Clifford algebras extend complex numbers by letting vectors square to plain numbers
Complex numbers rest on one strange rule: some quantity multiplied by itself gives minus one. Clifford algebras generalise that trick to whole spaces of vectors, demanding that every vector squared equals an ordinary number fixed by a chosen formula. From that single demand come the complex numbers, the quaternions and a family of tools used in physics.
The name honours William Kingdon Clifford, an English mathematician who lived from 1845 to 1879. The ingredients are a vector space and a quadratic form, a rule that assigns each vector a number, rather like a squared length. The algebra is then the most general way of multiplying vectors, associatively and with a unit, subject only to the condition that a vector times itself equals the number its quadratic form assigns. Formally, mathematicians build it by starting from the unrestricted tensor algebra and dividing out everything that breaks that one rule.
Over the real numbers the result is labelled by two counts: how many basis directions square to plus one and how many square to minus one. Different choices reproduce familiar number systems, which is why these algebras count as hypercomplex. A key consequence is that two perpendicular basis vectors anticommute, swapping sign when their order is reversed, which makes calculations with such bases pleasantly mechanical.
Set the quadratic form to zero, so every vector squares to nothing, and the construction collapses to the exterior algebra, the algebra of oriented areas and volumes. Clifford algebras can therefore be seen as a quantisation of the exterior algebra, carrying the extra information the quadratic form supplies, in the same way the Weyl algebra quantises the symmetric algebra. The two can even be combined as the odd and even parts of a single superalgebra.
One corner behaves badly. When the underlying number field has characteristic 2, where one plus one equals zero, a quadratic form no longer pins down a matching bilinear form, and many standard statements fail. Elsewhere the theory ties closely to orthogonal transformations and finds uses in geometry, theoretical physics and digital image processing.
Source: Clifford algebra
