Higher Spin Quaternion Waves in the Klein-Gordon Theory

Abstract

Electromagnetic interactions are discussed in the context of the Klein-Gordon fermion equation. The Mott scattering amplitude is derived in leading order perturbation theory and the result of the Dirac theory is reproduced except for an overall factor of sixteen. The discrepancy is not resolved as the study points into another direction. The vertex structures involved in the scattering calculations indicate the relevance of a modified Klein-Gordon equation, which takes into account the number of polarization states of the considered quantum field. In this equation the d'Alembertian is acting on quaternion-like plane waves, which can be generalized to representations of arbitrary spin. The method provides the same relation between mass and spin that has been found previously by Majorana, Gelfand, and Yaglom in infinite spin theories.

Conformal relativity with hypercomplex variables

“…The non-zero matrix elements of the metric tensor are proportional to the identity element of the algebra. The identity is represented in general in terms of a n × n matrix, where n corresponds to the dimension of the spin representation [17]. The antisymmetric contributions in Eq.…”

“…The explicit form of the antisymmetric tensor in terms of the basis elements and the pseudoscalar is displayed in [17] with a different sign convention. The pseudoscalar is interpreted in geometric algebra as a directed volume element or volume form, see Doran and Lasenby [47].…”

“…Furthermore, the vector and spin vector coefficient of the representation functions are orthogonal to each other. Despite the new notation and interpretation the calculation can be performed as in [17] △…”

“…In fact one has the choice, which parameters are considered to be physical as the interpretation of the constant is still unclear. Based on this choice one can reinterpret momentum and mass of the Poincaré group representation functions as given in [17] and see them only as coefficients in a series expansion. What is considered as energy, momentum, and mass has to be derived at a later stage, for example in correspondence with the Fourier transformation of the Green function of the considered system.…”

“…Sachs developed a unified theory of gravitation and electroweak interactions with a Maxwell-like representation of Einstein's general relativity [20,21]. The method is based on a quaternionic paravector algebra, which is congruent to the hyperbolic complex Pauli algebra applied in [17]. Sachs states that the Majorana equation is the most general irreducible expression of relativistic quantum mechanics, in special relativity.…”