Generalized Clifford algebra


In mathematics, a generalized Clifford algebra is a unital associative algebra that generalizes the Clifford algebra, and goes back to the work of Hermann Weyl, who utilized and formalized these clock-and-shift operators introduced by J. J. Sylvester, and organized by Cartan and Schwinger.
Clock and shift matrices find routine applications in numerous areas of mathematical physics, providing the cornerstone of quantum mechanical dynamics in finite-dimensional vector spaces. The concept of a spinor can further be linked to these algebras.
The term generalized Clifford algebra can also refer to associative algebras that are constructed using forms of higher degree instead of quadratic forms.

Definition and properties

Abstract definition

The -dimensional generalized Clifford algebra is defined as an associative algebra over a field, generated by
and
Moreover, in any irreducible matrix representation, relevant for physical applications, it is required that
, and gcd. The field is usually taken to be the complex numbers C.

More specific definition

In the more common cases of GCA, the -dimensional generalized Clifford algebra of order has the property, for all j,''k, and. It follows that
and
for all j'',k,''ℓ = 1,...,n'', and
is the th root of 1.
There exist several definitions of a Generalized Clifford Algebra in the literature.
; Clifford algebra
In the Clifford algebra, the elements follow an anticommutation rule, with.

Matrix representation

The Clock and Shift matrices can be represented by matrices in Schwinger's canonical notation as
Notably,,, and .
With, one has three basis elements which, together with, fulfil the above conditions of the Generalized Clifford Algebra.
These matrices, and, normally referred to as "shift and clock matrices", were introduced by J. J. Sylvester in the 1880s..

Specific examples

Case

In this case, we have = −1, and
thus
which constitute the Pauli matrices.

Case

In this case we have =, and
and may be determined accordingly.