# 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]

The term Generalized Clifford Algebras can also refer to associative algebras that are constructed using forms of higher degree instead of quadratic forms.[8][9][10][11]

## Definition and properties

### Abstract definition

The n-dimensional generalized Clifford algebra is defined as an associative algebra over a field F, generated by[12]

${\displaystyle e_{j}e_{k}=\omega _{jk}e_{k}e_{j}\,}$
${\displaystyle \omega _{jk}e_{l}=e_{l}\omega _{jk}\,}$
${\displaystyle \omega _{jk}\omega _{lm}=\omega _{lm}\omega _{jk}\,}$

and

${\displaystyle e_{j}^{N_{j}}=1=\omega _{jk}^{N_{j}}=\omega _{jk}^{N_{k}}\,}$

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

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

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

j,k = 1,...,n,   and ${\displaystyle N_{kj}=}$gcd${\displaystyle (N_{j},N_{k})}$. 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, ${\displaystyle N_{k}=p}$   for all j,k, and ${\displaystyle \nu _{kj}=1}$. It follows that

${\displaystyle e_{j}e_{k}=\omega \,e_{k}e_{j}\,}$
${\displaystyle \omega e_{l}=e_{l}\omega \,}$

and

${\displaystyle e_{j}^{p}=1=\omega ^{p}\,}$

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

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

is the pth root of 1.

There exist several definitions of a Generalized Clifford Algebra in the literature.[13]

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

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

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

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

thus

${\displaystyle e_{1}={\begin{pmatrix}0&1\\1&0\end{pmatrix}}}$ ,    ${\displaystyle e_{2}={\begin{pmatrix}0&-1\\1&0\end{pmatrix}}}$ ,    ${\displaystyle e_{3}={\begin{pmatrix}1&0\\0&-1\end{pmatrix}}}$ ,

which constitute the Pauli matrices.

Case n = p = 4,

In this case we have ω = i, and

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

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. ^ Tesser, Steven Barry (2011). "Generalized Clifford algebras and their representations". In Micali, A.; Boudet, R.; Helmstetter, J. Clifford algebras and their applications in mathematical physics. Dordrecht: Springer. pp. 133–141. ISBN 978-90-481-4130-2.
9. ^ Childs, Lindsay N. (30 May 2007). "Linearizing of n-ic forms and generalized Clifford algebras". Linear and Multilinear Algebra. 5 (4): 267–278. doi:10.1080/03081087808817206.
10. ^ Pappacena, Christopher J. (July 2000). "Matrix pencils and a generalized Clifford algebra". Linear Algebra and its Applications. 313 (1-3): 1–20. doi:10.1016/S0024-3795(00)00025-2.
11. ^ Chapman, Adam; Kuo, Jung-Miao (April 2015). "On the generalized Clifford algebra of a monic polynomial". Linear Algebra and its Applications. 471: 184–202. doi:10.1016/j.laa.2014.12.030.
12. ^ 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.
13. ^ See for example the review provided in: Tara L. Smith: Decomposition of Generalized Clifford Algebras
14. ^ 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