Nonlinear Phenomena in Complex Systems, 2015, Vol.18, No.1, pp.44-62

NONLINEAR PHENOMENA IN COMPLEX SYSTEMS
An Interdisciplinary Journal

2015, Vol.18, No.1, pp.44-62

Peculiarities of the Squaring Method Applied to Construct Solutions of the Dirac, Majorana, and Weyl Equations
O. V. Veko and V. M. Red'kov

It is shown that the known method to solve the Dirac equation by means of the squaring method, when relying on the scalar function of the form Φ= e^−iete^ik₁xe^ik₂ysin(kz+α) leads to a 4-dimensional space of the Dirac solutions. It is shown that such a constructed basis is equivalent to the space of the Dirac states relied on the use of quantum numbers k₁, k₂, ±k and helicity operator; linear transformations relating these two spaces are found. Application of the squaring method substantially depends on the choice of representation for the Dirac matrices, some features of this are considered. Peculiarities of applying the squaring method in Majorana representation are investigated as well. The constructed bases are relevant to describe the Casimir effect for Dirac and Weyl fields in the domain restricted by two planes.

Key words: Dirac, Majorana, Weyl fields, squaring method, Casimir effect, quantization

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