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

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

${\displaystyle F=\mathbf {E} +ic\mathbf {B} .}$

Alternatively, the fields can be derived from a paravector potential ${\displaystyle A}$ as

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

where:

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

and

${\displaystyle A=\phi /c+\mathbf {A} .}$

## Lorenz gauge

The Lorenz gauge (without t) is expressed as

${\displaystyle \langle \partial {\bar {A}}\rangle _{S}=0}$

The electromagnetic field ${\displaystyle F}$ is still invariant under a gauge transformation

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

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

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

where

${\displaystyle {\bar {\partial }}={\frac {\partial }{\partial x^{0}}}+\nabla }$

## Maxwell Equations

The Maxwell equations can be expressed in a single equation

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

where the current ${\displaystyle j}$ is

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

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

${\displaystyle L={\frac {1}{2}}\langle FF\rangle _{S}-\langle A{\bar {j}}\rangle _{S}}$

### Energy density and Poynting vector

The energy density and Poynting vector can be extracted from

${\displaystyle {\frac {\epsilon _{0}}{2}}FF^{\dagger }=\varepsilon +{\frac {1}{c}}S,}$

where energy density is

${\displaystyle \varepsilon ={\frac {\epsilon _{0}}{2}}(E^{2}+c^{2}B^{2})}$

and the Poynting vector is

${\displaystyle S={\frac {1}{\mu _{0}}}E\times B}$

## Lorentz Force

The electromagnetic field plays the role of a spacetime rotation with

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

The Lorentz force equation becomes

${\displaystyle {\frac {dp}{d\tau }}=\langle Fu\rangle _{V}}$

or equivalently

${\displaystyle {\frac {dp}{dt}}=\langle F(1+v)\rangle _{V}}$

and the Lorentz force in spinor form is

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

### Lorentz Force Lagrangian

The Lagrangian that gives the Lorentz Force is

${\displaystyle {\frac {1}{2}}mu{\bar {u}}+e\langle {\bar {A}}u\rangle _{S}}$

## Plane electromagnetic waves

The propagation paravector is defined as

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

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

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

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

${\displaystyle A=e^{is\mathbf {\hat {k}} }\mathbf {a} ,}$

where the phase is

${\displaystyle s=\left\langle k{\bar {x}}\right\rangle _{S}=\omega t-\mathbf {k} \cdot \mathbf {x} }$

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

${\displaystyle F=ickA_{{}_{}},}$

which is nilpotent

${\displaystyle FF_{{}_{}}=0}$