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, 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. The term generalized Clifford algebra can also refer to associative algebras that are constructed using forms of higher degree instead of quadratic forms.
Definition and properties
Abstract definition The n-dimensional generalized Clifford algebra is defined as an associative algebra over a field F, generated by
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_{\ell }&=e_{\ell }\omega _{jk}\\\omega _{jk}\omega _{\ell m}&=\omega _{\ell m}\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, 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_{\ell }&=e_{\ell }\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 = e 2 π i / p {\displaystyle \omega =e^{2\pi i/p}}
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.
Matrix representation
The Clock and Shift matrices can be represented 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 Generalizations of Pauli matrices DFT matrix Circulant matrix
References
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". Journal of Science of the Hiroshima University, Series A. 16: 13–41. doi:10.32917/hmj/1557367250. Morris, A.O. (1967). "On a Generalized Clifford Algebra". The Quarterly Journal of Mathematics. 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". The Quarterly Journal of Mathematics. 19 (1): 289–299. Bibcode:1968QJMat..19..289M. doi:10.1093/qmath/19.1.289.