Interpreting the Electric Field with Complexified Quaternions

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.