On Nelson-Type Hamiltonians and Abstract Boundary Conditions

Abstract

We construct Hamiltonians for systems of nonrelativistic particles linearly coupled to massive scalar bosons using abstract boundary conditions. The construction yields an explicit characterisation of the domain of self-adjointness in terms of boundary conditions that relate sectors with different numbers of bosons. We treat both models in which the Hamiltonian may be defined as a form perturbation of the free operator, such as Fröhlich's polaron, and renormalisable models, such as the massive Nelson model.

On a direct description of pseudorelativistic Nelson Hamiltonians

“…In the recent article [LS18], J. Lampart together with the author used abstract boundary conditions to characterise the domain and the action of certain otherwise ultraviolet-divergent Hamiltonians. Those Hamiltonians describe models where nonrelativistic scalar particles (often called nucleons) are linearly coupled to a field of massive scalar bosons, the most prominent of which is the so called Nelson model ( [Nel64]).…”

“…We will later assume that v p (k) is uniformly bounded by |k| −α for some α ∈ [0, d/2), as in [LS18]. Such form factors do not exhibit infrared-problems, because they are in L 2 loc (R d ).…”

“…In this note we will show that the abstract IBC method of [LS18] can be applied to Eckmann's (in d = 3) and to Gross' model (in d = 2). This will allow for the direct description of H ∞ as a self-adjoint operator on H .…”

“…In both models dicussed so far, the counter terms E Λ diverges for fixed p ∈ R d logarithmically when Λ → ∞, exactly as in the original Nelson model. With the method applied in [LS18] and in the present note, slightly more singular interactions can be treated (depending on various parameters and in a way to be made precise below). Recently it was shown in [Lam18] that the IBC approach, if modified in a suitable way, also allows for the definition of a more singular model.…”

Complex charges, time reversal asymmetry, and interior-boundary conditions in quantum field theory

“…For particle creation, one takes Q as in (3) and the IBC to relate boundary points of the n-particle sector to interior points in the (n − 1)particle sector, where the boundary configurations are those with two particles at the same location ("collision configurations"). We focus here on the spinless non-relativistic case based on the negative Laplacian operator as the free Hamiltonian of the bosons; for this case, IBCs were discussed in [20,21,10,13,12,11,25] after previous work in [14,15,22,27,23]. Bohmian trajectories associated with IBCs are defined in [6].…”