On the nonlinearity of Four-Dimensional Conformal Transformations in spinor representation

TL;DR

This work investigates the nonlinear nature of four-dimensional conformal transformations, focusing on how this nonlinearity manifests in spinor representations. It develops a four-dimensional spinor representation by generalizing two-dimensional linear fractional transformations with biquaternions and links inversions to discrete spacetime symmetries. Applying these transformations to Dirac spinors shows that translations and SCTs remain linear for the Yukawa term but introduce nonlinear terms in vector-spinor interactions, which also induce $CP$ violation in this sector. The approach provides a geometric bridge between conformal symmetry and spinor theory and suggests possible implications for early-universe dynamics and cosmology.

Abstract

The nonlinearity of the conformal group is an essential factor that ruins the global conformal invariance for interacting material fields. In this paper we attempt to track such nonlinearity from spacetime transformations to spinor representations. To this end we rederive the spinor representation by generalizing the linear fractional transformation from two dimensions to four dimensions via replacing complex numbers with biquaternions. To check the effect of the nonlinearity we apply the translations and special conformal transformations (SCTs) to Dirac spinors in certain interactions. These two transformations do not lead to nonlinear terms in Yukawa term, but do in vector-spinor interaction. And the nonlinear terms would definitely cause $CP$ violation.