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}. $$

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 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_{V} $$

or equivalently



\frac{d p}{d t} = \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} 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 $$