Application of Bi-Quaternions In Physics

Scientific Paper
Title	Application of Bi-Quaternions In Physics
Read in full	Link to paper
Author(s)	Andre Waser
Keywords	Maxwell equations, electrodynamics, bi-quaternions, Lorentz condition, special relativity, gravitation, Dirac equation
Published	2000
No. of pages	39

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Abstract

This paper introduces a new bi-quaternion notation and applies this notation to electrodynamics. A set of extended MAXWELL equations and other fundamental equations of electrodynamics are derived. By applying the LORENTZ condition, these equations reduce to the classical form. Additionally the bi-quaternion notation allows a compact formulation of SRT. Furthermore an application of bi-quaternions in other disciplines of physics as mechanics (dynamics) is shown.

Overview

André Waser, writing from Einsiedeln, Switzerland, first issued this paper on 29 July 2000 and revised it through 6 May 2007. It opens with a historical grievance: the late-nineteenth-century "emotional dispute" over notation, in which James Clerk Maxwell, William Rowan Hamilton and Peter Guthrie Tait favoured quaternions while Oliver Heaviside and Josiah Willard Gibbs independently decided they could do better with just a part of the quaternion — the three-dimensional vector. Vector notation won. Waser's contention is that something was lost in the process: Maxwell in the Treatise only translated finished results into "vectorized" quaternion form (prefixing scalar parts with S and vector parts with V) and never used quaternion calculus at all, so quaternions were never really given their chance.

The paper's programme is to redo electrodynamics with bi-quaternions — quaternions whose four coefficients are themselves complex, giving an eight-component number — and to show that the whole of Maxwell's Equations, the Lorentz transformation, the Lorentz Force, Poynting's theorem, a formally identical theory of gravitation, and the Dirac equation all follow from a handful of bi-quaternion products with no tensor calculus anywhere. The one substantive physical departure from textbook electrodynamics is that the derivation naturally produces an extra real scalar field s = −(1/c²)∂φ/∂t − ∇·A, which classical theory suppresses by fiat as the Lorenz gauge condition. Waser keeps it, obtains extended Maxwell equations containing it, and shows that setting s = 0 recovers the standard equations exactly. The extended theory is thus not in conflict with classical electrodynamics; it contains it as the s = 0 case. The paper is a companion to his work with Koen van Vlaenderen on "Generalisation of classical electrodynamics to admit a scalar field and longitudinal waves" (Hadronic Journal 24, 2001).

The argument

Quaternions and bi-quaternions

A quaternion Q = a + bi + cj + dk has Hamilton's units satisfying i² = j² = k² = ijk = −1. Waser sets up the calculus explicitly — conjugation, scalar product, the non-commuting product XY ≠ YX (they differ by 2x⃗ × y⃗), magnitude, inverse and division — then extends to bi-quaternions by letting the four coefficients be complex. He is careful that this is not the octonions of Lie algebra: no new imaginary units are introduced, and the Hamiltonian units remain valid. A bi-quaternion is a superposition of two four-dimensional numbers, and he distinguishes incomplete bi-quaternions (only one four-vector present) from complete ones. Physical four-vectors are assigned to the part that keeps a real term, and the mapping consistently forces the companion four-vector to vanish — a nullification whose origin, Waser argues, lies in physical conservation laws.

The key operators are the bi-quaternion nabla, ∇ = (1/c)∂/∂t + i·∇⃗, whose product with its conjugate is the d'Alembertian, and a total time-derivative operator D = ∂/∂t + c·∇ + (v·∇ + i(v × ∇)) in which the scalar part uses the dot product and the vector part the cross product.

Special relativity in one line

An event is ℰ = ct + i·x⃗ with magnitude ct² − x², invariant across inertial frames; dividing the differential by c gives dτ = dt√(1 − v²/c²) directly. The relativistic four-velocity V = dℰ/dτ has magnitude always exactly c. The Lorentz transformation then reduces to a single bi-quaternion product: ℰ′ = (γ/c)V*ℰ, with the inverse ℰ = (γ/c)Vℰ′, and comparing coefficients yields ct′ = γ(ct − v·x/c) and x′ = γ(x − vt). Waser proves in Appendix B that the second of these is identical to the vector Lorentz transformation of position usually written with the γ − 1 term. The velocity-addition law follows from the same operator once the product is normalised so that the scalar part remains c: v′ = (u + v′)/(1 + u·v′/c²).

