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]

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

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

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. (1927). "Quantenmechanik und Gruppentheorie". Zeitschrift für Physik. 46 (1–2): 1–46. Bibcode:1927ZPhy...46....1W. doi:10.1007/BF02055756.
— (1950) [1931]. The Theory of Groups and Quantum Mechanics. Dover. ISBN 9780486602691.
Sylvester, J. J. (1882), A word on Nonions, Johns Hopkins University Circulars, I, pp. 241–2, hdl:1774.2/32845; 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" (PDF). Annales de la Faculté des Sciences de Toulouse. 12 (1): B65–B99.
Schwinger, J. (April 1960). "Unitary operator bases". Proc Natl Acad Sci U S A. 46 (4): 570–9. Bibcode:1960PNAS...46..570S. doi:10.1073/pnas.46.4.570. PMC 222876. PMID 16590645.
— (1960). "Unitary transformations and the action principle". Proc Natl Acad Sci U S A. 46 (6): 883–897. Bibcode:1960PNAS...46..883S. doi:10.1073/pnas.46.6.883. PMC 222951. PMID 16590686.
Santhanam, T. S.; Tekumalla, A. R. (1976). "Quantum mechanics in finite dimensions". Foundations of Physics. 6 (5): 583. Bibcode:1976FoPh....6..583S. doi:10.1007/BF00715110.
See for example: Granik, A.; Ross, M. (1996). "On a new basis for a Generalized Clifford Algebra and its application to quantum mechanics". In Ablamowicz, R.; Parra, J.; Lounesto, P. (eds.). Clifford Algebras with Numeric and Symbolic Computation Applications. Birkhäuser. pp. 101–110. ISBN 0-8176-3907-1.
Kwaśniewski, A.K. (1999). "On generalized Clifford algebraC⁽ⁿ⁾₄ andGL_q(2;C) quantum group". AACA. 9 (2): 249–260. arXiv:math/0403061. doi:10.1007/BF03042380.
Tesser, Steven Barry (2011). "Generalized Clifford algebras and their representations". In Micali, A.; Boudet, R.; Helmstetter, J. (eds.). Clifford algebras and their applications in mathematical physics. 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. arXiv:1406.1981. 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 (3): 267–320. Bibcode:2004RPPh...67..267V. doi:10.1088/0034-4885/67/3/R03.
See for example the review provided in: Smith, Tara L. "Decomposition of Generalized Clifford Algebras" (PDF). Archived from the original (PDF) on 2010-06-12.
Ramakrishnan, Alladi (1971). "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 (PDF). Madras: Matscience. pp. 87–96.