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 later in quaternion algebra (Treatise 1873), and then 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.

Revision v5: Added two structural results in §2.4: the relation  between the two field biquaternions, requiring both conjugations. Second, the closed form of the quaternionic derivative on Maxwell solutions, which does not vanish but reduces to the source together with a term in the gauge scalar, placing the latter alongside the source rather than among the fields; Maxwell's equations are recovered unchanged for any value of the normalisation coefficient, and ∇̃b is shown to be local in the potentials. Related Work extended with modern quaternionic electrodynamics (Giardino, Liu et al., Kravchenko, Demir–Tanışlı). Source biquaternion now defined once, with the reversed-sign form written as its quaternion conjugate.

Version f (October 2026)

Sec. 5.2: the normalisation coefficient beta of the scalar sector is no longer
fixed to a single value. Rev. e reported beta = -1, obtained by requiring the
scalar relation to hold in the wavefront-dominated regime; that determination
is withdrawn. Beta is fixed separately at the two levels at which the relation
can be evaluated: beta = 0 on the Maxwell fields, where the scalar sector
contributes nothing beyond the Poynting theorem, and beta = -2 on the
potentials outside the sources, where the relation becomes an exact transport
identity for the diagnostic density u_b at every radius and for arbitrary
waveform.

Consequences. The diagnostic density is u_b = b^2/mu_0, not b^2/(2 mu_0) as the
time-derivative coefficient of rev. e implied; this is the normalisation for
which the shell integral returns CV^2/2, and rev. e was internally inconsistent
on this point. E_FW is now defined as the 1/r part of -grad(phi), with which
the kinematic identity b = -E_FW.rhat/c is exact rather than asymptotic. The
term rho c^2 b is identified as Gauss's law weighted by b, which cancels within
the relation and is not a source.

Sec. 4.1: a notation paragraph fixes the lower-case convention for
b = div A, by analogy with B = curl A, and distinguishes it from the
Nakanishi-Lautrup auxiliary field, for which the same letter is often used.

Sec. 5.2: the term "phase sector" is introduced for the gauge sector of the
representation, to which b, u_b and S_b belong. Its manifestations are phases,
never energy-momentum.

Sec. 5.3 retitled: E_FW is gauge-dependent, so "gauge-invariant form" was
misleading. Two statements in the introduction and conclusions are reworded:
the channel supplies the local, retarded representation of a gauge-invariant
measured quantity, rather than being an observable in its own right.

No result of Secs. 2-4 is affected.