Abstract
The fundamental solution of the Dirac equation for an electron in an electromagnetic field with harmonic dependence on space-time coordinates is obtained. The field is composed of three standing plane harmonic waves with mutually orthogonal phase planes and the same frequency. Each standing wave consists of two eigenwaves with different complex amplitudes and opposite directions of propagation. The fundamental solution is obtained in the form of the projection operator defining the subspace of solutions to the Dirac equation. It is illustrated by the analysis of the ground state and the spin precession of the Dirac electron in the field of two counterpropagating plane waves with left and right circular polarizations. Interrelations between the fundamental solution and approximate partial solutions is discussed and a criterion for evaluating accuracy of approximate solutions is suggested.
Highlights
Considerable recent attension has been focussed on the possibility of time and space-time crystals [1,2,3,4], analogous to ordinary crystals in space
The relations presented and appendix provide convenient means to find operators ρk(n) by making use the recurrent algorithm devised to minimize volumes of computations and data files [11, 12]. It begins with the selection of an infinite subsystem consisting from independent equations and the calculation of the projection operators ρ0(n) = P (n), n ∈ F0 ⊂ L, which uniquely define the fundamental solutions of these equations
The projection operator S′ (36) defines the exact fundamental solution of the finite subsystem (35) which expands with each new step of the recurrent process
Summary
Considerable recent attension has been focussed on the possibility of time and space-time crystals [1,2,3,4], analogous to ordinary crystals in space. The description of the motion of electrons in ESTCs by the Dirac equation takes into account both the space-time periodicity of the vector potential and the intrinsic electron properties (charge, spin, and magnetic moment). In this case, the Dirac equation reduces to an infinite system of matrix equations. The method of projection operators is very useful at problem solving in classical and quantum field theory [8,9,10] It was developed by Fedorov [8, 9] to treat finite systems of linear homogeneous equations. Additional information on the numerical implementation of the presented approach and some results of its computer simulation can be found in [11,12,13]
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