Gyromagnetic g_s factors of the spin-1/2 particles in the (1/2⁺-1/2^--3/2^-) triad of the four-vector spinor, ψ_μ, irreducibility and linearity

Abstract

The gauged Klein–Gordon equation, extended by a gsσμνFμν/4 interaction, the contraction of the electromagnetic field strength tensor, Fμν, with the generators, σμν/2, of the Lorentz group in (1/2, 0) ⊕ (0, 1/2), and gs being the gyromagnetic factor, is examined with the aim to find out as to what extent it qualifies as a wave equation for general relativistic spin-1/2 particles transforming as (1/2, 0) ⊕ (0, 1/2) and possibly distinct from the Dirac fermions. This equation can be viewed as the generalization of the gs = 2 case, known under the name of the Feynman–Gell-Mann equation, the only one which allows for a bilinearization into the gauged Dirac equation and its conjugate. At the same time, it is well-known a fact that a gs = 2 value can also be obtained upon the bilinearization of the nonrelativistic Schrödinger into nonrelativistic Pauli equations. The inevitable conclusion is that it must not be necessarily relativity which fixes the gyromagnetic factor of the electron to g(1/2) = 2, but rather the specific form of the primordial quadratic wave equation obeyed by it, that is amenable to a linearization. The fact is that space-time symmetries alone define solely the kinematic properties of the particles and neither fix the values of their interacting constants, nor do they necessarily prescribe linear Lagrangians. Information on such properties has to be obtained from additional physical inputs involving the dynamics. We here provide an example in support of the latter statement. Our case is that the spin-1/2- fermion residing within the four-vector spinor triad, ψμ ~ (1/2+-1/2--3/2-), whose sectors at the free particle level are interconnected by spin-up and spin-down ladder operators, does not allow for a description within a linear framework at the interacting level. Upon gauging, despite transforming according to the irreducible (1/2, 1) ⊕ (1, 1/2) building block of ψμ, and being described by 16-dimensional four-vector spinors, though of only four independent components each, its Compton scattering cross sections, both differential and total, result equivalent to those for a spin-1/2 particle described by the generalized Feynman–Gell-Mann equation from above (for which we provide an independent algebraic motivation) and with g(1/2-) = -2/3. In effect, the spin-1/2- particle residing within the four-vector spinor effectively behaves as a true relativistic "quadratic" fermion. The g(1/2-) = -2/3 value ensures in addition the desired unitarity in the ultraviolet. In contrast, the spin-1/2+ particle, in transforming irreducibly in the (1/2, 0) ⊕ (0, 1/2) sector of ψμ, is shown to behave as a truly linear Dirac fermion. Within the framework employed, the three spin sectors of ψμ are described on equal footing by representation- and spin-specific wave equations and associated Lagrangians which are of second-order in the momenta.

Lorentz Group Projector Technique for Decomposing Reducible Representations and Applications to High Spins

“…and in accord with Chapter 34 in [29]. This template example shows that the spin-Lorentz group projector technique suggested in [22,35] and advocated here provides results equivalent to representation reductions based on index symmetrization and anti-symmetrization. In the subsequent section, we shall decompose the direct product space of a four-vector with a Dirac spinor.…”

“…In the present, work we reviewed the technique suggested in [22,35] for decomposing products of Lorentz tensors into irreducible representation spaces that is based on covariant projectors built up in a transparent way from the Casimir invariants of the homogeneous spin-Lorentz group. Decomposition of product spaces by group projectors is a method fundamental to the construction of basis states in atomic and molecular physics [39], in which for the case of compact groups, use can be made of of Weyl's character formula.…”

“…are the generators of the four-vector space, (1/2, 1/2), while S µν ab are the generators in the Dirac spinor space, (1/2, 0) ⊕ (0, 1/2). The expressions are well known [35] and read,…”

“…a result due to [35]. When they are contracted with any general four-vector spinor ψ β , the resulting states transform according to,…”

“…This is the proof that the P (0,1/2) F projector correctly identifies the Dirac space in ψ µ . Moreover, in [35], the spinor in (99) was employed in the calculation of Compton scattering off a spin-1/2 particle, and the precise cross-sections typical for the traditional Dirac spinor were reproduced, thus providing an additional cross check for the correctness of the projector used. In a similar way, all the remaining identities can be verified.…”