Abstract
The asymptotic behavior of the effective mass meff(Λ) of the so-called Nelson model in quantum field theory is considered, where Λ is an ultraviolet cutoff parameter of the model. Let m be the bare mass of the model. It is shown that for sufficiently small coupling constant α of the model, meff(Λ)/m can be expanded as meff(Λ)/m=1+∑n=1∞an(Λ)α2n. A physical folklore is that an(Λ)=O(logΛ(n-1)) as Λ→∞. It is rigorously shown that 0<limΛ→∞a1(Λ)<C, C1≤limΛ→∞a2(Λ)/logΛ≤C2 with some constants C, C1, and C2.
Highlights
Introduction and Main ResultsThe model considered in this paper is the so-called Nelson model [1], which describes a nonrelativistic nucleon with bare mass m > 0 interacting with a quantized scalar field with mass ] > 0
The Fock vacuum Ω ∈ F is defined by Ω = {1, 0, 0, . . .}
The coupling of the nucleon and a scalar field is mediated through the Segal field operator Φ푠(g) defined by 1 √2 (a (g) + a (g)∗), (5)
Summary
The model considered in this paper is the so-called Nelson model [1], which describes a nonrelativistic nucleon with bare mass m > 0 interacting with a quantized scalar field with mass ] > 0. Let us first define the Nelson Hamiltonian. Let T be a self-adjoint operator on L2(R3). We define the self-adjoint operator dΓ(T) on F by dΓ(T) = ⨁∞ 푛=0T(푛), where. The coupling of the nucleon and a scalar field is mediated through the Segal field operator Φ푠(g) defined by. The Nelson Hamiltonian with total momentum p ∈ R3 is given by a self-adjoint operator on F as follows:. Let E(p, α) be the energymomentum relation (the infimum of the spectrum σ(H(p))) defined by. The effective mass and energy-momentum relation have been studied mainly in nonrelativistic electrodynamics. Hiroshima and Spohn [3] study a perturbative mass renormalization including fourth order in the coupling constant in the case of a spinless electron. Frohlich and Pizzo [7] investigate energy-momentum relation when infrared cutoff goes to 0
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