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

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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},


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


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

Lorenz gauge[edit]

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[edit]

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[edit]

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[edit]

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[edit]

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[edit]

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[edit]

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