Generalized Clifford algebra

This article is about the algebra called generalized Clifford algebra (GCA). For (orthogonal) Clifford algebra, see Clifford algebra. For symplectic Clifford algebra, see Weyl 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]

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}}}\,

$\forall j, k, l, m = 1,..., n$ .

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}}}}

$\forall j, k = 1,..., n$ , and $N_{{kj}}=$ gcd $(N_{j},N_{k})$ . The field $F$ is usually taken to be the complex numbers C.

More specific definition

Main article: Generalizations of Pauli matrices

In the more common cases of GCA,^[6] the $n$ -dimensional generalized Clifford algebra of order $p$ has the property $ω kj = ω$ , $N_{k}=p$ 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 $p$ th 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

Main article: Generalizations of Pauli matrices § Construction: The clock and shift matrices

The Clock and Shift matrices can be represented^[14] by $n\timesn$ 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, $V n = 1$ , $VU = ωUV$ (the Weyl braiding relations), and $W -1 VW = U$ (the Discrete Fourier transform). With $e 1 = V, e 2 = VU, and e 3 = U$ , 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 $e 1, e 2, e 3$ may be determined accordingly.

References

↑ 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)
↑ 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.
↑ 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
↑ 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
↑ Santhanam, T. S.; Tekumalla, A. R. (1976). "Quantum mechanics in finite dimensions". Foundations of Physics. 6 (5): 583. doi:10.1007/BF00715110.
1 2 3 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
↑ A. K. Kwaśniewski: On Generalized Clifford Algebra C4(n) and GLq(2;C) quantum group
↑ 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.
↑ 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.
↑ 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.
↑ 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.
↑ 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.
↑ See for example the review provided in: Tara L. Smith: Decomposition of Generalized Clifford Algebras
↑ 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