Quaternion Algebra as a Cross-Disciplinary Framework for Classical and Quantum Field Theories: From Special Relativity to the Electromagnetic b-Channel

Maxwell's original electromagnetic theory (1865) was formulated in quaternion algebra, later set aside in favour of the vector reformulation of Heaviside and Gibbs. This paper revisits quaternion algebra as an active cross-disciplinary tool: applying the same quaternion-nabla operator to the appropriate physical quaternion reconstructs, in compact form, known field equations across four independent domains — special-relativistic four-momentum conservation, ideal fluid dynamics (with the linear acoustic wave equation as a direct corollary), the algebraic structure underlying the Dirac equation, and Maxwell's equations together with the Lorenz gauge condition, recovered as a fifth field equation on equal footing rather than an auxiliary constraint.

In the electromagnetic case, this is shown to be more than reconstructive: the same formalism, applied to an extended Poynting theorem, exposes a scalar transport channel — governed by b ≡ ∇·A = −(1/c²)∂φ/∂t — invisible to the standard vector formulation and supported by direct experimental evidence reported elsewhere. The paper situates this result relative to the classical Conway–Silberstein–Lanczos biquaternion tradition (1911–1932) and modern quaternion gravi-electromagnetic literature, and tests, rather than assumes, the reach of the method: it does not extend to dissipative processes such as heat conduction, and no configuration checked motivates an analogous channel in linearized gravitoelectromagnetism.

This paper is intended as the methodological foundation for a wider research programme; it is self-contained and citable independently of the electromagnetic (b-channel) results developed in the companion papers.