Advertisement

Letters in Mathematical Physics

, Volume 92, Issue 3, pp 197–220 | Cite as

Spectral and Scattering Theory of Space-cutoff Charged \({P(\varphi)_{2}}\) Models

  • Christian Gérard
Article

Abstract

We consider in this paper space-cutoff charged \({P(\varphi)_{2}}\) models arising from the quantization of the non-linear charged Klein–Gordon equation:
$$(\partial_{t}+\i V(x))^{2}\phi(t, x)+ (-\Delta_{x}+ m^{2})\phi(t,x)+ g(x)\partial_{\overline{z}}P(\phi(t,x), \overline{\phi}(t,x))=0,$$
where V(x) is an electrostatic potential, g(x) ≥ 0 a space-cutoff, and \({P(\lambda, \overline{\lambda})}\) a real bounded below polynomial. We discuss various ways to quantize this equation, starting from different CCR representations. After describing the construction of the interacting Hamiltonian H we study its spectral and scattering theory. We describe the essential spectrum of H, prove the existence of asymptotic fields and of wave operators, and finally prove the asymptotic completeness of wave operators. These results are similar to the case when V = 0.

Mathematics Subject Classification (2000)

81T08 47N50 81Q10 81T10 

Keywords

quantum field theory scattering theory 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Baez, J., Segal, I., Zhou, S.: Introduction to Algebraic and Constructive Quantum Field Theory Princeton Series in Physics (1992)Google Scholar
  2. 2.
    Derezinski J., Gérard C.: Spectral and scattering theory of spatially cut-off \({P(\varphi)_{2}}\) Hamiltonians. Comm. Math. Phys. 213, 39–125 (2000)MATHCrossRefMathSciNetADSGoogle Scholar
  3. 3.
    Gérard C., Panati A.: Spectral and scattering theory for abstract QFT Hamiltonians. Rev. Math. Phys. 21, 373–437 (2009)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Glimm J., Jaffe A.: Boson quantum field theory models. In: Streater, R. (eds) Mathematics of Contemporary Physics, Academic Press, London (1972)Google Scholar
  5. 5.
    Palmer J.: Symplectic groups and the Klein–Gordon field. J. Funct. Anal. 27, 308–336 (1978)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Rosen L.: The \({(\phi^{2n})_{2}}\) quantum field theory: higher order estimates. Comm. Pure Appl. Math. 24, 417–457 (1971)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Simon B., Høgh-Krohn R.: Hypercontractive semigroups and two dimensional self-coupled bose fields. J. Funct. Anal. 9, 121–180 (1972)MATHCrossRefGoogle Scholar

Copyright information

© Springer 2010

Authors and Affiliations

  1. 1.Laboratoire de MathématiquesUniversité de Paris XIOrsay CedexFrance

Personalised recommendations