[Paper Review] Invariant solutions for equations of axion electrodynamics

Oksana Kuriksha, A. G. Nikitin|arXiv (Cornell University)|Jan 30, 2010

Galaxies: Formation, Evolution, Phenomena2 references3 citations

TL;DR

This paper presents an extended class of exact solutions for the field equations of axion electrodynamics using Lie group analysis of the Poincaré algebra. By classifying three-dimensional subalgebras of the Poincaré algebra, the authors derive solutions involving arbitrary functions and parameters, including bound, square-integrable solutions that propagate superluminally yet have energy velocities below light speed, demonstrating causality despite non-standard phase group velocities.

ABSTRACT

Using the three-dimensional subalgebras of the Lie algebra of Poincaré group an extended class of exact solutions for the field equations of the axion electrodynamics is obtained. These solutions include arbitrary parameters and arbitrary functions as well. The most general solutions include six arbitrary functions. Among them there are bound and square integrable solutions which propagate faster than light. However, their energy velocities are smaller than the velocity of light.

Motivation & Objective

• To construct a comprehensive class of exact solutions for axion electrodynamics using Lie group-theoretic methods.
• To classify invariant solutions through three-dimensional subalgebras of the Poincaré algebra.
• To analyze the physical properties of these solutions, particularly their propagation and energy characteristics.
• To explore connections between axion electrodynamics and integrable models in quantum mechanics and condensed matter physics.
• To provide a foundation for further study of symmetries and conservation laws in nonlinear field theories.

Proposed method

• Application of Sophus Lie's method of continuous symmetries to derive invariant solutions of the field equations.
• Classification of three-dimensional subalgebras of the Poincaré algebra p(1,3) to generate optimal systems of symmetry reductions.
• Reduction of the original system (2), (3) to lower-dimensional systems via symmetry-based ansätze.
• Construction of solutions involving arbitrary functions and parameters, including solutions with six arbitrary functions in the most general case.
• Use of gauge transformations to simplify system (4) into a form involving free Maxwell equations and a nonlinear scalar equation.
• Leveraging known exact solutions of the free Maxwell equations to build solutions for the axion-coupled system.

Experimental results

Research questions

• RQ1 What exact solutions can be derived for the field equations of axion electrodynamics using Lie symmetry analysis of the Poincaré algebra?
• RQ2 How do the symmetries of the system relate to the existence of solutions with arbitrary functions and parameters?
• RQ3 What are the physical implications of solutions that propagate faster than light but have subluminal energy velocities?
• RQ4 How do the solutions of axion electrodynamics relate to integrable models in quantum mechanics and topological insulators?
• RQ5 Can the structure of the solutions reveal deeper connections between relativistic field theories and non-relativistic symmetries?

Key findings

• The authors construct an extended class of exact solutions involving up to six arbitrary functions, representing the most general invariant solutions under the Poincaré algebra.
• Among the solutions are bound and square-integrable fields that propagate with group velocities exceeding the speed of light, yet their energy velocities remain below c, preserving causality.
• Solutions exist that satisfy the superposition principle and include arbitrary functions, making them suitable for initial and boundary value problems.
• Specific symmetry algebras (e.g., A9, A1, A17, A18, A28) generate dynamical contributions to the axion mass.
• The system (4), which describes a scalar axion field, can be reduced to a nonlinear scalar equation coupled to known Maxwell solutions via a gauge transformation.
• Certain solutions, such as those in (35), give rise to exactly solvable Dirac equations for charged particles, indicating potential applications in quantum systems with anomalous coupling.