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]
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.
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.
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).
^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.
^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.
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.)