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In mathematics, a generalized Clifford algebra (GCA) is a unital associative algebra that generalizes the Clifford algebra, and goes back to the work of Hermann Weyl, who utilized and formalized these clock-and-shift operators introduced by J. J. Sylvester (1882), and organized by Cartan (1898) and Schwinger.

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. The concept of a spinor can further be linked to these algebras.

The term generalized Clifford algebra can also refer to associative algebras that are constructed using forms of higher degree instead of quadratic forms.

Definition and properties

Abstract definition

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

${\displaystyle {\begin{aligned}e_{j}e_{k}&=\omega _{jk}e_{k}e_{j}\\\omega _{jk}e_{\ell }&=e_{\ell }\omega _{jk}\\\omega _{jk}\omega _{\ell m}&=\omega _{\ell m}\omega _{jk}\end{aligned}}}$

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, the n-dimensional generalized Clifford algebra of order p has the property ω_kj = ω, ${\displaystyle N_{k}=p}$   for all j,k, and ${\displaystyle \nu _{kj}=1}$. It follows that

${\displaystyle {\begin{aligned}e_{j}e_{k}&=\omega \,e_{k}e_{j}\,\\\omega e_{\ell }&=e_{\ell }\omega \,\end{aligned}}}$

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.

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 by n×n matrices in Schwinger's canonical notation as

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

Notably, V = 1, VU = ωUV (the Weyl braiding relations), and WVW = 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

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

thus

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

which constitute the Pauli matrices.

Case n = p = 4

In this case we have ω = i, and

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

and e₁, e₂, e₃ may be determined accordingly.

See also

• Clifford algebra
• Generalizations of Pauli matrices
• DFT matrix
• Circulant matrix

References

1. Weyl, H. (1927). "Quantenmechanik und Gruppentheorie". Zeitschrift für Physik. 46 (1–2): 1–46. Bibcode:1927ZPhy...46....1W. doi:10.1007/BF02055756. S2CID 121036548.
   — (1950) . The Theory of Groups and Quantum Mechanics. Dover. ISBN 9780486602691.
2. Sylvester, J. J. (1882), A word on Nonions, Johns Hopkins University Circulars, vol. I, pp. 241–2; 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" (PDF). Annales de la Faculté des Sciences de Toulouse. 12 (1): B65 – B99.
4. 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.
5. 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. S2CID 119936801.
6. ^ 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.
7. Kwaśniewski, A.K. (1999). "On generalized Clifford algebra C₄ and GL_q(2;C) quantum group". Advances in Applied Clifford Algebras. 9 (2): 249–260. arXiv:math/0403061. doi:10.1007/BF03042380. S2CID 117093671.
8. 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.
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. arXiv:1406.1981. doi:10.1016/j.laa.2014.12.030. S2CID 119280952.
12. For a serviceable review, see Vourdas, A. (2004). "Quantum systems with finite Hilbert space". Reports on Progress in Physics. 67 (3): 267–320. Bibcode:2004RPPh...67..267V. doi:10.1088/0034-4885/67/3/R03.
13. 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.
14. 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.

Further reading

• Fairlie, D. B.; Fletcher, P.; Zachos, C. K. (1990). "Infinite-dimensional algebras and a trigonometric basis for the classical Lie algebras". Journal of Mathematical Physics. 31 (5): 1088. Bibcode:1990JMP....31.1088F. doi:10.1063/1.528788.
• Jagannathan, R. (2010). "On generalized Clifford algebras and their physical applications". arXiv:1005.4300 . (In The legacy of Alladi Ramakrishnan in the mathematical sciences (pp. 465–489). Springer, New York, NY.)
• Morinaga, K.; Nono, T. (1952). "On the linearization of a form of higher degree and its representation". J. Sci. Hiroshima Univ. Ser. A. 16: 13–41. doi:10.32917/hmj/1557367250.
• Morris, A.O. (1967). "On a Generalized Clifford Algebra". Quart. J. Math (Oxford. 18 (1): 7–12. Bibcode:1967QJMat..18....7M. doi:10.1093/qmath/18.1.7.
• Morris, A.O. (1968). "On a Generalized Clifford Algebra II". Quart. J. Math (Oxford. 19 (1): 289–299. Bibcode:1968QJMat..19..289M. doi:10.1093/qmath/19.1.289.

• Algebras
• Clifford algebras
• Ring theory
• Quadratic forms
• Mathematical physics