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The progress made so far now allows us to propose a first mathematical face to the intuition presented on the "first step" page.

Each electric (resp. magnetic) field in an empty region of space-time can be associated with a purely imaginary quaternion:

E ≡ (0, < h,E >) ∈ Im(H)

H ≡ (0, < h,H >) ∈ Im(H)

Je veux maintenant affiner et explorer les conséquences de cette manière d'associer un champ spatial avec un quaternion imaginaire.

Assuming that the product of two quaternions represents the interaction of their respective vector parts, it becomes possible to get an idea of the interaction of an electric (resp. magnetic) field with itself; indeed:

(0, < h,E >). (0, < h, E >) = (-||E||^2, < h,0 >)

(0, < h,H >). (0, < h,H >) = (-||H||^2, < h,0 >)

And since I have postulated that only the real part of quaternions is perceptible, I deduce that the logic adopted leads us to say here that the interaction of a classical electric field with itself manifests as a number related to the volume energy density it carries with it.

To refine this assumption and harmonize it with the writings of this century, it is undoubtedly appropriate to reformulate the initial association by introducing the electric permeability coefficient ε0, and the magnetic permittivity μ0.

(ε₀, E) → q(ε₀, E) = (0, < h,√(ε₀/2). E >)

(μ₀, E) → q(μ₀, H) = (0, < h,√(1/2.μ₀). H >)

Which leads to:

 q^2(ε₀, E) = (- ε₀/2.||E||^2, < h,0 >)

q^2(μ₀, H) = (- (1/2.μ₀).||H||^2, < h,0 >)

 
The square of the purely imaginary quaternion representing an electric (resp. magnetic) field in a vacuum is purely real and is equal to minus one times the electric (resp. magnetic) field energy volume density carried by this field when measured in a vacuum.

The real part of the difference of the squares calculated previously coincides with the first invariant associated with the pair (E, H):

q²(μ₀, H) - q²(ε₀, E) = [Inv1(E, H), < h,0 >)

The real part of the sum of the squares calculated previously coincides with minus one times the volumetric density of electromagnetic energy carried by the pair (E, H) when measured in a vacuum.

q²(μ₀, H) + q²(ε₀, E) = (-ρ(E, H), < h,0 >)

Applying the same principle as before to an interaction between an electric field and a magnetic field (and vice versa), I find:

q(ε₀, E).q(μ₀, H) ∼ (- < E,H >), < h, E ∧ H >)

q(ε₀, H).q(μ₀, E) ∼ (- < H,E >), < h, H ∧ E >)

I infer that the perceptible (real) part of this interaction is the first invariant of the electromagnetic field, namely the scalar product < E,H > weighted by the ratio √(ε0/μ0).

But in a region of space empty of matter, this invariant is currently reputed to be almost zero because the field would not be polarized there and, consequently, there is no visible trace of the interaction of the electric part with the magnetic part of an (E, H) pair for a human eye.

A direct, simple and important consequence of these calculations is the relation:

q(ε₀, E).q(μ₀, H) + q(μ₀, H).q(ε₀, E) = 0

 
The anticommutativity of the purely imaginary quaternion pair (q(ε0, E), q(μ0, H)) representing the pair (E, H) appears here in a striking manner. The association made at the beginning of this approach has the characteristic of transforming the electric and magnetic branches into anticommutative vector variables.

In this context, the invisible (imperceptible) part of this interaction is proportional to the Poynting vector. Contemporary physics interprets it as the local spatial velocity of propagation of electromagnetic energy.

The subject of the matrix representation of electromagnetic fields in a context using quaternions has already been covered on the page introducing "electromagnetic fields".
 
The part of the matrix representation that is proportional to the identity matrix accounts for the real part of the quaternion (or quaternion product) it represents.
 
It is possible to verify that (up to you to calculate):

This calculation shows that the product of two matrix representations, each associated with a quaternion, is the matrix representation of this product of two quaternions ... provided we accept that the argument of the Φ-type matrix present at the core of this representation has the classical cross product as its argument:

In other words, the matrix provisionally denoted [...] is generically written as:

The logic of quaternion matrix representations adopted so far allows us to induce the existence of a third vector:

This observation paves the way for numerous physical applications. I will give a few examples later.

Anticommutativity is a very strange concept that the passing of time illustrates to perfection. This is the reason behind the choice of the image at the header of this page.
 
Time, like water rushing down a slope, flows inexorably in a single direction. Never does water spontaneously flow back up to the source, never does the flight of time bring us back to the cradle.