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
(
?
)
. 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=????(
1/c
)
?
t
A
plus a gauge-dependent scalar term
(
1/c
)
?
t
?+??A
. In a specific gauge where
A=??S
and
?=(
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.