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] [2]

Definition and properties

General definition

This definition is brought forward by R. Jagannathan, in line with the work of his teacher Alladi Ramakrishnan. The n-dimensional generalized Clifford algebra is defined as an associative algebra over a field F, generated by

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

and

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

for all j,k,l,m = 1,...,n. Additionally, in any irreducible matrix representation, relevant for physical applications, it is required that

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

for all j,k = 1,...,n, and $N_{kj} =$gcd$(N_j,N_k)$.[3] The field F is usually taken to be the complex numbers C.

More specific definition

In the more common cases of GCA,[4][5] the n-dimensional generalized Clifford algebra of order p has the property $\omega_{kj}=\omega$, $N_k=p$ for all j,k, and $\nu_{kj}=1$. From this 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.[6]

Clifford algebra

In the (orthogonal) Clifford algebra, the elements follow an anticommutation rule, with $\omega=-1$ and p = 2.

Matrix representation

GCA matrices can be represented as:[7]

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

Then $C^n = 1$ and $C B = \omega B C$ and $U^{-1} C U = B$. Then with $e_1=C$ and $e_2=CB$ and $e_3=B$ we have three basis elements which, together with $\omega$, fulfil the defining conditions of the Generalized Clifford Algebra (GCA).

These matrices $C$ and $B$ are often referred to as "shift and clock matrices", introduced by J. J. Sylvester in the 1880s. (Note that the matrices $C$ 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).

Numerical examples

Case n = p = 2

In this case we have $\omega=-1$ and

$C = \begin{pmatrix} 0&1\\ 1&0 \end{pmatrix}$,    $B = \begin{pmatrix} 1&0\\ 0&-1 \end{pmatrix}$,    $U = \begin{pmatrix} 1&1\\ 1&-1 \end{pmatrix}$

thus

$e_1 = \begin{pmatrix} 0&1\\ 1&0 \end{pmatrix}$,    $e_2 = \begin{pmatrix} 0&-1\\ 1&0 \end{pmatrix}$,    $e_3 = \begin{pmatrix} 1&0\\ 0&-1 \end{pmatrix}$
Case n = p = 4

In this case we have $\omega=i$ and

$C = \begin{pmatrix} 0&1&0&0\\ 0&0&1&0\\ 0&0&0&1\\ 1&0&0&0 \end{pmatrix}$,    $B = \begin{pmatrix} 1&0&0&0\\ 0&i&0&0\\ 0&0&-1&0\\ 0&0&0&-i \end{pmatrix}$,    $U = \begin{pmatrix} 1&1&1&1\\ 1&i&-1&-i\\ 1&-1&1&-1\\ 1&-i&-1&i \end{pmatrix}$

and $e_1$, $e_2$, $e_3$ can be determined accordingly.

Application to physics

Generalized Clifford algebras has been applied to quantum physics.[4][5][8] In particular, the concept of a spinor can be linked directly to these algebras.[4]

• R. Jagannathan, On generalized Clifford algebras and their physical applications
• K. Morinaga, T. Nono (1952): On the linearization of a form of higher degree and its representation, J. Sci. Hiroshima Univ. Ser. A, 16, pp. 13–41
• O. Morris (1967): On a Generalized Clifford Algebra, Quart. J. Math (Oxford), 18, pp. 7–12
• O. Morris (1968): On a Generalized Clifford Algebra II, Quart. J. Math (Oxford), 19, pp. 289–299

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. ^ 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
3. ^ R. Jagannathan: On generalized Clifford algebras and their physical applications, arXiv:1005.4300v1 (submitted 24 May 2010)
4. ^ 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
5. ^ a b See for ex.: E.H.El Kinani: Between Quantum Virasoro Algebra Lc and Generalized Clifford Algebras arXiv:math-ph/0310044v1 (submitted 22 October 2003)
6. ^ See for example the review provided in: Tara L. Smith: Decomposition of Generalized Clifford Algebras
7. ^ 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
8. ^ A. K. Kwaśniewski: On Generalized Clifford Algebra C4(n) and GLq(2;C) quantum group