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 Algebras 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
:$\begin e_j e_k &= \omega_ e_k e_j \\ \omega_ e_l &= e_l \omega_ \\ \omega_ \omega_ &= \omega_ \omega_ \end$

and
:$e_j^ = 1 = \omega_^ = \omega_^ \,$

.

Moreover, in any irreducible matrix representation, relevant for physical applications, it is required that
:$\omega_ = \omega_^ = e^$
,   and $N_ =$ gcd$(N_j, N_k)$. The field is usually taken to be the complex numbers C.

More specific definition

In the more common cases of GCA,See for example: the -dimensional generalized Clifford algebra of order has the property , $N_k=p$   for all ''j'',''k'', and $\nu_=1$. It follows that
:$\begin e_j e_k &= \omega \, e_k e_j \,\\ \omega e_l &= e_l \omega \, \end$

and
:$e_j^ = 1 = \omega^ \,$

for all ''j'',''k'',l = 1,...,''n'', and
:$\omega = \omega^ = e^$

is the th 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 .

Matrix representation

The Clock and Shift matrices can be represented by matrices in Schwinger's canonical notation as
:$\begin V &= \begin 0 & 1 & 0 & \cdots & 0\\ 0 & 0 & 1 & \cdots & 0\\ 0 & 0 & \ddots & 1 & 0\\ \vdots & \vdots & \vdots & \ddots & \vdots\\ 1 & 0 & 0 & \cdots & 0 \end, & U &= \begin 1 & 0 & 0 & \cdots & 0\\ 0 & \omega & 0 & \cdots & 0\\ 0 & 0 & \omega^2 & \cdots & 0\\ \vdots & \vdots & \vdots & \ddots & \vdots\\ 0 & 0 & 0 & \cdots & \omega^ \end, & W &= \begin 1 & 1 & 1 & \cdots & 1\\ 1 & \omega & \omega^2 & \cdots & \omega^\\ 1 & \omega^2 & (\omega^2)^2 & \cdots & \omega^\\ \vdots & \vdots & \vdots & \ddots & \vdots\\ 1 & \omega^ & \omega^ & \cdots & \omega^ \end \end$ .

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

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

In this case, we have = −1, and
:$\begin V &= \begin 0 & 1\\ 1 & 0 \end, & U &= \begin 1 & 0 \\ 0 & -1 \end, & W &= \begin 1 & 1 \\ 1 & -1 \end \end$

thus
:$\begin e_1 &= \begin 0 & 1 \\ 1 & 0 \end, & e_2 &= \begin 0 & -1 \\ 1 & 0 \end, & e_3 &= \begin 1 & 0 \\ 0 & -1 \end \end$ ,

which constitute the Pauli matrices.

Case

In this case we have = , and
:$\begin V &= \begin 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1\\ 1 & 0 & 0 & 0 \end, & U &= \begin 1 & 0 & 0 & 0\\ 0 & i & 0 & 0\\ 0 & 0 & -1 & 0\\ 0 & 0 & 0 & -i \end, & W &= \begin 1 & 1 & 1 & 1\\ 1 & i & -1 & -i\\ 1 & -1 & 1 & -1\\ 1 & -i & -1 & i \end \end$

and may be determined accordingly.

See also

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

References

Further reading

*
* (In ''The legacy of Alladi Ramakrishnan in the mathematical sciences'' (pp. 465–489). Springer, New York, NY.)
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{{DEFAULTSORT:Generalized Clifford Algebra
Algebras
Clifford algebras
Ring theory
Quadratic forms
Mathematical physics