Generalized Clifford algebra

Abstract definition

The n-dimensional generalized Clifford algebra is defined as an associative algebra over a field F, generated by

e j e k = ω j k e k e j ω j k e l = e l ω j k ω j k ω l m = ω l m ω j k

and

e j N j = 1 = ω j k N j = ω j k N k

∀ j,k,l,m = 1,...,n.

Moreover, in any irreducible matrix representation, relevant for physical applications, it is required that

ω j k = ω k j − 1 = e 2 π i ν k j / N k j

∀ j,k = 1,...,n,   and N k j = gcd ( N j , N k ) . The field F is usually taken to be the complex numbers C.

More specific definition

In the more common cases of GCA, the n-dimensional generalized Clifford algebra of order p has the property ω_kj = ω, N k = p   for all j,k, and ν k j = 1 . It follows that

e j e k = ω e k e j ω e l = e l ω

and

e j p = 1 = ω p

for all j,k,l = 1,...,n, and

ω = ω − 1 = e 2 π i / p

is the pth root of 1.

There exist several definitions of a Generalized Clifford Algebra in the literature.

Clifford algebra

In the (orthogonal) Clifford algebra, the elements follow an anticommutation rule, with ω = −1, and p = 2.

Matrix representation

The Clock and Shift matrices can be represented by n×n matrices in Schwinger's canonical notation as

V = ( 0 1 0 ⋯ 0 0 0 1 ⋯ 0 0 0 ⋯ 1 0 ⋯ ⋯ ⋯ ⋯ ⋯ 1 0 0 ⋯ 0 ) ,     U = ( 1 0 0 ⋯ 0 0 ω 0 ⋯ 0 0 0 ω 2 ⋯ 0 ⋯ ⋯ ⋯ ⋯ ⋯ 0 0 0 ⋯ ω ( n − 1 ) ) ,     W = ( 1 1 1 ⋯ 1 1 ω ω 2 ⋯ ω n − 1 1 ω 2 ( ω 2 ) 2 ⋯ ω 2 ( n − 1 ) ⋯ ⋯ ⋯ ⋯ ⋯ 1 ω n − 1 ω 2 ( n − 1 ) ⋯ ω ( n − 1 ) 2 ) .

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).

These matrices, V and U, normally referred to as "shift and clock matrices", were introduced by J. J. Sylvester in the 1880s. (Note that the matrices V are cyclic permutation matrices that perform a circular shift; they are not to be confused with upper and lower shift matrices which have ones only either above or below the diagonal, respectively).

Specific examples

Case n = p = 2.

In this case, we have ω = −1, and

V = ( 0 1 1 0 ) ,     U = ( 1 0 0 − 1 ) ,     W = ( 1 1 1 − 1 )

thus

e 1 = ( 0 1 1 0 ) ,     e 2 = ( 0 − 1 1 0 ) ,     e 3 = ( 1 0 0 − 1 ) ,

which constitute the Pauli matrices.

Case n = p = 4,

In this case we have ω = i, and

V = ( 0 1 0 0 0 0 1 0 0 0 0 1 1 0 0 0 ) ,     U = ( 1 0 0 0 0 i 0 0 0 0 − 1 0 0 0 0 − i ) ,     W = ( 1 1 1 1 1 i − 1 − i 1 − 1 1 − 1 1 − i − 1 i )

and e₁, e₂, e₃ may be determined accordingly.