Spinor calculus for q-deformed quantum spaces II

TL;DR

This paper develops a comprehensive q-deformed spinor calculus framework for quantum spaces like Euclidean and Minkowski, including Clifford algebras, Dirac matrices, and spinor transformations, relevant for quantum physics applications.

Contribution

It introduces explicit constructions of q-deformed Clifford algebras, Dirac matrices, and spinor structures for four-dimensional quantum spaces, advancing the mathematical tools for quantum space analysis.

Findings

Constructed bases for four-dimensional Clifford algebras

Derived q-deformed Dirac matrices and spin matrices

Analyzed behavior of spinors under q-deformed Lorentz transformations

Abstract

This is the second part of an article about q-deformed analogs of spinor calculus. The considerations refer to quantum spaces of physical interest, i.e. q-deformed Euclidean space in three or four dimensions as well as q-deformed Minkowski space. The Clifford algebras corresponding to these quantum spaces are treated. Especially, their commutation relations and their Hopf structures are written down. Bases of the four-dimensional Clifford algebras are constructed and their properties are discussed. Matrix representations of the Clifford algebras lead to q-deformed Dirac-matrices for the four-dimensional quantum spaces. Moreover, q-analogs of the four-dimensional spin matrices are presented. A very complete set of trace relations and rearrangement formulae concerning spin and Dirac-matrices is given. Dirac spinors together with their bilinear covariants are defined. Their behavior under q-deformed Lorentz transformation is discussed in detail.