Seminar - A brief introduction to Clifford’s Geometric Algebra

Speaker: Robert Forchheimer

Geometric Algebra was developed by William Clifford in 1878, partly in response to shortcomings in the early multidimensional algebras by William Hamilton (Quaternions, 1843) to describe in a concise way the newly found equations for electro-magnetic fields by James Maxwell (1864).

Clifford algebra never became wide-spread, partly due to the fact that Clifford died at an early age of 33, partly because vector analysis simultaneously developed by Oliver Heaviside and Josiah Gibbs (1884) became popularised through Gibbs lectures at MIT and his textbook on the topic (Gibbs/Wilson, 1901).

Today, Matrix Algebra has replaced these earlier algebras. However, when it comes to intuitive understanding of physical relations, we are still using complex numbers and Heaviside/Gibbs vectors (which are both subsets of Clifford algebra) to describe e.g. the behaviour of electrical circuits and electro-magnetic fields.

In this seminar, I will give a brief introduction to Clifford’s Geometric Algebra. I will particularly stress its relations to complex numbers, quaternions, vectors and objects in Euclidean space. As a final remark I will show how the traditional four Maxwell equations from vector analysis can be reduced to a single equation, revealing a simple interpretation of the electro-magnetic field theory.

• Seminar notes. PhD students who want to earn grades (suggested 1 hp) from this seminar will be asked to complete a short examination.