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

TL;DR

This paper investigates the nonlinearity of four-dimensional conformal transformations in spinor representations, revealing its potential role in causing CP violation in certain interactions.

Contribution

It generalizes the spinor representation using biquaternions and analyzes the impact of conformal transformations on Dirac spinors, highlighting nonlinear effects.

Findings

Nonlinearity affects vector-spinor interactions.

Translations and SCTs do not produce nonlinear terms in Yukawa interactions.

Nonlinear terms may lead to CP violation.

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.