Particle Creation at a Point Source by Means of Interior-Boundary Conditions

Abstract

We consider a way of defining quantum Hamiltonians involving particle creation and annihilation based on an interior-boundary condition (IBC) on the wave function, where the wave function is the particle-position representation of a vector in Fock space, and the IBC relates (essentially) the values of the wave function at any two configurations that differ only by the creation of a particle. Here we prove, for a model of particle creation at one or more point sources using the Laplace operator as the free Hamiltonian, that a Hamiltonian can indeed be rigorously defined in this way without the need for any ultraviolet regularization, and that it is self-adjoint. We prove further that introducing an ultraviolet cutoff (thus smearing out particles over a positive radius) and applying a certain known renormalization procedure (taking the limit of removing the cut-off while subtracting a constant that tends to infinity) yields, up to addition of a finite constant, the Hamiltonian defined by the IBC. MSC: 81T10, 81Q10, 47F05. Key words: Ultraviolet divergence problem; renormalization in quantum field theory; self-adjoint Hamiltonian; self-adjoint extensions of the Laplace operator; particle-position representation; ultraviolet cut-off.

On a direct description of pseudorelativistic Nelson Hamiltonians

“…In [LSTT17], a slightly different approach involving the adjoint L * 0 of the operator L 0 := L| ker(a(V )) and an extension A of a(V ) was used. The operator A is added to L * 0 and their sum is then restricted to a certain subspace of D(L * 0 ) which makes it a self-adjoint operator.…”

Bohmian Trajectories for Hamiltonians with Interior–Boundary Conditions

“…As pointed out already in [60] for Model 4 and in [59,38,65] for other models, other IBCs are possible that involve derivatives of ψ normal to the boundary. While the IBC (3) is of Dirichlet type in that it involves, like a Dirichlet boundary condition, the value but not the normal derivative of ψ on the boundary, an IBC of Neumann type involves the normal derivative but not the value of ψ, and one of Robin type involves both.…”

“…Dirichlet vs. Neumann vs. Robin conditions. We have used a Dirichlet-type IBC for Model 3, but Neumann-type or Robin-type conditions are equally possible [59,38], also with respect to the Bohmian dynamics. The version of the theory with the Dirichlet-type condition seems to be the physically most natural and relevant [59].…”

“…We will consider Models 1-4 in reverse order, the order of increasing complexity. For some of these models, the Hamiltonians of their IBC versions are known [38,35] to be bounded from below (as would be physically reasonable), although the IBC approach in general neither requires nor guarantees that Hamiltonians are bounded from below. The remainder of this paper is organized as follows.…”

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].…”