Spinor Structure and Modulo 8 Periodicity

TL;DR

This paper explores the algebraic structure of spinors as tensor products of biquaternion algebras, revealing a fractal pattern in Lorentz group representations driven by modulo 8 periodicity, with implications for quantum information.

Contribution

It introduces a novel algebraic framework linking spinor structures to Clifford algebra periodicity and uncovers a self-similar fractal organization of Lorentz representations.

Findings

Modulo 8 periodicity induces relations among Lorentz representations

Representation system exhibits a self-similar fractal structure

Connections between spinors, twistors, and qubits are discussed

Abstract

Spinor structure is understood as a totality of tensor products of biquaternion algebras, and the each tensor product is associated with an irreducible representation of the Lorentz group. A so-defined algebraic structure allows one to apply modulo 8 periodicity of Clifford algebras on the system of real and quaternionic representations of the Lorentz group. It is shown that modulo 8 periodic action of the Brauer-Wall group generates modulo 2 periodic relations on the system of representations, and all the totality of representations under this action forms a self-similar fractal structure. Some relations between spinors, twistors and qubits are discussed in the context of quantum information and decoherence theory.