In mathematics, a generalized Clifford algebra (GCA) is a unital associative algebra that generalizes the Clifford algebra, and goes back to the work of Hermann Weyl,[1] who utilized and formalized these clock-and-shift operators introduced by J. J. Sylvester (1882),[2] and organized by Cartan (1898)[3] and Schwinger.
[4] 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.
,n. Moreover, in any irreducible matrix representation, relevant for physical applications, it is required that ∀ j,k = 1, .
The field F is usually taken to be the complex numbers C. In the more common cases of GCA,[6] the n-dimensional generalized Clifford algebra of order p has the property ωkj = ω,
There exist several definitions of a Generalized Clifford Algebra in the literature.
[13] In the (orthogonal) Clifford algebra, the elements follow an anticommutation rule, with ω = −1, and p = 2.
The Clock and Shift matrices can be represented[14] by n×n matrices in Schwinger's canonical notation as Notably, Vn = 1, VU = ωUV (the Weyl braiding relations), and W−1VW = U (the discrete Fourier transform).
With e1 = V , e2 = VU, and e3 = U, one has three basis elements which, together with ω, fulfil the above conditions of the Generalized Clifford Algebra (GCA).