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, 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 Algebras 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

[math]\displaystyle{ \begin{align} 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{align} }[/math]

and

[math]\displaystyle{ e_j^{N_j} = 1 = \omega_{jk}^{N_j} = \omega_{jk}^{N_k} \, }[/math]

∀ j,k,l,m = 1,...,n.

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

[math]\displaystyle{ \omega_{jk} = \omega_{kj}^{-1} = e^{2\pi i \nu_{kj}/N_{kj}} }[/math]

∀ j,k = 1,...,n,   and [math]\displaystyle{ N_{kj} ={} }[/math]gcd[math]\displaystyle{ (N_j, N_k) }[/math]. 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 ω_kj = ω, [math]\displaystyle{ N_k=p }[/math]   for all j,k, and [math]\displaystyle{ \nu_{kj}=1 }[/math]. It follows that

[math]\displaystyle{ \begin{align} e_j e_k &= \omega \, e_k e_j \,\\ \omega e_l &= e_l \omega \, \end{align} }[/math]

and

[math]\displaystyle{ e_j^{p} = 1 = \omega^{p} \, }[/math]

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

[math]\displaystyle{ \omega = \omega^{-1} = e^{2\pi i /p} }[/math]

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

[math]\displaystyle{ \begin{align} 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{align} }[/math] .

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

[math]\displaystyle{ \begin{align} 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{align} }[/math]

thus

[math]\displaystyle{ \begin{align} 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{align} }[/math] ,

which constitute the Pauli matrices.

Case n = p = 4

In this case we have ω = i, and

[math]\displaystyle{ \begin{align} 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{align} }[/math]

and e₁, e₂, e₃ may be determined accordingly.

See also

• Clifford algebra
• Generalizations of Pauli matrices
• DFT matrix
• Circulant matrix

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. doi:10.1063/1.528788. Bibcode: 1990JMP....31.1088F. 
• 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. doi:10.1093/qmath/18.1.7. Bibcode: 1967QJMat..18....7M. 
• Morris, A.O. (1968). "On a Generalized Clifford Algebra II". Quart. J. Math (Oxford 19 (1): 289–299. doi:10.1093/qmath/19.1.289. Bibcode: 1968QJMat..19..289M.