[Paper Review] Maxwell's Theory on Non-Commutative Spaces and Quaternions

S. I. Kruglov|ArXiv.org|Oct 6, 2001

Algebraic and Geometric Analysis3 citations

TL;DR

This paper formulates Maxwell's electrodynamics on non-commutative (NC) spacetime using quaternions and spin-tensors, deriving non-linear second-order wave equations for electromagnetic fields. It identifies a trace anomaly in the energy-momentum tensor and shows that dual symmetry of electromagnetic fields is broken in NC spacetime, with plane waves remaining valid solutions under the new formalism.

ABSTRACT

The Maxwell theory on non-commutative spaces has been considered. The non-linear equations of electromagnetic fields on non-commutative spaces were obtained in the compact spin-tensor (quaternion) form. It was shown that the plane electromagnetic wave is the solution of the system of non-linear wave equations of the second order for the electric and magnetic induction fields. We have found the canonical and symmetrical energy-momentum tensors and their non-zero traces. So, the trace anomaly of the energy-momentum tensor was obtained in electrodynamics on non-commutative spaces. It was noted that the dual transformations of electromagnetic fields on non-commutative spaces are broken.

Motivation & Objective

• To extend classical Maxwell electrodynamics to non-commutative spacetime using quaternionic and spin-tensor formalisms.
• To derive the non-linear field equations governing electromagnetic waves in non-commutative geometry.
• To analyze the energy-momentum tensor and determine whether trace anomalies arise in NC electrodynamics.
• To investigate the validity of dual symmetry transformations in non-commutative spacetime.
• To explore the physical implications of non-commutativity on electromagnetic field dynamics and relativistic symmetries.

Proposed method

• The non-commutative structure is modeled via the commutation relation $[\widehat{x}_\mu, \widehat{x}_\nu] = i\theta_{\mu\nu}$, with $\theta_{\mu\nu}$ as a constant antisymmetric tensor.
• The Moyal-Weyl star-product $A(x) \star B(x)$ is used to represent non-commutative field interactions, encoding non-locality in the Lagrangian.
• The Seiberg-Witten map is applied to first-order in $\theta_{\mu\nu}$ to relate NC fields to their commutative counterparts.
• The field strength tensor is expressed as $\widehat{F}_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu - ie[A_\mu, A_\nu]_M$, with the Moyal bracket encoding non-commutativity.
• The equations are reformulated in compact spin-tensor (quaternion) form using Pauli matrices and biquaternion algebra.
• The energy-momentum tensor is derived in both canonical and symmetric forms, and its trace is computed to detect anomalies.

Experimental results

Research questions

• RQ1 How do the Maxwell equations transform when formulated on non-commutative spacetime using quaternionic formalism?
• RQ2 Do plane electromagnetic waves remain exact solutions of the non-linear wave equations in non-commutative electrodynamics?
• RQ3 Is the trace of the energy-momentum tensor non-zero in non-commutative electrodynamics, indicating a trace anomaly?
• RQ4 Are dual transformations of electromagnetic fields preserved in non-commutative spacetime?
• RQ5 What is the role of the non-commutative parameter $\theta_{\mu\nu}$ in breaking Lorentz and duality symmetries?

Key findings

• The non-commutative Maxwell theory yields non-linear second-order wave equations for the electric and magnetic induction fields, derived in compact quaternionic form.
• Plane electromagnetic waves are exact solutions of the derived non-linear wave equations in non-commutative spacetime.
• The canonical and symmetric energy-momentum tensors are computed, and both exhibit non-zero traces, confirming the presence of a trace anomaly in NC electrodynamics.
• Dual symmetry of electromagnetic fields is explicitly broken in non-commutative spacetime, as the field equations do not remain invariant under Hodge duality.
• The non-commutative parameter $\theta_{\mu\nu}$, though small ($\Lambda_{NC} \geq 10^3$ GeV), induces observable effects at high energies, including non-local interactions and modified field dynamics.
• The use of quaternions and spin-tensors provides a compact and symmetric formulation of NC electrodynamics, facilitating the analysis of field symmetries and conservation laws.