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

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

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

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

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

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

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

where

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

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

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

Energy density and Poynting vector

The energy density and Poynting vector can be extracted from

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

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

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

or equivalently

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

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

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

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

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

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

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

which is nilpotent

$FF_{{}_{}}=0$