Dirac equation with external potential and initial data on Cauchy surfaces

Abstract

With this paper we provide a mathematical review on the initial-value problem of the one-particle Dirac equation on space-like Cauchy hypersurfaces for compactly supported external potentials. We, first, discuss the physically relevant spaces of solutions and initial values in position and mass shell representation; second, review the action of the Poincaré group as well as gauge transformations on those spaces; third, introduce generalized Fourier transforms between those spaces and prove convenient Paley-Wiener-and Sobolev-type estimates. These generalized Fourier transforms immediately allow the construction of a unitary evolution operator for the free Dirac equation between the Hilbert spaces of square-integrable wave functions of two respective Cauchy surfaces. With a Picard-Lindelöf argument this evolution map is generalized to the Dirac evolution including the external potential. For the latter we introduce a convenient interaction picture on Cauchy surfaces. These tools immediately provide another proof of the well-known existence and uniqueness of classical solutions and their causal structure.

A Perspective on External Field QED

“…In other words, for A = 0, instantly infinitely many electron-positron pairs are created from the vacuum state Ω. Therefore, the picture is not nearly as peaceful as suggested by example state (5). However, if A is switched off at some later time one can expect that almost all of these pairs disappear again, and only a few excitations of the vacuum as in (5) will remain (hence, the name virtual pairs that is used by physicists).…”

“…A Lorentz boost, for example, tilts an equal-time hyperplane to a Cauchy surface Σ which requires a change from the standard Hilbert space H = L 2 (R 3 , C 4 ) to one that is attached to Σ, and likewise, for the corresponding Fock spaces. Hence, a Lorentz transform will naturally be described by a map from one Fock space into another [5]. In the special case of equaltime hyperplanes, parts of this program have been carried out in [21,22] and [4].…”

“…Since we avoid technicalities in this paper, we assume A is a smooth and compactly supported (although sufficient strong decay would be sufficient), and the following theorem will suffice to discuss the one-particle Dirac evolution. Theorem 2.3 (Theorem 2.23 in [5]). Let Σ, Σ ′ be two Cauchy surfaces and ψ Σ ∈ C Σ the initial data.…”

“…This space consists of the sea wave function ΛΦ, its excitations ΛΨ that form a generating set, and superpositions thereof. An example excitation analogous to (5) representing an electronpositron pair with electron wave function χ ∈ V ⊥ and positron wave function ϕ 1 ∈ V is given by…”

“…and also its corresponding adjoint a, which is called annihilation operator. The state Ψ from example in (5) can then be written as Ψ = a * (χ)a(ϕ 1 )Ω. With the help of a * , one-particle operators like the evolution operator U A generated by (4) can be lifted to an operator U on F in a canonical way by requiring that…”