Physics Using Geometric Algebra/Relativistic Classical Mechanics/The electromagnetic field

The electromagnetic field is defined in terms of the electric and magnetic fields as

$F=𝐄+ic𝐁.$

Alternatively, the fields can be derived from a paravector potential $A$ as

$F=c{⟨\partial \overline{A}⟩}_{V+BV},$

where:

$\partial =\frac{\partial }{\partial x^0}-\mathrm{\nabla }$

and

$A=\varphi /c+𝐀.$

Lorenz gauge

The Lorenz gauge (without t) is expressed as

$⟨\partial \overline{A}{⟩}_S=0$

The electromagnetic field $F$ is still invariant under a gauge transformation

$A\to A^{\prime }=A+\partial \chi ,$

where $\chi$ is a scalar function subject to the following condition

$\overline{\partial }\partial \chi =0$

where

$\overline{\partial }=\frac{\partial }{\partial x^0}+\mathrm{\nabla }$

Maxwell Equations

The Maxwell equations can be expressed in a single equation

$\overline{\partial }F=\frac{1}{cϵ}\overline{j},$

where the current $j$ is

$j=\rho c+𝐣$

Decomposing in parts we have

• Real scalar: Gauss's Law
• Real vector: Ampere's Law
• Imaginary scalar: No magnetic monopoles
• Imaginary vector: Faraday's law of induction

Electromagnetic Lagrangian

The electromagnetic Lagrangian that gives the Maxwell equations is

$L=\frac{1}{2}⟨FF{⟩}_S-⟨A\overline{j}{⟩}_S$

Energy density and Poynting vector

The energy density and Poynting vector can be extracted from

$\frac{{ϵ}_0}{2}F{F}^{†}=\epsilon +\frac{1}{c}S,$

where energy density is

$\epsilon =\frac{{ϵ}_0}{2}(E^2+c^2{B}^2)$

and the Poynting vector is

$S=\frac{1}{{\mu }_0}E\times B$

Lorentz Force

The electromagnetic field plays the role of a spacetime rotation with

$\mathrm{\Omega }=\frac{e}{mc}F$

The Lorentz force equation becomes

$\frac{dp}{d\tau }=⟨Fu{⟩}_V$

or equivalently

$\frac{dp}{dt}=⟨F(1+v){⟩}_V$

and the Lorentz force in spinor form is

$\frac{d\mathrm{\Lambda }}{d\tau }=\frac{e}{2mc}F\mathrm{\Lambda }$

Lorentz Force Lagrangian

The Lagrangian that gives the Lorentz Force is

$\frac{1}{2}mu\overline{u}+e⟨\overline{A}u{⟩}_S$

Plane electromagnetic waves

The propagation paravector is defined as

$k=\frac{\omega }{c}+𝐤,$

which is a null paravector that can be written in terms of the unit vector $𝐤$ as

$k=\frac{\omega }{c}(1+\widehat{𝐤}),$

A vector potential that gives origin to a polarization|circularly polarized plane wave of left helicity is

$A=e^{is\widehat{𝐤}}𝐚,$

where the phase is

$s={⟨k\overline{x}⟩}_S=\omega t-𝐤\cdot 𝐱$

and $𝐚$ is defined to be perpendicular to the propagation vector $𝐤$. This paravector potential obeys the Lorenz gauge condition. The right helicity is obtained with the opposite sign of the phase

The electromagnetic field of this paravector potential is calculated as

$F=ick{A}_{},$

which is nilpotent

$F{F}_{}=0$