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

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

$F = \mathbf{E} + i c\mathbf{B},$

which can be derived from a paravector potential $A$ as

$F = c \left\langle \partial \bar{A} \right\rangle _{V},$

where:

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

and

$A = \Phi/c + \mathbf{A}.$

Lorenz gauge

The Lorenz gauge (without t) is expressed as

$\langle \partial \bar{A} \rangle_S = 0$

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

$A \rightarrow A^\prime = A + \partial \chi,$

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

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

Maxwell Equations

The Maxwell equations can be expressed in a single equation

$\bar{\partial} F = \frac{1}{c \epsilon} \bar{j},$

where the current $j$ is

$j = \rho c + \mathbf{j}$

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

$L = \frac{1}{2} \langle F F \rangle_S - \langle A \bar{j} \rangle_S$

Energy density and Poynting vector

The energy density and Poynting vector can be extracted from

$\frac{\epsilon_0}{2} F F^\dagger = \varepsilon+ \frac{1}{c}S,$

where energy density is

$\varepsilon = \frac{\epsilon_0}{2}( E^2 + c^2 B^2 )$

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

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

The Lorentz force equation becomes

$\frac{d p}{d \tau} = \langle F u \rangle_{\Re}$

and the Lorentz force in spinor form is

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

Lorentz Force Lagrangian

The Lagrangian that gives the Lorentz Force is

$\frac{1}{2} m u \bar{u} + e \langle \bar{A}u \rangle_S$

Plane electromagnetic waves

The propagation paravector is defined as

$k = \frac{\omega}{c} + \mathbf{k},$

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

$k = \frac{\omega}{c}(1 +\mathbf{\hat k} ),$

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

$A = e^{i s \mathbf{\hat k}} \mathbf{a},$

where the phase is

$s=\left\langle k \bar{x} \right\rangle_S = \omega t - \mathbf{k} \cdot \mathbf{x}$

and $\mathbf{a}$ is defined to be perpendicular to the propagation vector $\mathbf{k}$. 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

$F = i c k A_{{}_{}},$

which is nilpotent

$F F_{{}_{}} = 0$