Consistency of multi-time Dirac equations with general interaction potentials

Abstract

In 1932, Dirac proposed a formulation in terms of multi-time wave functions as candidate for relativistic many-particle quantum mechanics. A well-known consistency condition that is necessary for existence of solutions strongly restricts the possible interaction types between the particles. It was conjectured by Petrat and Tumulka that interactions described by multiplication operators are generally excluded by this condition, and they gave a proof of this claim for potentials without spin-coupling. Under smoothness assumptions of possible solutions we show that there are potentials which are admissible, give an explicit example, however, show that none of them fulfills the physically desirable Poincaré invariance. We conclude that in this sense Dirac's multi-time formalism does not allow to model interaction by multiplication operators, and briefly point out several promising approaches to interacting models one can instead pursue.

Multi-time formulation of particle creation and annihilation via interior-boundary conditions

“…For a variable number of particles, a multi-time wave function becomes a so-called multitime Fock function (see [19] and also [5,6]). It can be represented as a sequence of n-particle wave functions ψ (n) , ψ = (ψ (0) , ψ (1) , ψ (2) , ψ (3) , ...).…”

“…The dynamics is defined as follows. ψ (2) obeys the free multi-time Dirac equations on S (here in Hamiltonian form):…”

“…3.2). Expressions involving ψ on boundary points q ∈ ∂S , such as ψ (2) +− (t, z, t, z), denote limits of ψ within S 1 towards the boundary, e.g., ψ…”

“…The form of the IBC (25) can be motivated as follows. 2 In the case of no coupling between the two sectors (i.e., A = 0, B = 0), the model corresponds to free motion for the 1-particle sector and pure delta interactions for the 2-particle sector. Such relativistic delta interactions for multi-time wave functions have been treated in [13] and the appropriate form of the boundary conditions is known from there.…”

“…where S[Λ] are matrices forming a representation of the Lorentz group. In the context of QFT, ψ can be represented as a sequence of n-particle wave functions, ψ = (ψ (0) , ψ (1) , ψ (2) , ...). The multi-time picture thus offers the covariant version of the particleposition representation that a relativistic treatment of IBCs requires.…”