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Infrared Problem for the Nelson Model on Static Space-Times

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Abstract

We consider the Nelson model on some static space-times and investigate the problem of existence of a ground state. Nelson models with variable coefficients arise when one replaces in the usual Nelson model the flat Minkowski metric by a static metric, allowing also the boson mass to depend on position. We investigate the existence of a ground state of the Hamiltonian in the presence of the infrared problem, i.e. assuming that the boson mass m(x) tends to 0 at spatial infinity. We show that if m(x) ≥ C |x|−1 at infinity for some C > 0 then the Nelson Hamiltonian has a ground state.

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References

  1. Ammari Z.: Asymptotic completeness for a renormalized non-relativistic Hamiltonian in quantum field theory: the Nelson model. Math. Phys. Anal. Geom. 3, 217–285 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  2. Arai A., Hirokawa M., Hiroshima F.: On the absence of eigenvectors of Hamiltonians in a class of massless quantum field models without infrared cutoff. J. Funct. Anal. 168, 470–497 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bach V., Fröhlich J., Sigal I.M.: Quantum electrodynamics of confined non-relativistic particles. Adv. Math. 137, 299–395 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bachelot A.: The Hawking effect. Ann. Inst. H. Poincaré Phys. Théor. 70, 41–99 (1999)

    MathSciNet  MATH  Google Scholar 

  5. Betz V., Hiroshima F., Lörinczi J., Minlos R.A., Spohn H.: Ground state properties of the Nelson Hamiltonian – a Gibbs measure-based approach. Rev. Math. Phys. 14, 173–198 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bruneau L., Dereziński J.: Pauli-Fierz Hamiltonians defined as quadratic forms. Rep. Math. Phys. 54, 169–199 (2004)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  7. Brunetti R., Fredenhagen K., Köhler M.: The microlocal spectrum condition and Wick polynomials of free fields on curved space-times. Commun. Math. Phys. 180, 633–652 (1996)

    Article  ADS  MATH  Google Scholar 

  8. Davies, E.B.: Heat Kernels and Spectral Theory. Cambridge Tracts in Mathematics 92. Cambridge: Cambridge University Press, 1989

    Google Scholar 

  9. de Bièvre S., Merkli M.: The Unruh effect revisited. Class. Quant. Grav. 23, 6525–6542 (2006)

    Article  MATH  Google Scholar 

  10. Derezinski, J., Gérard, C.: Scattering Theory of Classical and Quantum N. Particle Systems. Texts and Monographs in Physics, Berlin-Heidelberg-New York: Springer-Verlag, 1997

  11. Derezinski J., Gérard C.: Asymptotic completeness in quantum field theory. Massive Pauli-Fierz Hamiltonians. Rev. Math. Phys. 11, 383–450 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  12. Derezinski J., Gérard C.: Scattering theory of infrared divergent Pauli-Fierz Hamiltonians. Annales Henri Poincaré 5, 523–578 (2004)

    Article  ADS  MATH  Google Scholar 

  13. Fredenhagen K., Haag R.: On the derivation of Hawking radiation associated with the formation of a black hole. Commun. Math. Phys. 127, 273–284 (1990)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  14. Georgescu V., Gérard C., Moeller J.: Spectral theory of massless Nelson models. Commun. Math. Phys. 249, 29–78 (2004)

    Article  ADS  MATH  Google Scholar 

  15. Gérard C.: On the existence of ground states for massless Pauli-Fierz Hamiltonians. Ann. Henri Poincaré 1, 443–455 (2000)

    Article  MATH  Google Scholar 

  16. Gérard C., Hiroshima F., Panati A., Suzuki A.: Infrared Divergence of a Scalar Quantum Field Model on a Pseudo Riemannian Manifold. Interdisciplinary Information Sciences 15, 399–421 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  17. Gérard, C., Hiroshima, F., Panati, A., Suzuki, A.: Absence of ground state for the Nelson model on static space-times. http://arxiv.org/abs/1012.2655vI [math-ph], 2010

  18. Gérard, C., Hiroshima, F., Panati, A., Suzuki, A.: Removal of UV cutoff for the Nelson model on static space-times. In preparation

  19. Gérard C., Panati A.: Spectral and scattering theory for some abstract QFT Hamiltonians. Rev. Math. Phys. 21, 373–437 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  20. Griesemer M.: Exponential decay and ionization thresholds in non-relativistic quantum electrodynamics. J. Funct. Anal. 210, 321–340 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  21. Griesemer M., Lieb E., Loss M.: Ground states in non-relativistic quantum electrodynamics. Invent. Math. 145, 557–595 (2001)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  22. Hawking S.W.: Particle creation by black holes. Commun. Math. Phys. 43, 199–220 (1975)

