Nonminimal Spin-Field Interaction of the Classical Electron and Quantization of Spin
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HYSICS OF ELEMENTARY PARTICLES AND ATOMIC NUCLEI. THEORY
Nonminimal Spin-Field Interaction of the Classical Electron and Quantization of Spin A. A. Deriglazov* Depto. de Matemática, ICE, Universidade Federal de Juiz de Fora, MG, Brazil and Department of Physics, Tomsk State University, Tomsk, 634050 Russia *e-mail: [email protected] Received November 15, 2019; revised January 15, 2020; accepted February 28, 2020
Abstract—We shortly describe classical models of spinning electron and list a number of theoretical issues where these models turn out to be useful. Then we discuss the possibility to extend the range of applicability of these models by introducing an interaction,that forces the spin to align up or down relative to its precession axis. Keywords: Thomas precession, relativistic spin, noncommutative geometry DOI: 10.1134/S1547477120050131
The notion of spin of an elementary particle [1–4] has been developed in attempts to explain the energy levels of atomic spectra. This analysis culminated in quantum-mechanical expression for the energy, known as the Pauli Hamiltonian (E is the Coulomb electric field and B is a constant magnetic field) H = 1 (pˆ − e A )2 + eA 0 2m c (1) e 1 ˆ ˆ ˆ − (S, B) + (S, [E, p]) . mc 2mc Besides the position and momentum operators, the Hamiltonian contains operators proportional to the Pauli matrices, called the spin operators: Sˆi = σi . 2 They have discrete spectrum of eigenvalues, so their contribution to the energy turns out to be quantized. ˆ, B) has eigenvalues For instance, the operator (S λ = ± B , that is the measurement of spin in the 2 direction of vector B always gives one of the values ± . 2 The extra degrees of freedom contributes to the energy levels of an electron in a good agreement with experiments. The fine structure of Hydrogen-like atoms with one valent electron fixes the factor in front of S-E-p term, while Zeeman effect fixes the factor in front of S-B term. According the canonical quantization paradigm, the classical analogy of the theory (1) could be a point particle (x, p) which carries a vector S attached to it. The commutator [Sˆi , Sˆ j ] = i eijk Sˆ k implies the use of classical-mechanics bracket {S i , S j } = eijk S k . Then the Hamiltonian equations for spin are S = {S, H }, or (we
use the notation [A, B] and (A, B) for the vector and scalar product of three-dimensional vectors)
{
}
dS = [R, S], where R ≡ − e B − 1 [p, E] . (2) dt mc 2mc 1, the spin precesses When R is a constant vector around R : the end of S always lies in a plane orthogonal to R , and describe a circle around R with an angular velocity equal to the magnitude R of this vector. This, in essence, represents the classical model of non relativistic spin. It can be constructed on the base of a variational problem that implies both dynamical equations and the magnitude-of-spin 2 constraint S2 = 2 s(s + 1) = 3 (see [5]). 4 The relativistic generalization and systematic construction of the resulting model on the base of a variational problem presents an i
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