Generalized Clifford algebra

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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[edit]

Abstract definition[edit]

The n-dimensional generalized Clifford algebra is defined as an associative algebra over a field F, generated by[12]

and

j,k,l,m = 1,...,n.

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

j,k = 1,...,n,   and gcd. The field F is usually taken to be the complex numbers C.

More specific definition[edit]

In the more common cases of GCA,[6] the n-dimensional generalized Clifford algebra of order p has the property ωkj = ω,   for all j,k, and . It follows that

and

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

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[edit]

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

.

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[edit]

Case n = p = 2[edit]

In this case, we have ω = −1, and

thus

,

which constitute the Pauli matrices.

Case n = p = 4[edit]

In this case we have ω = i, and

and e1, e2, e3 may be determined accordingly.

See also[edit]

References[edit]

  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, 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.
  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 algebraC(n)4 and GLq(2;C) quantum group". AACA. 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[edit]