    Article  MathSciNet  ADS  Google Scholar 

  23. Hirokawa M.: Infrared catastrophe for Nelson’s model, non-existence of ground state and soft-boson divergence. Publ. RIMS, Kyoto Univ. 42, 897–922 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  24. Lörinczi J., Minlos R.A., Spohn H.: The infrared behavior in Nelson’s model of a quantum particle coupled to a massless scalar field. Ann. Henri Poincaré 3, 1–28 (2002)

    Article  Google Scholar 

  25. Milman P.D., Semenov Y.A.: Global heat kernel bounds via desingularizing weights. J. Funct. Anal. 212, 373–398 (2004)

    Article  MathSciNet  Google Scholar 

  26. Nelson E.: Interaction of non-relativistic particles with a quantized scalar field. J. Math. Phys. 5, 1190–1997 (1964)

    Article  ADS  Google Scholar 

  27. Panati A.: Existence and nonexistence of a ground state for the massless Nelson model under binding condition. Rep. Math. Phys. 63, 305–330 (2009)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  28. Porper F.O., Eidel’man S.D.: Two sided estimates of fundamental solutions of second order parabolic equations and some applications. Russ. Math. Surv. 39, 119–178 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  29. Radzikowski M.: Micro-local approach to the Hadamard condition in quantum field theory on curved space-time. Commun. Math. Phys. 179, 529–553 (1996)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  30. Radzikowski M.: A local-to-global singularity theorem for quantum field theory on curved space-time. Commun. Math. Phys. 180, 1–22 (1996)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  31. Reed, M., Simon, B.: Methods of Modern Mathematical Physics, I: Functional Analysis. New York: Academic Press, 1975

  32. Reed, M., Simon, B.: Methods of Modern Mathematical Physics, II: Fourier Analysis, Self-adjointness. New York: Academic Press, 1975

  33. Sanders K.: Equivalence of the (generalized) Hadamard and microlocal spectrum condition for (generalized) free fields in curved space-time. Commun. Math. Phys. 295, 485–501 (2010)

    Article  ADS  MATH  Google Scholar 

  34. Semenov Y.A.: Stability of l p−spectrum of generalized Schrödinger operators and equivalence of Green’s functions. IMRN 12, 573–593 (1997)

    Article  Google Scholar 

  35. Simon B.: Functional Integration and Quantum Physics. Academic Press, New York (1979)

    MATH  Google Scholar 

  36. Spohn H.: Ground state of a quantum particle coupled to a scalar boson field. Lett. Math. Phys. 44, 9–16 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  37. Unruh W.G.: Notes on black hole evaporation. Phys. Rev. D 14, 870–892 (1976)

    Article  ADS  Google Scholar 

  38. Unruh W.G, Wald R.: What happens when an accelerating observer detects a rindler particle. Phys. Rav. D 29, 1047–1056 (1984)

    Article  ADS  Google Scholar 

  39. Zhang Q.S.: Large time behavior of Schroedinger heat kernels and applications. Commun. Math. Phys. 210, 371–398 (2000)

    Article  ADS  MATH  Google Scholar 

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Authors and Affiliations

  1. Département de Mathématiques, Université de Paris XI, 91405, Orsay Cedex, France

    Christian Gérard

  2. Department of Mathematics, University of Kyushu, 6-10-1, Hakozaki, Fukuoka, 812-8581, Japan

    Fumio Hiroshima

  3. UMR 6207, Université Toulon-Var, 83957, La Garde Cedex, France

    Annalisa Panati

  4. Department of Mathematics, Faculty of Engineering, Shinshu University, 4-17-1 Wakasato, Nagano, 380-8553, Japan

    Akito Suzuki

Authors
  1. Christian Gérard
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  2. Fumio Hiroshima
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  3. Annalisa Panati
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  4. Akito Suzuki
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Correspondence to Christian Gérard.

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Communicated by I.M. Sigal

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Gérard, C., Hiroshima, F., Panati, A. et al. Infrared Problem for the Nelson Model on Static Space-Times. Commun. Math. Phys. 308, 543–566 (2011). https://doi.org/10.1007/s00220-011-1289-7

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  • DOI: https://doi.org/10.1007/s00220-011-1289-7

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