Extended electrodynamics and the scalar field s

Defining the bi-quaternion potential 𝒜 = φ + ic·A and taking ∇𝒜 produces four terms, into which Waser substitutes s = −(1/c²)∂φ/∂t − ∇·A, E = −∇φ − ∂A/∂t, and B = ∇ × A. He notes that s is exactly the Lorenz condition, and reads its vanishing not as a gauge choice but as a conservation law for the scalar potential: ∂φ/∂t + ∇·(φv) = 0, formally the same statement as charge conservation ∂ρ/∂t + ∇·(ρv) = 0, which appears as the corresponding scalar term of ∇𝒥 for the current density bi-quaternion 𝒥 = ρc + i·J.

Applying the d'Alembertian to the potential, Δ𝒜 = ∇∇*𝒜 = μ𝒥, gives the extended Maxwell set:

∇·B = 0
Ampère's law: ∂B/∂t + ∇ × E = 0
Extended Coulomb law: ∇·E − ∂s/∂t = ρ/ε
Extended Faraday law: ∇ × B − (1/c²)∂E/∂t + ∇s = μJ

With s = 0 the last two collapse to the textbook forms. Waser notes similar equations were published by W. M. Honig in 1977. The same machinery yields the wave equations for potentials and for the force fields, including a wave equation for s itself, □s = −(∂ρ/∂t + ∇·J) = 0.

Lorentz force, Poynting theorem, field transformations

Taking the bi-quaternion force density ℱ = P/c + i·F and the identity ℱ = −ρ∇𝒜, the components separate into an extended power density ρ(v·E + c²s) and an extended Lorentz Force density ρ(v × B + E + vs), both reducing to the standard forms when s = 0. Two identities fall out along the way: v·B = 0 and B = (v × E)/c². A long computation in Appendix C gives the extended Poynting theorem ∂w/∂t + ∇·S + P = 0 with modified energy density w = ½(εE·E + B·B/μ + s²/μ) and flux S = (E × B − Es)/μ, while the imaginary vector part reproduces the divergence of the Maxwell stress tensor plus the time derivative of the expanded Poynting vector. The Lorentz transformations of φ, A, ρ, J, E and B all follow by the same operator sandwich and coefficient comparison, with the new field transforming as s′ = γ(s − u·E/c²).

The same structure for gravitation

Waser then simply relabels. A gravity potential φ (units m²/s²) and a velocity field U (m/s) replace φ and A; the gravity/acceleration field G replaces E and a "rotations field" or dynamic induction T = ∇ × U (units s⁻¹) replaces B. Setting Δ𝒜 = (g/c²)𝒥_m with g = 6.67 × 10⁻¹¹ N m²/kg² and the mass–momentum bi-quaternion 𝒥_m = cm + i·p_m, he obtains "Maxwell equations of dynamics": ∇·T = 0, an "Ampère law" ∂T/∂t + ∇ × G = 0, and extended "Coulomb" and "Faraday" laws that reduce, when the gravity-potential conservation s_m = 0 holds, to ∇·G = gm and ∇ × T − (1/c²)∂G/∂t = (g/c²)p. In vacuum these give transverse gravity waves propagating at c. The reaction (inertial) force density is m(v × T + G + vs_m). Running the same operator on the coordinate velocity instead of an external field yields the action force F = ma with the acceleration in the form ∂v/∂t + ∇(v²/2) − v × (∇ × v) — which Waser notes "is known from fluid mechanics" — together with a newly identified "flow rate" e = ∇·v + (v/c²)·∂v/∂t with units s⁻¹.

Self-energy and the classical electron radius

Forming the scalar product of the current-density and potential bi-quaternions gives a self-energy density; for the static case this integrates to ∫mc²dV = ε∫E·EdV, and for a spherical potential field to m = q²/(4πεc²r). Inserting the elementary charge and the electron mass returns r_e = 2.818 × 10⁻¹⁵ m.

