Bohmian Trajectories for Hamiltonians with Interior–Boundary Conditions

Abstract

Recently, there has been progress in developing interior-boundary conditions (IBCs) as a technique of avoiding the problem of ultraviolet divergence in nonrelativistic quantum field theories while treating space as a continuum and electrons as point particles. An IBC can be expressed in the particle-position representation of a Fock vector ψ as a condition on the values of ψ on the set of collision configurations, and the corresponding Hamiltonian is defined on a domain of vectors satisfying this condition. We describe here how Bohmian mechanics can be extended to this type of Hamiltonian. In fact, part of the development of IBCs was inspired by the Bohmian picture. Particle creation and annihilation correspond to jumps in configuration space; the annihilation is deterministic and occurs when two particles (of the appropriate species) meet, whereas the creation is stochastic and occurs at a rate dictated by the demand for the equivariance of the |ψ| 2 distribution, time reversal symmetry, and the Markov property. The process is closely related to processes known as Bell-type quantum field theories.

Interior-boundary conditions for Schrödinger operators on codimension-1 boundaries

“…We begin with a special case that features many elements of the general discussion that will follow. Let the configuration space Q = Q (1) ∪ Q (2) be the union of Q (1) = R d−1 (where d is any natural number, not related to the dimension of physical space) and a half-space Q (2)…”

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

“…Now we can introduce Bohmian trajectories for H ϕ g [7,8] and H IBC g [6]. Except for particle creation and annihilation, the actual configuration Q t ∈ Q moves according to Bohm's equation of motion,…”

“…. From the point Y on the boundary of the (n + 1)-particle sector of Q, Q t moves into the interior according to (24) [21,6].…”

“…In both versions of the theory, with cut-off or with IBC, it follows from the defining laws that if Q t is |ψ(t)| 2 distributed for t = 0, then Q t is |ψ(t)| 2 distributed also for t > 0 [8,6]. One thus obtains a stochastic process (Q ψ t ) t≥0 associated with every (normalized) initial wave function ψ.…”

“…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]. We find that (7) applies again and more generally that, in many situations with IBCs, a generic choice of coefficients creates a time asymmetry.…”