**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*^{−iet}e^{ik1x}e^{ik2y}sin(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
k_{1}, k_{2}, ±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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Last updated: *May 19, 2015*