Non-Relativistic Matter and Quantized Radiation

  • M. Griesemer
Chapter
Part of the Lecture Notes in Physics book series (LNP, volume 695)

Abstract

This is a didactic review of spectral and dynamical properties of atoms and molecules at energies below the ionization threshold, the focus being on recent work in which the author was involved. As far as possible, the results are described using a simple model with one electron only, and with scalar bosons. The main ideas are explained but no complete proofs are given. The full-fledged standard model of non-relativistic QED and various of its aspects are described in the appendix.

Keywords

Annihilation Operator Quantum Electrodynamic Rayleigh Scattering Ionization Threshold Dense Subspace 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Download to read the full chapter text

References

  1. 1.
    Robert A. Adams. Sobolev spaces. Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975. Pure and Applied Mathematics, Vol. 65.Google Scholar
  2. 2.
    Shmuel Agmon. Lectures on exponential decay of solutions of second-order elliptic equations: bounds on eigenfunctions of N-body Schrödinger operators, volume 29 of Mathematical Notes. Princeton University Press, Princeton, NJ, 1982.MATHGoogle Scholar
  3. 3.
    Asao Arai. Rigorous theory of spectra and radiation for a model in quantum electrodynamics. J. Math. Phys., 24(7):1896–1910, 1983.MATHMathSciNetCrossRefADSGoogle Scholar
  4. 4.
    Asao Arai and Masao Hirokawa. On the existence and uniqueness of ground states of a generalized spin-boson model. J. Funct. Anal., 151(2):455–503, 1997.MATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Asao Arai and Masao Hirokawa. Ground states of a general class of quantum field Hamiltonians. Rev. Math. Phys., 12(8):1085–1135, 2000.MATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    V. Bach, J. Fröhlich, and A. Pizzo. private communication by Jürg Fröhlich, July 2004.Google Scholar
  7. 7.
    Volker Bach, Jürg Fröhlich, and Israel Michael Sigal. Quantum electrodynamics of confined nonrelativistic particles. Adv. Math., 137(2):299–395, 1998.MATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Volker Bach, Jürg Fröhlich, and Israel Michael Sigal. Renormalization group analysis of spectral problems in quantum field theory. Adv. Math., 137(2):205–298, 1998.MATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Volker Bach, Jürg Fröhlich, and Israel Michael Sigal. Spectral analysis for systems of atoms and molecules coupled to the quantized radiation field. Comm. Math. Phys., 207(2):249–290, 1999.MATHMathSciNetCrossRefADSGoogle Scholar
  10. 10.
    Volker Bach, Jürg Fröhlich, Israel Michael Sigal, and Avy Soffer. Positive commutators and the spectrum of Pauli-Fierz Hamiltonian of atoms and molecules. Comm. Math. Phys., 207(3):557–587, 1999.MATHMathSciNetCrossRefADSGoogle Scholar
  11. 11.
    Ola Bratteli and Derek W. Robinson. Operator algebras and quantum statistical mechanics. 2. Texts and Monographs in Physics. Springer-Verlag, Berlin, second edition, 1997. Equilibrium states. Models in quantum statistical mechanics.Google Scholar
  12. 12.
    R. Carmona and B. Simon. Pointwise bounds on eigenfunctions and wave packets in N-body quantum systems. V. Lower bounds and path integrals. Comm. Math. Phys., 80(1):59–98, 1981.MATHMathSciNetCrossRefADSGoogle Scholar
  13. 13.
    Claude Cohen-Tannoudji, Jacques Dupont-Roc, and Gilbert Grynberg. Photons and Atoms - Introduction to Quantum Electrodynamics. Wiley-Interscience, February 1997.Google Scholar
  14. 14.
    J. Dereziński and C. Gérard. Asymptotic completeness in quantum field theory. Massive Pauli-Fierz Hamiltonians. Rev. Math. Phys., 11(4):383–450, 1999.MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    J. Dereziński and C. Gérard. Spectral scattering theory of spatially cut-off P(Ф)2 Hamiltonians. Comm. Math. Phys., 213(1):39–125, 2000.MathSciNetCrossRefADSMATHGoogle Scholar
