TITLE:
Interpreting the Electric Field with Complexified Quaternions
AUTHORS:
Daniel Kovach
KEYWORDS:
Biquaternions, Complexified Quaternions, Electromagnetic Potentials, Lorentz Gauge
JOURNAL NAME:
International Journal of Modern Nonlinear Theory and Application,
Vol.15 No.2,
June
23,
2026
ABSTRACT: This paper revisits classical electromagnetism using complexified quaternions (biquaternions), an associative algebra isomorphic to M 2 ( ℂ ) . Building on Maxwell’s original quaternionic insights, we define a generalized four-gradient ∂ r and four-potential A in biquaternionic form. The electric field E and magnetic field B emerge from the anticommutator and commutator of ∂ r and A , respectively. Explicit computation yields the standard expressions for B=∇×A and E=−∇ϕ−( 1/c ) ∂ t A plus a gauge-dependent scalar term ( 1/c ) ∂ t ϕ+∇⋅A . In a specific gauge where A=−∇S and ϕ=( 1/c ) ∂ t S , this reduces to the d’Alembertian wave equation □S=0 under the Lorentz condition. The biquaternionic framework unifies scalar and vector components more symmetrically than pure real-quaternionic or vector formulations, offering compact notation and potential insights into wave propagation, invariants, and extensions to relativistic or chiral contexts.