The Lorentz Group with Dual-Translations and the Conformal Group

TL;DR

This paper explores an extension of the Lorentz group with dual-translations, leading to the conformal group, and finds that the Dirac 4-spinor formalism uniquely represents this extended symmetry with spin $(A,B)(C,D)$.

Contribution

It introduces dual-translations into Lorentz group representations, resulting in a novel connection to the conformal group and identifying the Dirac 4-spinor as the unique matrix representation.

Findings

Dual-translations expand Lorentz group representations.

The conformal group emerges with one additional scale transformation.

Dirac 4-spinor formalism uniquely fits the extended symmetry.

Abstract

For those finite-matrix representations of the Lorentz group of rotations/boosts with spin $(A,B)\oplus(C,D)$ that can also represent translations, two possible translation subgroups qualify. Of these two, one must be selected, and one discarded, to represent the Poincar\'{e} group of rotations/boosts with translations in spacetime. Instead, let us discard the requirement that there be just one translation subgroup. With dual-translations, one gives up agreement with simple macroscopic observations of spacetime. Now the transformations of both possible translation subgroups combine with those of the Lorentz group. The resulting commutation relations require new transformations and generators to satisfy the linearity requirement of a Lie algebra. Special cases of spins are sought to restrict the influx of new transformations. One finds that the Dirac 4-spinor formalism is the only viable solution. The slightly expanded group it represents is the conformal group with just one new transformation, scale change. It follows as a corollary that the Dirac 4-spinor formalism is the only matrix representation of the conformal group with spin $(A,B)\oplus(C,D).$