  16. 16.
    J. Fröhlich, M. Griesemer, and B. Schlein. Asymptotic completeness for Comp-ton scattering, to appear in Communications in Mathematical Physics.Google Scholar
  17. 17.
    J. Fröhlich, M. Griesemer, and B. Schlein. Asymptotic electromagnetic fields in models of quantum-mechanical matter interacting with the quantized radiation field. Adv. Math., 164(2):349–398, 2001.MATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    J. Fröhlich, M. Griesemer, and B. Schlein. Asymptotic completeness for Rayleigh scattering. Ann. Henri Poincaré, 3(1):107–170, 2002.MATHCrossRefGoogle Scholar
  19. 19.
    C. Gérard. On the existence of ground states for massless Pauli-Fierz Hamiltonians. Ann. Henri Poincaré, 1(3):443–459, 2000.MATHCrossRefGoogle Scholar
  20. 20.
    Christian Gerard. On the scattering theory of massless nelson models, mp-arc 01–103, March 2001.Google Scholar
  21. 21.
    David Gilbarg and Neil S. Trudinger. Elliptic partial differential equations of second order. Classics in Mathematics. Springer-Verlag, Berlin, 2001. Reprint of the 1998 edition.Google Scholar
  22. 22.
    Gian Michele Graf and Daniel Schenker. Classical action and quantum N-body asymptotic completeness. In Multiparticle quantum scattering with applications to nuclear, atomic and molecular physics (Minneapolis, MN, 1995), pp. 103–119. Springer, New York, 1997.Google Scholar
  23. 23.
    G.M. Graf. Asymptotic completeness for N-body short-range quantum systems: A new proof. Comm. Math. Phys., 132:73–102, 1990.MATHMathSciNetCrossRefADSGoogle Scholar
  24. 24.
    M. Griesemer. Exponential decay and ionization thresholds in non-relativistic quantum electrodynamics. J. Funct. Anal., 210(2):321–340, 2004.MATHMathSciNetCrossRefGoogle Scholar
  25. 25.
    Marcel Griesemer, Elliott H. Lieb, and Michael Loss. Ground states in non-relativistic quantum electrodynamics. Invent. Math., 145(3):557–595, 2001.MATHMathSciNetCrossRefADSGoogle Scholar
  26. 26.
    Christian Hainzl. Enhanced binding through coupling to a photon field. In Mathematical results in quantum mechanics (Taxco, 2001), volume 307 of Con-temp. Math., pp. 149–154. Amer. Math. Soc., Providence, R.I. 2002.Google Scholar
  27. 27.
    Christian Hainzl, Vitali Vougalter, and Semjon A. Vugalter. Enhanced binding in non-relativistic QED. Comm. Math. Phys., 233(1):13–26, 2003.MATHMathSciNetCrossRefADSGoogle Scholar
  28. 28.
    F. Hiroshima. Self–adjointness of the Pauli-Fierz Hamiltonian for arbitrary values of coupling constants. Ann. Henri Poincaré, 3(1):171–201, 2002.MATHMathSciNetCrossRefGoogle Scholar
  29. 29.
    F. Hiroshima and H. Spohn. Enhanced binding through coupling to a quantum field. Ann. Henri Poincaré, 2(6):1159–1187, 2001.MATHMathSciNetCrossRefGoogle Scholar
  30. 30.
    Fumio Hiroshima. Ground states of a model in nonrelativistic quantum electrodynamics. I. J. Math. Phys., 40(12):6209–6222, 1999.MATHMathSciNetCrossRefADSGoogle Scholar
  31. 31.
    Fumio Hiroshima. Ground states of a model in nonrelativistic quantum electrodynamics. II. J. Math. Phys., 41(2):661–674, 2000.MATHMathSciNetCrossRefADSGoogle Scholar
  32. 32.
    R. Hoegh-Krohn. Asymptotic fields in some models of quantum field theory, ii, iii. J. Mathematical Phys., 11:185–188, 1969.MathSciNetCrossRefADSGoogle Scholar
  33. 33.
    Matthias Hubner and Herbert Spohn. The spectrum of the spin-boson model. In Mathematical results in quantum mechanics (Blossin, 1993), volume 70 of Oper. Theory Adv. Appl, pp. 233–238. Birkhauser, Basel, 1994.Google Scholar
  34. 34.
    Matthias Hiibner and Herbert Spohn. Radiative decay: nonperturbative approaches. Rev. Math. Phys., 7(3):363–387, 1995.MathSciNetCrossRefGoogle Scholar
  35. 35.
    Matthias Hiibner and Herbert Spohn. Spectral properties of the spin-boson Hamiltonian. Ann. Inst. H. Poincare Phys. Theor., 62(3):289–323, 1995.MathSciNetGoogle Scholar
