Quaternions, Maxwell Equations and Lorentz Transformations
| Scientific Paper | |
|---|---|
| Title | Quaternions, Maxwell Equations and Lorentz Transformations |
| Read in full | Link to paper |
| Author(s) | Jose Luis Lopez-Bonilla |
| Keywords | Maxwell equations, rotations, electromagnetic field |
| Published | 2005 |
| Journal | Apeiron |
| Volume | 12 |
| Number | 4 |
| No. of pages | 14 |
Read the full paper here
Abstract
In this work: a) We show that the invariance of the Maxwell equations under duality rotations brings into scene to the complex vector (cB + iE), whose components allow to construct a quaternionic equation for the electromagnetic field in vacuo. b) For any analytic function f of the complex variable z, it is possible to prove that is a Debye potential for itself, which permits to reformulate the corresponding Cauchy-Riemann relations. Here we show that the Fueter conditions- when z is a quaternion- also accept a similar reformulation and a very compact quaternionic expression. c) We exhibit how the rotations in three and four dimensions can be described through a complex matrix relation or equivalently by a quaternionic formula.
Overview
M. Acevedo M., J. López-Bonilla and M. Sánchez-Meraz, all of the Instituto Politécnico Nacional in Mexico City, published this three-part note in Apeiron 12(4) in October 2005. It is a mathematical-physics paper rather than a foundational one: the authors are not disputing Maxwell's equations or the Lorentz transformation, but showing how compactly both can be written when quaternions rather than vectors are used as the underlying algebra.
The three parts share a single theme, that Hamilton's algebra is the natural home of source-free electrodynamics and of rotations in three and four dimensions alike. Part (a) derives a one-line quaternionic Maxwell equation from duality-rotation invariance. Part (b) extends a Debye-potential reformulation of the Cauchy–Riemann relations to Fueter's quaternionic analyticity conditions — a result the authors state is "not explicitly found in the literature". Part (c) collects the quaternionic and complex 2×2 constructions of the Lorentz and rotation groups, with explicit component formulae. Where the paper touches dissident concerns, it does so obliquely: the revival of the quaternion notation that Heaviside and Gibbs displaced is a recurring theme in the alternative-electrodynamics literature.
The argument
Duality rotations and the quaternionic Maxwell equation
The source-free Maxwell equations ∇·B = 0, ∇·E = 0, ∇ × B = (1/c2)∂E/∂t, ∇ × E = −∂B/∂t are invariant under the duality rotation E′ = Ecos α + cBsin α, cB′ = −Esin α + cBcos α. The authors note, citing Noether's theorem, that this invariance yields the continuity equation for electromagnetic energy, ∂/∂t[(ε0/2)(E2 + c2B2)] + ∇·(E × B/μ0) = 0 — that is, Poynting's theorem read as a conservation law associated with duality rather than with time translation.
Rewriting the duality rotation as cB′ + iE′ = eiα(cB + iE) exhibits the complex Riemann–Silberstein vector F = cB + iE, in terms of which all four Maxwell equations collapse to ∇·F = 0 and ∂F/∂t − ic∇ × F = 0. Building the quaternionic vector F = FXI + FYJ + FZK and the quaternionic operator ∇ = (i/c)∂/∂t + I∂/∂x + J∂/∂y + K∂/∂z, the whole system becomes the single equation ∇F = 0. The authors are careful about priority: Conway and Silberstein introduced quaternions into special relativity, and Silberstein and Lanczos were the first to deduce this form. They remark on its resemblance to the Weyl equation for massless spin-½ particles, and argue that since unitary complex quaternions generate proper Lorentz transformations, using them for the electromagnetic field is "a natural fact".
Fueter's conditions as Debye expressions
For an analytic f(z) = u + iv of a complex variable, the Cauchy–Riemann relations force u and v to be harmonic. The authors show that those relations can be recast in a form structurally identical to the Debye representation of the electromagnetic potentials — u = (1/r)[(r·∇)ru − [r × ∇]u], with the corresponding expression for v obtained by noting that if is also analytic. They present this as "a strong motivation for the existence of Debye generators in electromagnetic theory", recalling that the source-free field can be written from two real scalar generators ψE and ψM satisfying the wave equation.
