Algebra of Physical Space - Classical Electrodynamics

Classical Electrodynamics

The electromagnetic field is represented as a bi-paravector, with the Hermitian part representing the Electric field and the anti-Hermitian part representing the magnetic field. In the standard Pauli matrix representation, the electromagnetic field is

F = \mathbf{E}+ i \mathbf{B} \rightarrow \begin{pmatrix} E_3 & E_1 -i E_2 \\ E_1 +i E_2 & -E_3 \end{pmatrix} + i \begin{pmatrix} B_3 & B_1 -i B_2 \\ B_1 +i B_2 & -B_3 \end{pmatrix}

The electromagnetic field is obtained from the paravector potential as

F = \langle \partial \bar{A} \rangle_V.

and the electromagnetic field is invariant under a gauge transformation of the form

A \rightarrow A + \partial \chi,

where is a scalar function.

The Electromagnetic field is covariant under Lorentz transformations according to the law

F \rightarrow F^\prime = L F \bar{L}

The Maxwell equations can be expressed in a single equation as follows

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

where the overbar represents the Clifford conjugation and the four-current is defined as

j = \rho + \mathbf{j}.

The electromagnetic Lagrangian is

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

which is evidently a real scalar invariant.

The Lorentz force equation takes the form

\frac{d p}{d \tau} = e \langle F u \rangle_{R}

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