Bohr's Correspondence Principle for the Renormalized Nelson Model

Abstract

In the mid Sixties Edward Nelson proved the existence of a consistent quantum field theory that describes the Yukawa-like interaction of a non-relativistic nucleon field with a relativistic meson field. Since then it is thought, despite the renormalization procedure involved in the construction, that the quantum dynamics should be governed in the classical limit by a Schrödinger-Klein-Gordon system with Yukawa coupling. In the present paper we prove this fact in the form of a Bohr correspondence principle. Besides, our result enlighten the nature of the renormalization method employed in this model which we interpret as a strategy that allows to put the related classical Hamiltonian PDE in a normal form suitable for a canonical quantization.

Mean-Field Dynamics for the Nelson Model with Fermions

“…In the theorem we also provide bounds for the specific initial state N j=1 ϕ 0 j ⊗W (N 2/3 α 0 )Ω. Since for this state γ (1,0)…”

Mean-Field Limits of Particles in Interaction with Quantized Radiation Fields

“…A comparison between his result and Theorem III.3 is given in Remark III.4. Making use of a Wigner measure approach Ammari and Falconi [1] were able to establish the classical limit (without quantitative bounds on the rate of convergence) of the renormalized Nelson model without cutoff. Teufel [27] considered the adiabatic limit of the Nelson model and showed that the interaction mediated by the quantized radiation field is well approximated by a direct Coulomb interaction.…”

“…These involve fewer degrees of freedom, are less exact but easier to investigate. Effective evolution equations for particles that interact with quantized radiation fields have rigorously been derived for example in [9,6,1,27,7,8,14,10]. The general setting in these works is given by the tensor product of two Hilbert spaces…”

“…The state W (γ 1/2 α 0 )Ω ∈ F denotes gauge bosons in the coherent state α 0 with a mean particle number γ ||α 0 || 2 , see (16). Hereby, γ is a model dependent scaling parameter, for instance the number of particles [6,1,14] or the strong coupling parameter in the Polaron model [7,8,10]. From physics literature it is commonly known that coherent states with a high occupation number of gauge bosons can approximately be described by a classical radiation field [4,Chapter III.C.4].…”

“…We refer to this limit as mean-field limit, because it implies that the source term of the radiation field is replaced by its mean value in the effective description. So far, such kind of limits have been studied either by the coherent state approach [9,5,6] or by means of Wigner measures [1]. 1 While the method of Wigner measures allows us to derive limiting equations for an extensive class of initial states it does in contrast to the coherent state approach not provide quantitative bounds on the rate of 1 These approaches usually embed the N particle states of H (N) p in a bosonic Fock space for the particles Fp and consider the Hilbert space Fp ⊗ F. convergence.…”