Quaternions and Small Lorentz Groups in Noncommutative Electrodynamics

TL;DR

This paper investigates the structure of non-linear electrodynamics in noncommutative space-time using quaternion formalism, revealing specific small Lorentz groups and their physical interpretations, and analyzing dual symmetry invariance.

Contribution

It introduces a quaternion-based approach to analyze noncommutative electrodynamics and identifies two specific small Lorentz groups governing form-invariance.

Findings

Identifies two Abelian 2-parametric small groups, SO(2) d7 O(1.1) and T_{2}.

Shows nonlinear constitutive equations are invariant only under discrete dual transformations.

Provides a physical interpretation of Lorentz transformations in this context.

Abstract

Non-linear electrodynamics arising in the frames of field theories in noncommutative space-time is examined on the base of quaternion formalism. The problem of form-invariance of the corresponding nonlinear constitutive relations governed by six noncommutativity parameters $\theta_{kl}$ or quaternion \underline{K} = \underline{\theta} - i \underline{\epsilon} is explored in detail. Two Abelian 2-parametric small groups, SO(2) \otimes O(1.1) or T_{2}, depending on invariant length \underline{K}^{2}\neq 0 or \underline{K}^{2}= 0 respectively, have been found. The way to interpret both small groups in physical terms consists in factorizing corresponding Lorentz transformations into Euclidean rotations and Lorentzian boosts. In the context of general study of various dual symmetries in noncommutative field theory, it is demonstrated explicitly that the nonlinear constitutive equations under consideration are not invariant under continuous dual rotations, instead only invariance under discrete dual transformation exists.