Generalized Clifford algebra

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, who utilized and formalized these clock-and-shift operators introduced by J. J. Sylvester (1882), and organized by Cartan (1898) 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 Template:Mvar-dimensional generalized Clifford algebra is defined as an associative algebra over a field Template:Mvar, generated by

$\begin{array}{cc}{e}_j{e}_k& ={\omega }_{jk}{e}_k{e}_j\\ {\omega }_{jk}{e}_{\ell }& =e_{\ell }{\omega }_{jk}\\ {\omega }_{jk}{\omega }_{\ell m}& ={\omega }_{\ell m}{\omega }_{jk}\end{array}$

and

$e_j^{N_j}=1={\omega }_{jk}^{N_j}={\omega }_{jk}^{N_k}$

Template:Math.

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

${\omega }_{jk}={\omega }_{kj}^{-1}=e^{2\pi i{\nu }_{kj}/N_{kj}}$

Template:Math,   and $N_{kj}=$gcd$(N_j,N_k)$. The field Template:Mvar is usually taken to be the complex numbers C.

More specific definition

Template:Main article

In the more common cases of GCA, the Template:Mvar-dimensional generalized Clifford algebra of order Template:Mvar has the property Template:Math, $N_k=p$   for all j,k, and ${\nu }_{kj}=1$. It follows that

$\begin{array}{cc}{e}_j{e}_k& =\omega e_k{e}_j\\ \omega e_{\ell }& =e_{\ell }\omega \end{array}$

and

$e_j^p=1={\omega }^p$

for all j,k,ℓ = 1, . . . ,n, and

$\omega ={\omega }^{-1}=e^{2\pi i/p}$

is the Template:Mvarth 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 Template:Math.

Matrix representation

Template:Main article The Clock and Shift matrices can be represented by Template:Math matrices in Schwinger's canonical notation as

$\begin{array}{cccccc}V& =\left(\begin{array}{ccccc}0& 1& 0& \cdots & 0\\ 0& 0& 1& \cdots & 0\\ 0& 0& \ddots & 1& 0\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ 1& 0& 0& \cdots & 0\end{array}\right),& U& =\left(\begin{array}{ccccc}1& 0& 0& \cdots & 0\\ 0& \omega & 0& \cdots & 0\\ 0& 0& {\omega }^2& \cdots & 0\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ 0& 0& 0& \cdots & {\omega }^{(n-1)}\end{array}\right),& W& =\left(\begin{array}{ccccc}1& 1& 1& \cdots & 1\\ 1& \omega & {\omega }^2& \cdots & {\omega }^{n-1}\\ 1& {\omega }^2& ({\omega }^2{)}^2& \cdots & {\omega }^{2(n-1)}\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ 1& {\omega }^{n-1}& {\omega }^{2(n-1)}& \cdots & {\omega }^{(n-1{)}^2}\end{array}\right)\end{array}$ .

Notably, Template:Math, Template:Math (the Weyl braiding relations), and Template:Math (the discrete Fourier transform). With Template:Math, one has three basis elements which, together with Template:Mvar, fulfil the above conditions of the Generalized Clifford Algebra (GCA).

These matrices, Template:Mvar and Template:Mvar, normally referred to as "shift and clock matrices", were introduced by J. J. Sylvester in the 1880s. (Note that the matrices Template:Mvar 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 Template:Math

In this case, we have Template:Mvar = −1, and

$\begin{array}{cccccc}V& =\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right),& U& =\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right),& W& =\left(\begin{array}{cc}1& 1\\ 1& -1\end{array}\right)\end{array}$

thus

$\begin{array}{cccccc}{e}_1& =\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right),& e_2& =\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right),& e_3& =\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right),\end{array}$

which constitute the Pauli matrices.

Case Template:Math

In this case we have Template:Mvar = Template:Mvar, and

$\begin{array}{cccccc}V& =\left(\begin{array}{cccc}0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\\ 1& 0& 0& 0\end{array}\right),& U& =\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& i& 0& 0\\ 0& 0& -1& 0\\ 0& 0& 0& -i\end{array}\right),& W& =\left(\begin{array}{cccc}1& 1& 1& 1\\ 1& i& -1& -i\\ 1& -1& 1& -1\\ 1& -i& -1& i\end{array}\right)\end{array}$

and Template:Math may be determined accordingly.

See also

• Clifford algebra
• Generalizations of Pauli matrices
• DFT matrix
• Circulant matrix

Further reading

• Template:Cite journal
• Template:Cite arXiv (In The legacy of Alladi Ramakrishnan in the mathematical sciences (pp. 465–489). Springer, New York, NY.)
• Template:Cite journal
• Template:Cite journal
• Template:Cite journal