Numerical Calculations in Clifford Algebra

10Electromagnetic Fields

10.1 Theory

In Clifford algebra, electromagnetic fields are represented¹ by identifying the temporal unit vector and the spatial unit vectors with particular Clifford units:

(10.1)

in a context, where the signature is all .

Four dimensions are required, with the extra dimension used for time or frequency. The Clifford unit for time is imaginary rather than real so that the Clifford geometry with uniform signatures behaves like the Minkowskian geometry, which has mixed signatures.²

Electromagnetic fields in four dimensions form a particular subset of four‐dimensional Clifford numbers. Although four‐dimensional Clifford numbers have 5 grades and 16 components, electromagnetic fields retain only one of the grades (grade 2) and six of the components. Three of these components are used to accommodate the magnetic field, and the other three are used for the electric field.

10.1.1 Time and Frequency

Electromagnetic fields are in general functions of both three‐dimensional ...