Generalized Clifford algebra explained

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

The term Generalized Clifford Algebras can also refer to associative algebras that are constructed using forms of higher degree instead of quadratic forms.^{[7] [8] [9] [10]}

Definition and properties

Abstract definition

The -dimensional generalized Clifford algebra is defined as an associative algebra over a field, generated by^[11]

\begin{align} 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} \end{align}

and

e ^{N _{j}} _{j} = 1 = \omega ^{N _{j}} _{jk} = \omega ^{N _{k}} _{jk}

.

Moreover, in any irreducible matrix representation, relevant for physical applications, it is required that

\omega _{jk} = \omega ^{-1} _{kj} = e ^{2 \pi i \nu _{kj} /N _{kj}} ,   and N _{kj} = {} gcd(N _{j,} N _{k)} . The field is usually taken to be the complex numbers C.

More specific definition

See main article: article and Generalizations of Pauli matrices. In the more common cases of GCA,^[12] the -dimensional generalized Clifford algebra of order has the property,

N _{k=p}   for all j,k, and \nu _{kj} =1 . It follows that \begin{align} e _{j} e _{k} & = \omega e _{k} e _{j} \\ \omega e _{l} & = e _{l} \omega \end{align}

and

e ^{p} _{j} = 1 = \omega ^{p}

for all j,k,l = 1,...,n, and

\omega = \omega ^{-1} = e ^{2\pi}

is the th root of 1.

There exist several definitions of a Generalized Clifford Algebra in the literature.^[13]

Clifford algebraIn the (orthogonal) Clifford algebra, the elements follow an anticommutation rule, with .

Matrix representation

See main article: article. The Clock and Shift matrices can be represented^[14] by matrices in Schwinger's canonical notation as

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

Notably,, (the Weyl braiding relations), and (the discrete Fourier transform). With, one has three basis elements which, together with, fulfil the above conditions of the Generalized Clifford Algebra (GCA).

These matrices, and, normally referred to as "shift and clock matrices", were introduced by J. J. Sylvester in the 1880s. (Note that the matrices 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

In this case, we have = −1, and

\begin{align} 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{align}

thus

\begin{align} 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{align} ,

which constitute the Pauli matrices.

Case

In this case we have =, and

\begin{align} 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{align}

and may be determined accordingly.

See also

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

Further reading

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

Notes and References

1. Hermann Weyl . H. . Weyl . Quantenmechanik und Gruppentheorie . Zeitschrift für Physik . 46 . 1–2. 1–46 . 1927 . 10.1007/BF02055756 . 1927ZPhy...46....1W . 121036548 .
   Book: Weyl, H. . 1 . The Theory of Groups and Quantum Mechanics . Dover . 1931 . 1950 . 9780486602691 . registration .

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. E. . Cartan . Les groupes bilinéaires et les systèmes de nombres complexes . Annales de la Faculté des Sciences de Toulouse . 12 . 1 . B65–B99 . 1898 .
4. Julian Schwinger . J. . Schwinger . Unitary operator bases . Proc Natl Acad Sci U S A . 46 . 4 . 570–9 . April 1960 . 10.1073/pnas.46.4.570 . 16590645 . 222876. 1960PNAS...46..570S . free .
   1 . J. . Schwinger . Unitary transformations and the action principle . Proc Natl Acad Sci U S A . 46 . 6 . 883–897 . 1960 . 10.1073/pnas.46.6.883 . 16590686 . 222951. 1960PNAS...46..883S . free .
5. Santhanam . T. S. . Tekumalla . A. R. . 10.1007/BF00715110 . Quantum mechanics in finite dimensions . Foundations of Physics . 6 . 5 . 583 . 1976 . 1976FoPh....6..583S . 119936801 .
6. A.K. . Kwaśniewski . On generalized Clifford algebra C⁽ⁿ⁾₄ and GL_q(2;C) quantum group . Advances in Applied Clifford Algebras. 9 . 2. 249–260 . 1999 . 10.1007/BF03042380 . math/0403061 . 117093671 .
7. Book: Tesser. Steven Barry. Micali. A.. Boudet. R.. Helmstetter. J.. Clifford algebras and their applications in mathematical physics. limited. 2011. Springer . 978-90-481-4130-2. 133–141. Generalized Clifford algebras and their representations.
8. Childs. Lindsay N.. Linearizing of n-ic forms and generalized Clifford algebras. Linear and Multilinear Algebra. 30 May 2007. 5. 4. 267–278. 10.1080/03081087808817206.
9. Pappacena. Christopher J.. Matrix pencils and a generalized Clifford algebra. Linear Algebra and Its Applications. July 2000. 313. 1–3. 1–20. 10.1016/S0024-3795(00)00025-2. free.
10. Chapman. Adam. Kuo. Jung-Miao. On the generalized Clifford algebra of a monic polynomial. Linear Algebra and Its Applications. April 2015. 471. 184–202. 10.1016/j.laa.2014.12.030. 1406.1981. 119280952.
11. For a serviceable review, see A. . Vourdas . Quantum systems with finite Hilbert space . Reports on Progress in Physics . 67 . 3. 267–320 . 2004 . 10.1088/0034-4885/67/3/R03 . 2004RPPh...67..267V .
12. See for example: Book: A. . Granik . M. . Ross . R. . Ablamowicz . J. . Parra . P. . Lounesto . On a new basis for a Generalized Clifford Algebra and its application to quantum mechanics . https://books.google.com/books?id=OpbY_abijtwC&pg=PA101 . Clifford Algebras with Numeric and Symbolic Computation Applications . Birkhäuser . 1996 . 0-8176-3907-1 . 101–110 .
13. See for example the review provided in: Web site: Tara L. . Smith . Decomposition of Generalized Clifford Algebras . https://web.archive.org/web/20100612050907/http://math.uc.edu/~tsmith/papers/CliffAlg.pdf . 2010-06-12 .
14. Book: Ramakrishnan, Alladi . 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 . . Madras . 1971 . 87–96 .