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Proceedings of the National Academy of Sciences of Belarus Physics and Mathematics Series

A spin 1/2 particle with anomalous magnetic and electric dipole moments

Elena M. Ovsiyuk, A. Safronov, А. Д. Коральков, V. V. Kisel

Published Apr 4, 2022• 0 citations
Publisher

Abstract

It is Petras who first developed the P-symmetric theory for a spin 1/2 particle with an anomalous magnetic moment within the general Gel’fand – Yaglom approach. Recently, similarly it was introduced a P-asymmetric wave equation for a spin 1/2 particle which describes a particle with an electric dipole moment. In this paper, we study solutions of the equation for the P-asymmetric particle in presence of external magnetic fields. It turns out that the energy spectra are the same for P-asymmetric and P-symmetric particles. To clarify this coincidence, we demonstrate that there exists a simple transformation relating these two models, by which one wave equation can be reduced to the form of the other. Meanwhile, expressions for wave functions and P-reflection operators are different in these two theories. We extend this approach to the model in which both P-symmetric and P-asymmetric sectors are presented. The main result is the same, namely, there exists a simple, more general as compared with the mentioned above transformation relating the P-symmetric model and the model with two sectors, and expressions for wave functions and P-reflection operators are different in these two bases. We demonstrate that in the presence of an external uniform magnetic field, the energy spectra in the model with two sectors coincide with those in the P-symmetric theory. Thus, we develop a general theory for the P-asymmetric model and the model with two sectors within the Petras approach.

DOI
10.29235/1561-2430-2022-58-1-76-89

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It is Petras who first developed the P-symmetric theory for a spin 1/2 particle with an anomalous magnetic moment within the general Gel’fand – Yaglom approach. Recently, similarly it was introduced a P-asymmetric wave equation for a spin 1/2 particle which describes a particle with an electric dipole moment. In this paper, we study solutions of the equation for the P-asymmetric particle in presence of external magnetic fields. It turns out that the energy spectra are the same for P-asymmetric and P-symmetric particles. To clarify this coincidence, we demonstrate that there exists a simple transformation relating these two models, by which one wave equation can be reduced to the form of the other. Meanwhile, expressions for wave functions and P-reflection operators are different in these two theories. We extend this approach to the model in which both P-symmetric and P-asymmetric sectors are presented. The main result is the same, namely, there exists a simple, more general as compared with the mentioned above transformation relating the P-symmetric model and the model with two sectors, and expressions for wave functions and P-reflection operators are different in these two bases. We demonstrate that in the presence of an external uniform magnetic field, the energy spectra in the model with two sectors coincide with those in the P-symmetric theory. Thus, we develop a general theory for the P-asymmetric model and the model with two sectors within the Petras approach.
In this chapter, within the general method of Gel'fand–Yaglom, starting with an extended set of representations of the Lorentz group, we have constructed a relativistic generalized system of the first-order equations for a spin 1 particle with two additional characteristics, anomalous magnetic moment and polarizability. We have taken into account the presence of external electromagnetic fields. After eliminating the accessory variables of the complete wave function, we derive the minimal 10-component form of equations, the last includes two additional interaction terms which are interpreted as related to anomalous magnetic and moment and polarizability of the particle. This approach is extended to space-time models with pseudo-Riemannian structure, within the tetrad method. We specify the basic equations to the cylindrical coordinates, and also in presence of external uniform magnetic field. After separating the variables, we get the system of ten first-order-differential equations in polar coordinate. To resolve this system, we apply the method of Fedorov–Gronskiy; and the complete 10-component wave function is decomposed into the sum of three projective constituents, where dependence of each component on the polar coordinate is determined by only one function, https://www.w3.org/1998/Math/MathML" display="inline"> F i ( r ) , i = 1 , 2 , 3 . We construct them in terms of confluent hypergeometric equations; at this a quantization rule for some spectral parameter exists. Additionally arises an algebraic homogeneous system of ten equations, which completely determines the structure of 10-component solutions. By vanishing the determinant of the linear system, we derive a cubic algebraic equation with respect to energy parameter ϵ 2 . Its solutions are found in analytical form, and studied numerically. For different values of parameters, anomalous magnetic moment and polarizability, we have found three series of physically interpretable energy levels. For more details see [ 111 ]–[ 119 ].
The generalized Duffin–Kemmer equation for a spin 1 particle with the anomalous magnetic moment in the external uniform magnetic field is investigated. The separation of variables in the wave equation is performed on the basis of projective operator techniques. The problem is reduced to a system of differential equations for three independent functions that have been solved in terms of the confluent hypergeometric functions. Three series of the energy levels are found. To assign them the physical sense at all values of the main quantum number n = 0,1, 2, , special restrictions on anomalous magnetic moment values must be imposed – they are formulated in explicit form. Otherwise, only some part of the energy levels corresponds to the bound states. The neutral spin 1 particle is considered as well. In this case, no bound states exist in the systems. The main qualitative manifestation of the anomalous magnetic moment is the space scaling of the arguments of the wave functions in comparison with a particle without such a moment.
In recent times there has been a renewed interest in the force experienced by a charged-particle with anomalous magnetic moment in the presence of external fields. In this paper we address the basic question of the force experienced by a spin-1/2 point-like charged-particle with magnetic and electric moments in the presence of space-and time-dependent external electromagnetic fields, when derived from the Dirac equation {\em via} the Foldy-Wouthuysen transformation technique. It is interesting to note that the force thus derived differs from the ones obtained by various other prescriptions.
The St¨uckelberg equation for a particle with two spin states, S = 1 and S = 0, is studied in the presence of an external uniform magnetic field. In relativistic case, the particle is described by an 11-component wave function. On the solutions of the equation, the operators of energy, the third projection of the total angular momentum, and the third projection of the linear momentum along the direction of the magnetic field are diagonalized. After separation of variables, we derive a system for 11 functions depending on one variable. We perform the nonrelativistic approximation in this system. For this we apply the known method of deriving nonrelativistic equations from relativistic ones, which is based on projective operators related to the matrix Γ0 of the relativistic equation. The nonrelativistic wave function turns out to be 4-dimensional. We derive the system for 4 functions. It is solved in terms of confluent hypergeometric functions. There arise three series of energy levels with corresponding solutions. This result agrees with that obtained for the relativistic St¨uckelberg equation.
In this chapter, 16-component system of equations describing the Dirac–Kähler particle in presence of the external uniform magnetic field has been studied. This equation describes a multi-spin boson field equivalent to the scalar, pseudoscalar, vector, pseudovector, and anti-symmetric tensor. On the searched solutions, we diagonalize operators of the energy, third projection of the total angular momentum, and the third projection of the linear momentum. After separating the variable, we derive the system of 16 first-order differential equations in polar coordinate. To resolve this system, we apply the method of Fedorov–Gronskiy. We have found five independent solutions. The number 5 for independent solutions correlates with five different tensors entering the complete wave function for Dirac–Kähler particle [ 67 ].