ABSTRACT
In order to see the description of e.m. waves in terms of geometric algebra, it is useful to express the bivector F of e.m. field in terms of the even subalgebra of the space-time algebra (STA) (formed by the scalar, bivector, and pseudoscalar). We remember that Equation (7.61) allows us to write F in terms of the current j (due to the fact that the ∇ operator, the same ∇ = γ µ∂µ that appears in the Dirac equation, is invertible; see, for instance, Reference [1], in which it is given that the Green function allows to solve F in terms of j). We have, in fact, from (7.60):
F = ∇ ∧ A, (8.1) where A is the vector potential of the e.m. field, which is invariant through a gauge transformation
A′(x) = A(x) + ∇α(x), (8.2) where α(x) is an arbitrary scalar function. Using the γ0 frame, i.e., the laboratory system, we can write the bivector F in terms of the more familiar electric and magnetic fields E and B. In fact, Maxwell equations can be written as [2]:
(c−1∂i + ∇)F = 0, (8.3) where
F = E + iB, (8.4) where E = Ekσk and B = Bkσk are the relative vectors. Multiplying the bivector F by itself, we have the two Lorentz-invariant quantities (one scalar and one pseudoscalar):
F 2 = (|E |2 − |B|2) + i(E · B) (8.5) In particular, for a plane e.m. wave that propagates in vacuum, we have F 2 = 0 and, then, F can be considered a light-type bivector. Notice that the
94 Geometric Algebra and Applications to Physics
expression (8.4) is actually more than a complex vector; in fact, “i” is more than a unit imaginary; it is the unit pseudoscalar that we have introduced (see Equation 7.35, where i = σ1σ2σ3 = σ1 ∧ σ2 ∧ σ3), and it appears in Equation 8.4 because the magnetic field is correctly described by the bivector iB and not by its dual B.
