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.

Version v1: Fixed a cross-reference error between Sec. 2.4 and Sec. 3.1 (wrong conjugation identified in the text, mislabeled equation); the physics and all results are unchanged.

Version v2: Sections 4.1 and 4.2 described the sphere as fed by a real wire while assuming exact spherical symmetry (B field zero everywhere). These are two different configurations: a real single-point current injection breaks spherical symmetry and gives a small but non-zero B field; B is exactly zero only in the idealised, synchronous-charging limit. This revision makes that distinction explicit, and notes that the physical stress-energy tensor likewise vanishes exactly only in the idealised case — in the real case it is small but non-zero. The gauge-dependent status of the b-channel diagnostic quantities is unaffected, since it follows from their construction, not from the size of the stress-energy tensor. No other result in the paper changes.

Version v3: Corrected the historical attribution of quaternion notation: the 1865 paper is in component form, condensed quaternion notation appearing in the 1873 Treatise. Dating of the vector reformulation corrected to the 1880s. Two references added. No change to results.

Revision v4
The scalar parts of the electromagnetic biquaternions carry a normalisation coefficient that the defining relation leaves undetermined, since the scalar part of 2∇̃Ã is the Lorenz condition and vanishes identically. It has been re-determined as β = −1, so that B̃ = βb + B − (i/c)E and Ẽ = −icβb + E + icB (vector parts unchanged). The b-channel relation consequently reads
−½ ∂ₜ(b²/μ₀) + ∇·(b E_FW/μ₀) = ρc²b;
the flux and source terms are unchanged, the coefficient appearing in the time-derivative term alone. Its regime of validity is now stated explicitly: exact for a plane longitudinal wavefront, asymptotically exact for the retarded spherical wavefront with relative residual decaying as 1/r, and not an identity of Maxwell's equations for arbitrary field configurations. Minor typographical corrections included. The experimental results, the kinematic identity b = −E_FW/c, and the reconstruction of Maxwell's equations are unaffected.