Abstract
We consider the semi-relativistic Pauli–Fierz model for a single free electron interacting with the quantized radiation field. Employing a variant of Pizzo’s iterative analytic perturbation theory we construct a sequence of ground state eigenprojections of infra-red cutoff, dressing transformed fiber Hamiltonians and prove its convergence, as the cutoff goes to zero. Its limit is the ground state eigenprojection of a certain renormalized fiber Hamiltonian. The ground state energy is an exactly twofold degenerate eigenvalue of the renormalized Hamiltonian, while it is not an eigenvalue of the original fiber Hamiltonian unless the total momentum is zero. These results hold true, for total momenta inside a ball about zero of arbitrary radius \({\mathfrak{p} > 0}\), provided that the coupling constant is sufficiently small depending on \({\mathfrak{p}}\) and the ultra-violet cutoff. Along the way we prove twice continuous differentiability and strict convexity of the ground state energy as a function of the total momentum inside that ball.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Abdesselam A., Hasler D.: Analyticity of the ground state energy for massless Nelson models. Comm. Math. Phys. 310, 511–536 (2012)
Bach, V., Chen, T., Faupin, J., Fröhlich, J., Sigal, I.M.: Effective dynamics of an electron coupled to an external potential in non-relativistic QED. Ann. Henri Poincaré arXiv:1202.3189 (2012). doi:10.1007/s00023-012-0222-8
Bach V., Chen T., Fröhlich J., Sigal I.M.: The renormalized electron mass in non-relativistic quantum electrodynamics. J. Funct. Anal. 243, 426–535 (2007)
Bach V., Fröhlich J., Pizzo A.: Infrared-finite algorithms in QED: the groundstate of an atom interacting with the quantized radiation field. Commun. Math. Phys. 264, 145–165 (2006)
Bach V., Fröhlich J., Pizzo A.: An infrared-finite algorithm for Rayleigh scattering amplitudes, and Bohr’s frequency condition. Commun. Math. Phys. 274, 457–486 (2007)
Bach V., Fröhlich J., Pizzo A.: Infrared-finite algorithms in QED. II. The expansion of the groundstate of an atom interacting with the quantized radiation field. Adv. Math. 220, 1023–1074 (2009)
Bach V., Könenberg M.: Construction of the ground state in nonrelativistic QED by continuous flows. J. Differ. Equs. 231, 693–713 (2006)
Bachmann S., Deckert D.A., Pizzo A.: The mass shell of the Nelson model without cut-offs. J. Funct. Anal. 263, 1224–1282 (2012)
Chen T.: Infrared renormalization in non-relativistic QED and scaling criticality. J. Funct. Anal. 254, 2555–2647 (2008)
Chen, T., Fröhlich, J.: Coherent infrared representations in non-relativistic QED. In: Spectral theory and mathematical physics: a Festschrift in honor of Barry Simon’s 60th birthday. (Proceedings of the Symposium on Pure Mathematics), vol. 76, pp. 25–45. Am. Math. Soc., Providence (2007)
Chen T., Fröhlich J., Pizzo A.: Infraparticle scattering states in non-relativistic quantum electrodynamics. II. Mass shell properties. J. Math. Phys. 50(012103), 34 (2009)
Chen T., Fröhlich J., Pizzo A.: Infraparticle scattering states in non-relativistic QED. I. The Bloch-Nordsieck paradigm. Commun. Math. Phys. 294, 761–825 (2010)
Deckert, D.A., Pizzo, A.: Ultraviolet properties of the spinless, one-particle Yukawa Model. Preprint, arXiv:1208.2646 (2012)
Fröhlich J.: On the infrared problem in a model of scalar electrons and massless, scalar bosons. Ann. Inst. Henri Poincaré Sect. A (N.S.) 19, 1–103 (1973)
Fröhlich J.: Existence of dressed one electron states in a class of persistent models. Fortschritte Phys. 22, 159–198 (1974)
Fröhlich J., Griesemer M., Schlein B.: Asymptotic electromagnetic fields in models of quantum-mechanical matter interacting with the quantized radiation field. Adv. Math. 164, 349–398 (2001)
Fröhlich J., Pizzo A.: Renormalized electron mass in nonrelativistic QED. Commun. Math. Phys. 294, 439–470 (2010)
Hasler D., Herbst I.: Absence of ground states for a class of translation invariant models of non-relativistic QED. Commun. Math. Phys. 279, 769–787 (2008)
Hiroshima F., Sasaki I.: On the ionization energy of the semi-relativistic Pauli–Fierz model for a single particle. RIMS Kokyuroku Bessatsu. 21, 25–34 (2010)
Hiroshima F., Spohn H.: Ground state degeneracy of the Pauli–Fierz Hamiltonian with spin. Adv. Theor. Math. Phys. 5, 1091–1104 (2001)
Kato T.: Perturbation theory for linear operators, Classics in Mathematics. Springer, Berlin (1995)
Könenberg, M., Matte, O.: Ground states of semi-relativistic Pauli–Fierz and no-pair Hamiltonians in QED at critical Coulomb coupling. J. Operator Theory 70(1) (2013)
Könenberg, M., Matte, O.: On enhanced binding and related effects in the non- and semi-relativistic Pauli–Fierz models. arXiv:1207.5638 to appear in Commun. Math. Phys. (2012)
Könenberg M., Matte O., Stockmeyer E.: Existence of ground states of hydrogen-like atoms in relativistic quantum electrodynamics I: The semi-relativistic Pauli–Fierz operator. Rev. Math. Phys. 23, 375–407 (2011)
Könenberg M., Matte O., Stockmeyer E.: Existence of ground states of hydrogen-like atoms in relativistic quantum electrodynamics II: The no-pair operator. J. Math. Phys. 52, 123501 (2011)
Matte O.: On higher order estimates in quantum electrodynamics. Documenta Math. 15, 207–234 (2010)
Matte O., Stockmeyer E.: Exponential localization of hydrogen-like atoms in relativistic quantum electrodynamics. Commun. Math. Phys. 295, 551–583 (2010)
Miyao T., Spohn H.: Spectral analysis of the semi-relativistic Pauli–Fierz Hamiltonian. J. Funct. Anal. 256, 2123–2156 (2009)
Møller J.S.: The translation invariant massive Nelson model. I. The bottom of the spectrum. Ann. Henri Poincaré. 6, 1091–1135 (2005)
Pizzo A.: One-particle (improper) states in Nelson’s massless model. Ann. Henri Poincaré. 4, 439–486 (2003)
Pizzo A.: Scattering of an Infraparticle: the one particle sector in Nelson’s massless model. Ann. Henri Poincaré. 6, 553–606 (2005)
Reed M., Simon B.: Methods of modern mathematical physics. II. Fourier analysis, self-adjointness. Academic Press, New York (1975)
Schroer B.: Infrateilchen in der Quantenfeldtheorie. Fortschritte Phys. 11, 1–32 (1963)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Jan Derezinski.
Rights and permissions
About this article
Cite this article
Könenberg, M., Matte, O. The Mass Shell in the Semi-Relativistic Pauli–Fierz Model. Ann. Henri Poincaré 15, 863–915 (2014). https://doi.org/10.1007/s00023-013-0268-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00023-013-0268-2