Nonlinear Phenomena in Complex Systems, 2024, Vol.27, No.4, pp.341

NONLINEAR PHENOMENA IN COMPLEX SYSTEMS
An Interdisciplinary Journal

2024, Vol.27, No.4, pp.341 - 365

Fradkin Equation for a Spin 3/2 Particle in an External Magnetic Field

A. V. Ivashkevich, E. M. Ovsiyuk, V. V. Kisel, and V. M. Red'kov

he first investigations of a spin 3/2 field were performed by Pauli and Fierz. Later, within the general approach of Gelfand and Yaglom to the theory of relativistic wave equations, a more general equation for spin 3/2 particle was proposed by Fradkin. This equation contains one additional parameter to present time, its physical interpretation is not understood, also no solutions for this equation are not known as well. In the present paper, the Fradkin equation has been solved in the presence of the uniform magnetic field.We apply the covariant tetrad formalism and use the system of cylindrical coordinates, the wave function transforms as a vector-bispinor under the local Lorentz group. On searched solutions, we diagonalize operators of the energy, the third projection of lineal momentum, and the third projection of the total angular momentum. After separating the variables, we derive the system of 16 differential equations in polar coordinate r. To resolve this system, we apply the Fedorov–Gronskiy method, which is based on four projective operator constructed from the 16×16 spin matrix S₃ for vector-bispinor wave function. According to this approach, each projective constituent is determined by only one corresponding function F_i(r), i = 1, 2, 3, 4. Solutions for these basic variables F_i are found in terms of the confluent hypergeometric functions; due to the presence of the external magnetic field, there arises the definite quantization rule for basic spectral parameter, which will be related with the possible values of energy of the particle. Within the used approach, there exist possibility to transform the system of 16 differential equations to homogeneous system of algebraic equations. From vanishing its determinant, we derive an equation of sixth order with respect to an energy parameter ε². Its solutions are studied numerically; in this way we arrive at four physically interpretable positive roots, and two complex-valued conjugate roots which relate to anomalous solutions.

Key words: spin 3/2 particle, Fradkin equation, tetrad formalism, external magnetic field, projective operator, exact solution

DOI: https://doi.org/10.5281/zenodo.14512040

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