# Generalized Clifford algebra

In mathematics, a Generalized Clifford algebra (GCA) is an 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.[5][6][7] The concept of a spinor can further be linked to these algebras.[6]

## Definition and properties

### Abstract definition

The n-dimensional generalized Clifford algebra is defined as an associative algebra over a field F, generated by[8]

$e_j e_k = \omega_{jk} e_k e_j \,$
$\omega_{jk} e_l = e_l \omega_{jk} \,$
$\omega_{jk} \omega_{lm} = \omega_{lm} \omega_{jk} \,$

and

$e_j^{N_j} = 1 = \omega_{jk}^{N_j} = \omega_{jk}^{N_k} \,$

j,k,l,m = 1,...,n.

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

$\omega_{jk} = \omega_{kj}^{-1} = e^{2\pi i \nu_{kj}/N_{kj}}$

j,k = 1,...,n,   and $N_{kj} =$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,[6] the n-dimensional generalized Clifford algebra of order p has the property ωkj = ω, $N_k=p$   for all j,k, and $\nu_{kj}=1$. It follows that

$e_j e_k = \omega \, e_k e_j \,$
$\omega e_l = e_l \omega \,$

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 pth root of 1.

There exist several definitions of a Generalized Clifford Algebra in the literature.[9]

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[10] by n×n matrices in Schwinger's canonical notation as

$V = \begin{pmatrix} 0&1&0&\cdots&0\\ 0&0&1&\cdots&0\\ 0&0&\cdots&1&0\\ \cdots&\cdots&\cdots&\cdots&\cdots\\ 1&0&0&\cdots&0 \end{pmatrix}$ ,    $U = \begin{pmatrix} 1&0&0&\cdots&0\\ 0&\omega&0&\cdots&0\\ 0&0&\omega^2&\cdots&0\\ \cdots&\cdots&\cdots&\cdots&\cdots\\ 0&0&0&\cdots&\omega^{(n-1)} \end{pmatrix}$ ,    $W = \begin{pmatrix} 1&1&1&\cdots&1\\ 1&\omega&\omega^2&\cdots&\omega^{n-1}\\ 1&\omega^2&(\omega^2)^2&\cdots&\omega^{2(n-1)}\\ \cdots&\cdots&\cdots&\cdots&\cdots\\ 1&\omega^{n-1}&\omega^{2(n-1)}&\cdots&\omega^{(n-1)^2} \end{pmatrix}$ .

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

$V = \begin{pmatrix} 0&1\\ 1&0 \end{pmatrix}$ ,    $U = \begin{pmatrix} 1&0\\ 0&-1 \end{pmatrix}$ ,    $W = \begin{pmatrix} 1&1\\ 1&-1 \end{pmatrix}$

thus

$e_1 = \begin{pmatrix} 0&1\\ 1&0 \end{pmatrix}$ ,    $e_2 = \begin{pmatrix} 0&-1\\ 1&0 \end{pmatrix}$ ,    $e_3 = \begin{pmatrix} 1&0\\ 0&-1 \end{pmatrix}$ ,

which constitute the Pauli matrices.

Case n = p = 4,

In this case we have ω = i, and

$V = \begin{pmatrix} 0&1&0&0\\ 0&0&1&0\\ 0&0&0&1\\ 1&0&0&0 \end{pmatrix}$ ,    $U = \begin{pmatrix} 1&0&0&0\\ 0&i&0&0\\ 0&0&-1&0\\ 0&0&0&-i \end{pmatrix}$ ,    $W = \begin{pmatrix} 1&1&1&1\\ 1&i&-1&-i\\ 1&-1&1&-1\\ 1&-i&-1&i \end{pmatrix}$

and e1, e2, e3 may be determined accordingly.

## References

1. ^ Weyl, H., "Quantenmechanik und Gruppentheorie", Zeitschrift für Physik, 46 (1927) pp. 1–46, doi:10.1007/BF02055756. Weyl, H., The Theory of Groups and Quantum Mechanics (Dover, New York, 1931)
2. ^ Sylvester, J. J., (1882), Johns Hopkins University Circulars I: 241-242; ibid II (1883) 46; ibid III (1884) 7–9. Summarized in The Collected Mathematics Papers of James Joseph Sylvester (Cambridge University Press, 1909) v III . online and further.
3. ^ Cartan, E. (1898). "Les groupes bilinéaires et les systèmes de nombres complexes." Annales de la faculté des sciences de Toulouse 12.1 B65-B99. online
4. ^ Schwinger, J. (1960), "Unitary operator bases", Proc Natl Acad Sci U S A, April; 46(4): 570–579, PMCID: PMC222876; ibid, "Unitary transformations and the action principle", 46(6): 883–897, PMCID: PMC222951
5. ^ Santhanam, T. S.; Tekumalla, A. R. (1976). "Quantum mechanics in finite dimensions". Foundations of Physics 6 (5): 583. doi:10.1007/BF00715110.
6. ^ a b c See for example: A. Granik, M. Ross: On a new basis for a Generalized Clifford Algebra and its application to quantum mechanics, in: Rafal Ablamowicz, Joseph Parra, Pertti Lounesto (eds.): Clifford Algebras with Numeric and Symbolic Computation Applications, Birkhäuser, 1996, ISBN 0-8176-3907-1, p. 101–110
7. ^ A. K. Kwaśniewski: On Generalized Clifford Algebra C4(n) and GLq(2;C) quantum group
8. ^ For a serviceable review, see Vourdas A. (2004), "Quantum systems with finite Hilbert space", Rep. Prog. Phys. 67 267. doi: 10.1088/0034-4885/67/3/R03.
9. ^ See for example the review provided in: Tara L. Smith: Decomposition of Generalized Clifford Algebras
10. ^ Alladi Ramakrishnan: Generalized Clifford Algebra and its applications – A new approach to internal quantum numbers, Proceedings of the Conference on Clifford algebra, its Generalization and Applications, January 30 – February 1, 1971, Matscience, Madras 20, pp. 87–96