Bilinear Covariants and Spinor Fields Duality in Quantum Clifford Algebras

TL;DR

This paper explores the classification and duality of quantum spinor fields within quantum Clifford algebras on Minkowski spacetime, introducing deformed bilinear covariants and a new duality concept that extends classical spinor classifications.

Contribution

It introduces a novel classification of quantum spinor fields, incorporates arbitrary bilinear forms in spacetime, and establishes a new duality between standard and quantum spinor fields.

Findings

Quantum algebraic spinor fields can be constructed with arbitrary bilinear forms.

Deformed bilinear covariants extend classical spinor classifications.

A new duality between standard and quantum spinor fields is established.

Abstract

Classification of quantum spinor fields according to quantum bilinear covariants is introduced in a context of quantum Clifford algebras on Minkowski spacetime. Once the bilinear covariants are expressed in terms of algebraic spinor fields, the duality between spinor and quantum spinor fields is thus discussed. Hence, by endowing the underlying spacetime with an arbitrary bilinear form with a antisymmetric part in addition to a symmetric spacetime metric, quantum algebraic spinor fields and deformed bilinear covariants can be constructed. They are therefore compared to the classical (non quantum) ones. Classes of quantum spinor fields are introduced and compared with Lounesto's spinor field classification. A physical interpretation of the deformed parts and the underlying Z-grading is proposed. The existence of an arbitrary bilinear form endowing the spacetime already has been explored in the literature in the context of quantum gravity. Here, it is shown further to play a prominent role in the structure of Dirac, Weyl, and Majorana spinor fields, besides the most general flagpoles, dipoles and flag-dipoles ones as well. We introduce a new duality between the standard and the quantum spinor fields, by showing that when Clifford algebras over vector spaces endowed with an arbitrary bilinear form are taken into account, a mixture among the classes does occur. Consequently, novel features regarding the spinor fields can be derived.