• DOI:10.1063/1.4902376
• Corpus ID: 119273327

Dirac equation with external potential and initial data on Cauchy surfaces

@article{Deckert2014DiracEW,
  title={Dirac equation with external potential and initial data on Cauchy surfaces},
  author={Dirk-Andr{\'e} Deckert and Franz Merkl},
  journal={Journal of Mathematical Physics},
  year={2014},
  volume={55},
  pages={122305}
}

• D. Deckert, F. Merkl
• Published 4 April 2014
• Mathematics
• Journal of Mathematical Physics

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 Poincare 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-Lindelof 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.