The Mass Shell in the Semi-Relativistic Pauli–Fierz Model

2014

Commun. Math. Phys.

We consider the one-particle sector of the spinless Yukawa model, which describes the interaction of a nucleon with a real field of scalar massive bosons (neutral mesons). The nucleon as well as the mesons have relativistic dispersion relations. In this model we study the dependence of the nucleon mass shell on the ultraviolet cut-of . For any finite ultraviolet cut-off the nucleon one-particle states are constructed in a bounded region of the energy-momentum space. We identify the dependence of the ground state energy on and the coupling constant. More importantly, we show that the model considered here becomes essentially trivial in the limit → ∞ regardless of any (nucleon) mass and self-energy renormalization. Our results hold in the small coupling regime

Section: Introduction and Definition Of The Modelmentioning

confidence: 99%

Ultraviolet Properties of the Spinless, One-Particle Yukawa Model

Deckert

2014

Commun. Math. Phys.

“…The following theorem collects some results of the companion paper [DP17], which are relevant for the present investigation. We remark that regularity of P → E P,σ in infrared singular models was studied before in particular in [AH12, KM12,FP10]. Analyticity of ground state projections was established in some infrared regular models in [FFS14].…”

Section: Preliminaries and Resultsmentioning

confidence: 96%

Coulomb Scattering in the Massless Nelson Model III: Ground State Wave Functions and Non-commutative Recurrence Relations

Dybalski

2017

Ann. Henri Poincaré

Let H P,σ be the single-electron fiber Hamiltonians of the massless Nelson model at total momentum P and infrared cut-off σ > 0. We establish detailed regularity properties of the corresponding n-particle ground state wave functions f n P,σ as functions of P and σ. In particular, we show thatwhere c is a numerical constant, λ 0 → δ λ 0 is a positive function of the maximal admissible coupling constant which satisfies lim λ 0 →0 δ λ 0 = 0 and χ [σ,κ) is the (approximate) characteristic function of the energy region between the infrared cut-off σ and the ultraviolet cut-off κ. While the analysis of the first derivative is relatively straightforward, the second derivative requires a new strategy. By solving a non-commutative recurrence relation we derive a novel formula for f n P,σ with improved infrared properties. In this representation ∂ P j ′ ∂ P j f n P,σ is amenable to sharp estimates obtained by iterative analytic perturbation theory in part II of this series of papers. The bounds stated above are instrumental for scattering theory of two electrons in the Nelson model, as explained in part I of this series.

“…Properties (A.5)-(A.8) date back to [Pi03] and (A.9) even to [Fr73,Fr74]. Estimate (A.10), which is slightly stronger than strict convexity stated in (A.1), was shown in [KM12,FP10]…”

Section: Variation Of the Infrared Cut-offmentioning

confidence: 95%

Coulomb Scattering in the Massless Nelson Model I. Foundations of Two-Electron Scattering

Dybalski

2013

J Stat Phys

We construct two-electron scattering states and verify their tensor product struc- ture in the infrared-regular massless Nelson model. The proof follows the lines of Haag- Ruelle scattering theory: Scattering state approximants are defined with the help of two time-dependent renormalized creation operators of the electrons acting on the vacuum. They depend on the ground state wave functions of the (single-electron) fiber Hamiltonians with infrared cut-off. The convergence of these approximants as t → ∞ is shown with the help of Cook’s method combined with a non-stationary phase argument. The removal of the infrared cut-off in the limit t → ∞ requires sharp estimates on the derivatives of these ground state wave functions w.r.t. electron and photon momenta, with mild dependence on the infrared cut-off. These key estimates, which carry information about the localization of the electrons in space, are obtained in a companion paper with the help of iterative analytic perturbation theory. Our results hold in the weak coupling regime