Generalized Clifford 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 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 = ω j k e k e j ω j k e l = e l ω j k ω j k ω l m = ω l m ω j k {\displaystyle {\begin{aligned}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{aligned}}}

and

e j N j = 1 = ω j k N j = ω j k N k {\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

ω j k = ω k j − 1 = e 2 π i ν k j / N k j {\displaystyle \omega _{jk}=\omega _{kj}^{-1}=e^{2\pi i\nu _{kj}/N_{kj}}}

∀ j,k = 1,...,n,   and N k j = {\displaystyle N_{kj}={}} gcd ( N j , N k ) {\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,[6] the n-dimensional generalized Clifford algebra of order p has the property ωkj = ω, N k = p {\displaystyle N_{k}=p}   for all j,k, and ν k j = 1 {\displaystyle \nu _{kj}=1} . It follows that

e j e k = ω e k e j ω e l = e l ω {\displaystyle {\begin{aligned}e_{j}e_{k}&=\omega \,e_{k}e_{j}\,\\\omega e_{l}&=e_{l}\omega \,\end{aligned}}}

and

e j p = 1 = ω p {\displaystyle e_{j}^{p}=1=\omega ^{p}\,}

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

ω = ω − 1 = e 2 π i / p {\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

The Clock and Shift matrices can be represented[14] by n×n matrices in Schwinger's canonical notation as

V = ( 0 1 0 ⋯ 0 0 0 1 ⋯ 0 0 0 ⋱ 1 0 ⋮ ⋮ ⋮ ⋱ ⋮ 1 0 0 ⋯ 0 ) , U = ( 1 0 0 ⋯ 0 0 ω 0 ⋯ 0 0 0 ω 2 ⋯ 0 ⋮ ⋮ ⋮ ⋱ ⋮ 0 0 0 ⋯ ω ( n − 1 ) ) , W = ( 1 1 1 ⋯ 1 1 ω ω 2 ⋯ ω n − 1 1 ω 2 ( ω 2 ) 2 ⋯ ω 2 ( n − 1 ) ⋮ ⋮ ⋮ ⋱ ⋮ 1 ω n − 1 ω 2 ( n − 1 ) ⋯ ω ( n − 1 ) 2 ) {\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

V = ( 0 1 1 0 ) , U = ( 1 0 0 − 1 ) , W = ( 1 1 1 − 1 ) {\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

e 1 = ( 0 1 1 0 ) , e 2 = ( 0 − 1 1 0 ) , e 3 = ( 1 0 0 − 1 ) {\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

V = ( 0 1 0 0 0 0 1 0 0 0 0 1 1 0 0 0 ) , U = ( 1 0 0 0 0 i 0 0 0 0 − 1 0 0 0 0 − i ) , W = ( 1 1 1 1 1 i − 1 − i 1 − 1 1 − 1 1 − i − 1 i ) {\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

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. ^ a b c 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 GLq(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.