Diffraction for the Dirac–Coulomb Propagator

Møller operators and Hadamard states for Dirac fields with MIT boundary conditions

“…The requirement in Theorem 1.1 is also known as "propagation of singularity theorem" and it has been used for the scalar wave equation in various settings e.g. [31,47,61,62,63,74,75,76,77,9,58]. We expect a similar result to hold since the propagation of singularity for Dirac operators reduces to the propagation of singularity for the scalar wave operator, even though it is acting on vector-valued quantities whose boundary values are coupled.…”

“…We expect the propagation of singularities to hold true because there are already positive results in this direction, see e.g. [31,47,62,63,77] for the scalar wave equation, [59,74] for first order systems, and [9] for the Dirac-Coulomb system. We postpone this investigation to a forthcoming paper.…”

“…where n = − 1 β ∂ t is the past-directed unit normal vector to Σ while n ♭ = g(n, •) = βdt. In the next lemma, we shall prove that, if S is skew-adjoint, then the scalar product (9) does not depend on the choice of the Cauchy hypersurface Σ ⊂ M. Lemma 2.31. Let Σ ⊂ M be a smooth spacelike Cauchy hypersurface with its past-oriented unit normal vector field n and its induced volume element vol Σ .…”

“…On the one hand, the physical system classically described by Sol sc,mit (D) is quantized by introducing a unital * -algebra A, whose elements are interpreted as observables for the system under investigation. In a second stage, the description of possible physical states of the system is described through the choice of a suitable subclass of linear, positive and normalized functionals ω : A → C. By extending the analogous definition for a spacetime without boundary, we shall now introduce the * -algebra A associated with the space Sol sc,mit (D) of solutions with spacelike compact support of the Dirac operator D coupled with MIT boundary conditions and endowed with the positive definite Hermitian scalar product (9). To this avail we shall profit from the results and definition already present in the literature, see [2].…”