Dirac equation for particles with arbitrary half-integral spin

Abstract

The sets of 2ð2s þ 1Þ-component irreducible and Clifford algebraic Hermitian and unitary matrices through the two-component Pauli matrices are suggested, where s ¼ 1/2, 3/2, 5/2, . . . . Using these matrix sets, the eigenvalues of which are AE1, the 2ð2s þ 1Þ-component generalized Dirac equation for a description of arbitrary half-integral spin particles is constructed. In accordance with the correspondence principle, the generalized Dirac equation suggested arises from the condition of relativistic invariance. This equation is reduced to the sets of two-component matrix equations the number of which is equal to 2s þ 1. The new relativistic invariant equation of motion leads to an equation of continuity with a positive-definite probability density and also to the Klein-Gordon equation. This relativistic equation is causal in the presence of an external electromagnetic field interaction. It is shown that, in the case of nonrelativistic limit, the relativistic equation presented is reduced to the Pauli equation describing the motion of half-integral spin particle in the electromagnetic field.

Unified treatment of basis sets of relativistic exponential-type spinor orbitals for particles with arbitrary half-integral spin in position, momentum and four-dimensional spaces

“…All of these studies, which yield an adequate description of spin-s free particles, have many intrinsic contradictions and difficulties when an electromagnetic field interaction is introduced (see [9] and references therein). The generalized Dirac equations presented in our previous works [10] and [11] for the particles with half-integral and arbitrary spins, respectively, are consistent and causal in the presence of an electromagnetic field interaction.…”

Quantum Self-Frictional Relativistic Nucleoseed Spinor-Type Tensor Field Theory of Nature

“…where ≥ 1, ≤ ≤ + − 1, − ≤ ≤ min( + , − 1),̃= + 2| − |. See (53) and [21,22,[24][25][26][27][28][29][30][31][32] for the exact definition of functions occurring in ((43a) and (43b))-((45a) and (45b)). The relativistic orbitals of arbitrary spin ( = 0, 1/2, 1, 3/2, 2, .…”

Israfil I. Guseinov: A pioneer of the quantum theory of atomic, molecular, and nuclear systems*

“…He established a generalized relativistic equation of arbitrary mass and spin and basic sets of spinor functions for its solution in position, momentum, and 4D spaces . It was shown that the spinless particles ( s = 0) obey the Pauli principle; accordingly, they behave as fermions . We believe that this fact will play a leading role in natural sciences.…”

“… and references quoted therein). In study, Guseinov proposed a combined relativistic wave equation for the spin‐s particles, where s = 1/2, 3/2, 5/2,… directly by analogy with the relativistic equation for particles with s = 1/2, and the half‐integral spins are, therefore, described by spinors of multiple order. Thus, a generalized relativistic equation for the arbitrary half‐integral spin particles has been constructed.…”

“…Guseinov focused on the generalization of the Dirac equation for arbitrary mass and spin by using the condition of relativistic covariance, group theory, and Clifford algebra. He established a generalized relativistic equation of arbitrary mass and spin and basic sets of spinor functions for its solution in position, momentum, and 4D spaces . It was shown that the spinless particles ( s = 0) obey the Pauli principle; accordingly, they behave as fermions .…”