  36. 36.
    W. Hunziker and I. M. Sigal. The quantum N-body problem. J. Math. Phys., 41(6):3448–3510, 2000.MATHMathSciNetCrossRefADSGoogle Scholar
  37. 37.
    V. Jakš1ić and C.-A. Pillet. On a model for quantum friction. I. Fermi's golden rule and dynamics at zero temperature. Ann. Inst. H. Poincaré Phys. Theoré, 62(1):47–68, 1995.Google Scholar
  38. 38.
    V. Jakš1ić and C.-A. Pillet. On a model for quantum friction. II. Fermi's golden rule and dynamics at positive temperature. Comm. Math. Phys., 176(3):619–644, 1996.MathSciNetCrossRefADSGoogle Scholar
  39. 39.
    Vojkan Jakš1ić and Claude-Alain Pillet. On a model for quantum friction. III. Ergodic properties of the spin-boson system. Comm. Math. Phys., 178(3):627–651, 1996.MathSciNetCrossRefADSGoogle Scholar
  40. 40.
    Vojkan Jakš1ić and Claude-Alain Pillet. Spectral theory of thermal relaxation. J. Math. Phys., 38(4):1757–1780, 1997. Quantum problems in condensed matter physics.MathSciNetCrossRefADSGoogle Scholar
  41. 41.
    Tosio Kato. Fundamental properties of Hamiltonian operators of Schrodinger type. Trans. Amer. Math. Soc., 70:195–211, 1951.MATHMathSciNetCrossRefGoogle Scholar
  42. 42.
    E. Lieb and M. Loss. A note on polarization vectors in quantum electrodynamics. math-ph/0401016, Jan. 2004.Google Scholar
  43. 43.
    Elliott H. Lieb and Michael Loss. Existence of atoms and molecules in nonrelativistic quantum electrodynamics. Adv. Theor. Math. Phys., 7(4):667–710, 2003.MathSciNetMATHGoogle Scholar
  44. 44.
    J. Lorinczi, R. A. Minlos, and H. Spohn. The infrared behaviour in Nelson's model of a quantum particle coupled to a massless scalar field. Ann. Henri Poincare, 3(2):269–295, 2002.MathSciNetCrossRefGoogle Scholar
  45. 45.
    Matthias Mück. Construction of metastable states in quantum electrodynamics. Rev. Math. Phys., 16(1):1–28, 2004.MathSciNetCrossRefMATHGoogle Scholar
  46. 46.
    Arne Persson. Bounds for the discrete part of the spectrum of a semi-bounded Schrödinger operator. Math. Scand., 8:143–153, 1960.MATHMathSciNetGoogle Scholar
  47. 47.
    Michael Reed and Barry Simon. Methods of modern mathematical physics. II. Fourier analysis, self-adjointness. Academic Press [Harcourt Brace Jovanovich Publishers], New York, 1975.Google Scholar
  48. 48.
    Michael Reed and Barry Simon. Methods of modem mathematical physics. IV. Analysis of operators. Academic Press [Harcourt Brace Jovanovich Publishers], New York, 1978.Google Scholar
  49. 49.
    Michael Reed and Barry Simon. Methods of modern mathematical physics. III. Academic Press [Harcourt Brace Jovanovich Publishers], New York, 1979. Scattering theory.Google Scholar
  50. 50.
    I.M. Sigal and A. Softer. The N-particle scattering problem: asymptotic completeness for short-range systems. Annals of Math., 126:35–108, 1987.MATHMathSciNetCrossRefGoogle Scholar
  51. 51.
    Herbert Spohn. Asymptotic completeness for Rayleigh scattering. J. Math. Phys., 38(5):2281–2296, 1997.MATHMathSciNetCrossRefADSGoogle Scholar
  52. 52.
    Herbert Spohn. Ground state of a quantum particle coupled to a scalar Bose field. Lett. Math. Phys., 44(1):9–16, 1998.MATHMathSciNetCrossRefGoogle Scholar
  53. 53.
    Herbert Spohn. Dynamics of Charged Particles and their Radiation Field. Cambridge University Press, August 2004.Google Scholar
  54. 54.
    Walter Thirring. A course in mathematical physics. Vol. 3. Springer-Verlag, New York, 1981. Quantum mechanics of atoms and molecules, Translated from the German by Evans M. Harrell, Lecture Notes in Physics, 141.Google Scholar
  55. 55.
    D. Yafaev. Radiation condition and scattering theory for N-particle Hamilto-nians. Comm. Math. Phys., pp. 523–554, 1993.Google Scholar

Copyright information

© Springer 2006

Authors and Affiliations

  • M. Griesemer
    • 1
  1. 1.Department of MathematicsUniversity of Alabama at BirminghamBirminghamUSA

Personalised recommendations