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 a unital 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 algebra 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]

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

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

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, ${\displaystyle N_{k}=p}$  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. ^[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×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, 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

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

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) [1931]. 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 C⁽ⁿ⁾₄ 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 [math-ph]. (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.