On a direct description of pseudorelativistic Nelson Hamiltonians

Abstract

interior-boundary conditions (IBC's) allow for the direct description of the domain and the action of Hamiltonians for a certain class of ultraviolet-divergent models in Quantum Field Theory. The method was recently applied to models where nonrelativistic scalar particles are linearly coupled to a quantised field, the best known of which is the Nelson model. Since this approach avoids the use of ultraviolet-cutoffs, there is no need for a renormalisation procedure. Here, we extend the IBC method to pseudorelativistic scalar particles that interact with a real bosonic field. We construct the Hamiltonians for such models via abstract boundary conditions, describing their action explicitly. In addition, we obtain a detailed characterisation of their domain and make the connection to renormalisation techniques. As an example, we apply the method to two relativistic variants of Nelson's model, which have been renormalised for the first time by J. P. Eckmann and A. D. Sloan in 1970 and 1974, respectively.

A Nonrelativistic Quantum Field Theory with Point Interactions in Three Dimensions

“…Examples include the Nelson model and the two-dimensional variant of our problem. These results were generalised by Schmidt to general dispersion relations for the x-particles [Sch18], and massless models [Sch19]. We expect that our analysis can be similarly generalised.…”

Hamiltonians without ultraviolet divergence for quantum field theories

“…This equation differs from the expression (26) for H orig only in the second line, where ψ(y n , 0) has been replaced by a more complicated expression involving the behavior of ψ near the configuration (y n , 0); after all, ψ diverges at this configuration by virtue of the IBC, so the expression ψ(y n , 0) does not make sense. The great similarity between H IBC and H orig adds to the suggestion that H IBC is the right Hamiltonian because it is a mathematical interpretation of the formal expression H orig .…”

“…Let us point out a connection between the second line of (29), the line that differed from the original Hamiltonian (26), and the corresponding line in (26). Think of ψ y n , rω as a function of r; as r → 0, a ψ from the domain of H IBC can be expanded in the form…”

“…Here is a reason for thinking that, among the many different IBCs that are mathematically possible corresponding to different choices of the constants α, β, γ, δ, only the Dirichlet-type IBC (27), corresponding to α = −4π/g, β = 0 = γ, and δ = g/4π, is physically relevant as a replacement of the original (UV divergent) Hamiltonian (26): It is the only choice that leads to a term in H IBC that reproduces the Dirac delta function terms in H orig , i.e., for which the last line of (49) agrees with the last line of (26). That is because the Dirichlet case is the only case in which the last line of (49) can be expressed in terms of ψ (n−1) .…”

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

“…, so e −i2θn times the conjugate of (22) yields (22) for T ψ). Then, both H IBC g T and T H IBC g yield the right-hand side of (23) with ψ replaced by ψ * and each term multiplied by the same phase factor as the corresponding term in (18), thus proving (17) for (23).…”