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.
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December 2014
Research Article|
December 22 2014
Dirac equation with external potential and initial data on Cauchy surfaces
D.-A. Deckert;
D.-A. Deckert
a)
1
Mathematical Department of the University of California Davis
, One Shield Ave, Davis, California 95616, USA
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J. Math. Phys. 55, 122305 (2014)
Article history
Received:
May 11 2014
Accepted:
November 08 2014
Citation
D.-A. Deckert, F. Merkl; Dirac equation with external potential and initial data on Cauchy surfaces. J. Math. Phys. 1 December 2014; 55 (12): 122305. https://doi.org/10.1063/1.4902376
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