Dirac's equation without matrices

Appendix D, following A. W. Conway's 1937 quaternionic treatment, starts from 𝒫 = m𝒱 and E ≡ c𝒫, from which the quaternion magnitude gives E² = m²c⁴ + c²Σp_k². Waser argues this should be read as a "static average" over many energy oscillators, since quantum experiments show energy is oscillatory; substituting E → −iħ∂/∂t and p_k → iħ∂/∂x_k acting on a bi-quaternion wave function Ψ gives the Klein–Gordon form directly. Taking instead E ≡ ±c𝒫 avoids Dirac's square root and its 4 × 4 matrices, yielding a first-order equation in the Hamiltonian units; multiplying out gives Dirac's four coupled equations for the free particle, and the substitution p → p − qA, E → E − qφ gives the equation for a particle in an external field. Waser observes that the two possible signs are what motivated Dirac to postulate antiparticles and the Positron, and Appendix E exhibits the Hamiltonian units as the 4 × 4 matrices they are equivalent to.

Assessment

Judged as what it announces itself to be — a notation paper — this is careful, competent and largely successful. The algebra is set out from first principles, the appendices carry the long computations rather than hiding them, and the derivations are checkable. Several of the results are genuinely elegant: the Lorentz transformation as a single sandwich product ℰ′ = (γ/c)V*ℰ, with the awkward γ − 1 term of the vector formulation proved equivalent rather than waved at; the velocity-addition law emerging from normalising the scalar part to c; and Dirac's equation obtained without ever writing a gamma matrix, which is a real reduction in machinery even though Appendix E honestly concedes the Hamiltonian units are those matrices in another dress. The paper is also historically well grounded — Bork's 1966 account of the vectors-versus-quaternions correspondence, Conway 1937, Silberstein 1912, Honig 1977 — and it does not overstate its lineage.

The physics claims are more modest than the framing suggests, and the paper is mostly honest about this. Every "extended" equation reduces to the classical one when s = 0, so nothing here contradicts any measurement; the content is the proposal that s might not be zero. That proposal is where the difficulties are. Waser asserts that the Lorenz condition is "a demand for conservation of scalar potentials, i.e. a conservation law", and treats its formal parallel with charge conservation as evidence that s is a physical field. But the parallel is formal only: ρ is gauge-invariant and measurable, whereas φ and A are not, and the Lorenz condition is standardly a restriction on the choice of potentials, not on the fields. If s ≠ 0, then E and B as defined no longer exhaust the observable content of the theory, and the extended Coulomb and Faraday laws are no longer gauge-invariant. The paper never confronts this — it does not ask what happens to s under a gauge transformation, nor exhibit an experiment in which s would be observable, and it explicitly defers the question: "An interpretation of the scalar field s beyond the interpretation as pure Lorentz condition shall be discussed at another place."

That matters because a non-zero s entails longitudinal electromagnetic waves, and these are constrained by measurement. The absence of a longitudinal photon mode is tied to the photon's masslessness, which is bounded experimentally at below roughly 10⁻¹⁸ eV by Coulomb's-law tests and solar-wind magnetic-field measurements; any observable s field would have to hide beneath those bounds, and the paper gives no estimate of its magnitude. The modified Poynting theorem, with s²/2μ added to the energy density and −Es/μ to the flux, is likewise untested against the extremely well-measured radiation-pressure and energy-balance results of classical optics.

Two further steps are asserted rather than derived. The identities v·B = 0 and B = (v × E)/c² emerge from the force-density construction as though they were general truths, but they are properties of the particular field configuration produced by a charge moving with velocity v and do not hold for arbitrary superposed fields — a plane wave crossing a charge's own field violates both. And the gravitational section is pure formal transcription: the field equations are obtained by relabelling symbols, with Waser stating outright that the "deeper meaning shall not be discussed now". The resulting theory is a linear gravitomagnetic analogue, which is a known weak-field limit of General Relativity and therefore not new, and being linear it carries a factor-of-two discrepancy in the gravitomagnetic coupling relative to the GR limit — a discrepancy the paper does not address because it never compares its predictions to any gravitational measurement. Its transverse gravity waves at speed c are consistent with the Gravitational Waves detections, but so is the general-relativistic limit it duplicates, so nothing is discriminated.

The self-energy result deserves a word: recovering r_e = 2.818 × 10⁻¹⁵ m from m = q²/4πεc²r is presented as a check, but that expression is the definition of the classical electron radius, so the arithmetic confirms only that the substitutions were made correctly.

On its own terms the paper delivers what its abstract promises. It is a demonstration that bi-quaternions can carry the whole of classical electrodynamics and special relativity compactly, with a suggestive but undeveloped extension attached. Waser does not claim more, and readers should not read more into it: the extended equations are a hypothesis awaiting an interpretation of s, not a rival to Maxwell's theory.