Generalized Clifford algebra
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, 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.
Definition and properties
The n-dimensional generalized Clifford algebra is defined as an associative algebra over a field F, generated by
∀ 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
In the more common cases of GCA, the n-dimensional generalized Clifford algebra of order p has the property ωkj = ω, for all j,k, and . It follows that
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.
- Clifford algebra
In the (orthogonal) Clifford algebra, the elements follow an anticommutation rule, with ω = −1, and p = 2.
The Clock and Shift matrices can be represented 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).
- Case n = p = 2.
In this case, we have ω = −1, and
- , ,
- , , ,
which constitute the Pauli matrices.
- Case n = p = 4,
In this case we have ω = i, and
- , ,
and e1, e2, e3 may be determined accordingly.
- Weyl, H., "Quantenmechanik und Gruppentheorie", Zeitschrift für Physik, 46 (1927) pp. 1–46, doi:10.1007/BF02055756. Weyl, H., The Theory of Groups and Quantum Mechanics (Dover, New York, 1931)
- Sylvester, J. J., (1882), Johns Hopkins University Circulars I: 241-242; 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.
- Cartan, E. (1898). "Les groupes bilinéaires et les systèmes de nombres complexes." Annales de la faculté des sciences de Toulouse 12.1 B65-B99. online
- Schwinger, J. (1960), "Unitary operator bases", Proc Natl Acad Sci U S A, April; 46(4): 570–579, PMCID: PMC222876; ibid, "Unitary transformations and the action principle", 46(6): 883–897, PMCID: PMC222951
- Santhanam, T. S.; Tekumalla, A. R. (1976). "Quantum mechanics in finite dimensions". Foundations of Physics 6 (5): 583. doi:10.1007/BF00715110.
- See for example: A. Granik, M. Ross: On a new basis for a Generalized Clifford Algebra and its application to quantum mechanics, in: Rafal Ablamowicz, Joseph Parra, Pertti Lounesto (eds.): Clifford Algebras with Numeric and Symbolic Computation Applications, Birkhäuser, 1996, ISBN 0-8176-3907-1, p. 101–110
- A. K. Kwaśniewski: On Generalized Clifford Algebra C4(n) and GLq(2;C) quantum group
- For a serviceable review, see Vourdas A. (2004), "Quantum systems with finite Hilbert space", Rep. Prog. Phys. 67 267. doi: 10.1088/0034-4885/67/3/R03.
- See for example the review provided in: Tara L. Smith: Decomposition of Generalized Clifford Algebras
- 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, Matscience, Madras 20, pp. 87–96
- R. Jagannathan, On generalized Clifford algebras and their physical applications
- K. Morinaga, T. Nono (1952): On the linearization of a form of higher degree and its representation, J. Sci. Hiroshima Univ. Ser. A, 16, pp. 13–41
- O. Morris (1967): On a Generalized Clifford Algebra, Quart. J. Math (Oxford), 18, pp. 7–12
- O. Morris (1968): On a Generalized Clifford Algebra II, Quart. J. Math (Oxford), 19, pp. 289–299