The new result generalises this to Fueter's conditions — the four coupled first-order equations on the components u0, u1, u2, u3 of a function of a quaternionic variable q = x0 + y1I + y2J + y3K, which extend Cauchy–Riemann to four dimensions and are equivalent to the compact statement ∇G = 0. The authors observe the striking similarity between this and the quaternionic Maxwell equation ∇F = 0, of which it is a generalisation, and cite Imaeda's demonstration that the Fueter conditions connect to Maxwell's equations and so to a reformulation of classical electrodynamics. Using the trick that G(q)I, G(q)J and G(q)K are analytic whenever G is, they generate four Debye-type identities, one for each component.
Rotations in three and four dimensions
The final section is expository. A real 4×4 matrix L with LTL = I effects a Lorentz transformation x′j = Ljkxk preserving x2 + y2 + z2 − c2t2. Equivalently, packing the event into a complex 2×2 matrix X with det X = −xjxj and acting by X′ = UXU† with det U = 1 reproduces the same transformation; the authors reproduce the sixteen explicit component formulae of Synge, Rumer and Aharoni in terms of the unimodular parameters α, β, γ, δ. They then give Lanczos's quaternionic version, R'′ = A*R'A with A̅A = 1, and a second parameterisation yielding the Greenberg–Knauer expressions for L; they note that Mendel Sachs obtained special cases and that the formulae have been applied to the Newman–Penrose formalism in general relativity.
Setting x4 = 0 reduces the four-dimensional case to a rotation of 3-space, with the orthogonal matrix R written out in terms of two complex numbers and U an element of SU(2). Since U and −U give the same R, SU(2) is a two-valued representation of O(3) — as ±U likewise give a two-valued representation of the Lorentz group. The formal association 1 → I, I → −iσx, J → −iσy, K → −iσz with the Pauli matrices makes the correspondence between unit quaternions and SU(2) explicit. The authors close by noting the representation of a 3-rotation by a unit real quaternion was already known to Euler.
Assessment
Judged as what it is — a compact technical note — the paper is competent and honest. The derivation of ∇F = 0 from duality invariance is clean, the priority attributions to Conway, Silberstein and Lanczos are scrupulous, and the bibliography of sixty-odd items is a genuinely useful entry point into the quaternionic electrodynamics literature, which is scattered across Irish, Mexican and Indian journals and easily missed. The one item the authors advance as new — the Debye-type reformulation of the Fueter conditions, equations (20), (22) and (23) — is stated with appropriate modesty as "relations not explicitly found in the literature", not as a discovery.
The limitations are equally clear. Nothing here is a new physical claim. The Riemann–Silberstein vector, the quaternionic Maxwell equation, the SU(2)–O(3) and SL(2,C)–Lorentz double covers, and the Synge–Rumer–Aharoni component formulae are all standard results, reassembled rather than derived afresh; the paper's own citations make this plain. The interesting suggestion — Imaeda's, that Fueter analyticity leads to "a new formulation of classical electrodynamics" — is mentioned in a single sentence and not pursued, so the reader is left without any indication of what such a formulation would predict differently, if anything. Since the quaternionic form is by construction algebraically equivalent to the vector form, no observational consequence can follow from the change of notation alone, and the paper does not claim one.
Two smaller points weaken the presentation. The Debye-potential parallel in section 2 is offered as motivation for the existence of Debye generators in electromagnetic theory, but the argument runs the wrong way round: the Debye representation of source-free fields is a theorem with an independent proof, and its formal resemblance to a rewriting of Cauchy–Riemann is suggestive rather than explanatory. And section 3 is a compilation whose sixteen-component displays are reproduced without derivation or worked example, which makes the section hard to check and hard to use. The claimed intimacy between quaternions and Lorentz transformations is real but also well understood as the accident that the even subalgebra of the relevant Clifford algebra happens to be quaternionic in these low dimensions — a structural fact the paper does not mention, and which somewhat deflates the suggestion that quaternions are uniquely natural for electromagnetism rather than merely convenient.