR, UMR-CNRS 6625, Universite de Rennes 1, campus de Beaulieu, 35042 Rennes Cedex, France.

ENS de Cachan, Antenne de Bretagne, Campus de Ker Lann, Av. R. Schuman,35170 Bruz, France. email: sebastien.breteaux@ens-cachan. org


The Hepp method is the coherent state approach to the mean field dynamics for bosons or to the semiclassical propagation. A key point is the asymptotic evolution of Wick observables under the evolution given by a time-dependent quadratic Hamiltonian. This article provides a complete expansion with respect to the small parameter which makes sense within the infinite-dimensional setting and fits with finite-dimensional formulae.

Keywords and Phrases: Mean field limit, semiclassical limit, coherent states, squeezed states


El metodo de Hepp describe en terminos de estados coherentes la dinamica en campo medio de bosones o la propagación semiclasica. Un punto clave es la evolución asintótica de observables de Wick bajo la evolucion dada por un Hamiltoniano cuadratico dependiente del tiempo. Este artículo proporciona una expansion completa con respecto al parámetro pequeno valido en dimension infinita y que corresponde a formulas en dimension finita conocidas.




[1] Z. Ammari and F. Nier. Mean field limit for bosons and infinite dimensional phase-space analysis. Ann. Henri Poincare',9(8):1503-1574, 2008.         [ Links ]

[2] J. C. Baez, I. E. Segal, and Z.-F. Zhou. Introduction to algebraic and constructive quantum feld theory. Princeton Series in Physics. Princeton University Press, Princeton, NJ, 1992.         [ Links ]

[3] F. A. Berezin. The method of second quantization. Academic Press, New York, 1966.         [ Links ]

[4] F. A. Berezin and M. A. Shubin. The Schrodinger equation, volume 66 of Mathematics and its Applications (Soviet Series). Kluwer Academic Publishers Group, Dordrecht, 1991.         [ Links ]

[5] L. Bruneau and J. Dereziński. Bogoliubov Hamiltonians and one-parameter groups of Bogoli-ubov transformations. J. Math. Phys., 48(2):022101, 24, 2007.         [ Links ]

[6] M. Combescure, J. Ralston, and D. Robert. A proof of the Gutzwiller semiclassical trace formula using coherent states decomposition. Comm. Math. Phys., 202(2):463-480, 1999.         [ Links ]

[7] M. Combescure and D. Robert. Semiclassical spreading of quantum wave packets and applications near unstable fixed points of the classical flow. Asymptot. Anal., 14(4):377-404,1997.         [ Links ]

[8] M. Combescure and D. Robert. Quadratic quantum Hamiltonians revisited. Cubo, 8(1):61-86,2006.         [ Links ]

[9] H. L. Cycon, R. G. Froese, W. Kirsch, and B. Simon. Schrodinger operators ,with application to quantum mechanics and global geometry. Texts and Monographs in Physics. Springer-Verlag, Berlin, study edition, 1987.         [ Links ]

[10] W. G. Faris and R. B. Lavine. Commutators and self-adjointness of Hamiltonian operators.         [ Links ]

[11] J. Fröhlich, S. Graffi, and S. Schwarz. Mean-field- and classical limit of many-body Schrodinger dynamics for bosons. Comm. Math. Phys., 271(3):681-697, 2007.         [ Links ]

[12] J. Ginibre and G. Velo. The classical field limit of scattering theory for nonrelativistic many-boson systems. I. Comm. Math. Phys., 66(1):37-76, 1979.         [ Links ]

[13] J. Ginibre and G. Velo. The classical field limit of scattering theory for nonrelativistic many-boson systems. II. Comm. Math. Phys., 68(1):45-68, 1979.         [ Links ]

[14] J. Ginibre and G. Velo. The classical field limit of nonrelativistic bosons. I. Borel summability for bounded potentials. Ann. Physics, 128(2):243-285, 1980.         [ Links ]

[15] J. Ginibre and G. Velo. The classical field limit of nonrelativistic bosons. II. Asymptotic expansions for general potentials. Ann. Inst. H. Poincare Sect. A (N.S.), 33(4):363-394, 1980.         [ Links ]

[16] M. G. Grillakis, M. Machedon, and D. Margetis. Second-order corrections to mean field evolution of weakly interacting bosons. I. Comm. Math. Phys., 294(1):273-301, 2010.         [ Links ]

[17] G. A. Hagedorn. Raising and lowering operators for semiclassical wave packets. Ann. Physics, 269(1):77-104, 1998.         [ Links ]

[18] G. A. Hagedorn and A. Joye. Semiclassical dynamics with exponentially small error estimates.Comm. Math. Phys., 207(2):439-465, 1999.         [ Links ]

[19] G. A. Hagedorn and A. Joye. Exponentially accurate semiclassical dynamics: propagation, localization, Ehrenfest times, scattering, and more general states. Ann. Henri Poincare,1(5):837-883, 2000.         [ Links ]

[20] K. Hepp. Phys., The classical limit for quantum mechanical correlation functions. Comm.Math. 35:265-277, 1974.         [ Links ]

[21] E. Hille. Lectures on ordinary differential equations. Addison-Wesley Publ. Co., Reading, Mass.-London-Don Mills, Ont., 1969.         [ Links ]

[22] L. Hörmander. Symplectic classifcation of quadratic forms, and general Mehler formulas. Math.Z., 219(3):413-449, 1995.         [ Links ]

[23] T. Kato. Linear evolution equations of "hyperbolic" type. J. Fac. Sci. Univ. Tokyo Sect. I,17:241-258, 1970.         [ Links ]

[24] J. Kisyński. Sur les opńerateurs de Green des problmes de Cauchy abstraits. Studia Math., 23:285-328, 1963/1964.         [ Links ]

[25] T. Paul and A. Uribe. A construction of quasi-modes using coherent states. Ann. Inst. H. Poincare Phys. Theor., 59(4):357-381, 1993.         [ Links ]

[26] L. Polley, G. Reents, and R. F. Streater. Some covariant representations of massless boson fields. J. Phys. A, 14(9):2479-2488, 1981.         [ Links ]

[27] M. Reed and B. Simon. Methods of modern mathematical physics. II. Fourier analysis, self-adjointness. Academic Press [Harcourt Brace Jovanovich Publishers], New York, 1975.         [ Links ]

[28] I. Rodnianski and B. Schlein. Quantum fuctuations and rate of convergence towards mean field dynamics. Comm. Math. Phys., 291(1):31-61, 2009.         [ Links ]

[29] D. Shale. Linear symmetries of free boson fields. Trans. Amer. Math. Soc., 103:149-167, 1962.         [ Links ]

Received: October 2011. Revised: November 2011.

Creative Commons License Todo el contenido de esta revista, excepto dónde está identificado, está bajo una Licencia Creative Commons