Generalized Clifford algebra

This article is about the algebra called generalized Clifford algebra (GCA). For (orthogonal) Clifford algebra, see Clifford algebra. For symplectic Clifford algebra, see Weyl 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,^[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 algebra 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]

${\displaystyle {\begin{aligned}e_{j}e_{k}&=\omega _{jk}e_{k}e_{j}\\\omega _{jk}e_{\ell }&=e_{\ell }\omega _{jk}\\\omega _{jk}\omega _{\ell m}&=\omega _{\ell m}\omega _{jk}\end{aligned}}}$

and

${\displaystyle e_{j}^{N_{j}}=1=\omega _{jk}^{N_{j}}=\omega _{jk}^{N_{k}}\,}$

∀ j,k,ℓ,m = 1, . . . ,n.

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

${\displaystyle \omega _{jk}=\omega _{kj}^{-1}=e^{2\pi i\nu _{kj}/N_{kj}}}$

∀ j,k = 1, . . . ,n,   and ${\displaystyle N_{kj}={}}$gcd${\displaystyle (N_{j},N_{k})}$. The field F is usually taken to be the complex numbers C.

More specific definition

Main article: Generalizations of Pauli matrices

In the more common cases of GCA,^[6] the n-dimensional generalized Clifford algebra of order p has the property ω_kj = ω, ${\displaystyle N_{k}=p}$   for all j,k, and ${\displaystyle \nu _{kj}=1}$. It follows that

${\displaystyle {\begin{aligned}e_{j}e_{k}&=\omega \,e_{k}e_{j}\,\\\omega e_{\ell }&=e_{\ell }\omega \,\end{aligned}}}$

and

${\displaystyle e_{j}^{p}=1=\omega ^{p}\,}$

for all j,k,ℓ = 1, . . . ,n, and

${\displaystyle \omega =\omega ^{-1}=e^{2\pi i/p}}$

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

Main article: Generalizations of Pauli matrices § Construction: The clock and shift matrices

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

${\displaystyle {\begin{aligned}V&={\begin{pmatrix}0&1&0&\cdots &0\\0&0&1&\cdots &0\\0&0&\ddots &1&0\\\vdots &\vdots &\vdots &\ddots &\vdots \\1&0&0&\cdots &0\end{pmatrix}},&U&={\begin{pmatrix}1&0&0&\cdots &0\\0&\omega &0&\cdots &0\\0&0&\omega ^{2}&\cdots &0\\\vdots &\vdots &\vdots &\ddots &\vdots \\0&0&0&\cdots &\omega ^{(n-1)}\end{pmatrix}},&W&={\begin{pmatrix}1&1&1&\cdots &1\\1&\omega &\omega ^{2}&\cdots &\omega ^{n-1}\\1&\omega ^{2}&(\omega ^{2})^{2}&\cdots &\omega ^{2(n-1)}\\\vdots &\vdots &\vdots &\ddots &\vdots \\1&\omega ^{n-1}&\omega ^{2(n-1)}&\cdots &\omega ^{(n-1)^{2}}\end{pmatrix}}\end{aligned}}}$ .

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

${\displaystyle {\begin{aligned}V&={\begin{pmatrix}0&1\\1&0\end{pmatrix}},&U&={\begin{pmatrix}1&0\\0&-1\end{pmatrix}},&W&={\begin{pmatrix}1&1\\1&-1\end{pmatrix}}\end{aligned}}}$

thus

${\displaystyle {\begin{aligned}e_{1}&={\begin{pmatrix}0&1\\1&0\end{pmatrix}},&e_{2}&={\begin{pmatrix}0&-1\\1&0\end{pmatrix}},&e_{3}&={\begin{pmatrix}1&0\\0&-1\end{pmatrix}},\end{aligned}}}$

which constitute the Pauli matrices.

Case n = p = 4

In this case we have ω = i, and

${\displaystyle {\begin{aligned}V&={\begin{pmatrix}0&1&0&0\\0&0&1&0\\0&0&0&1\\1&0&0&0\end{pmatrix}},&U&={\begin{pmatrix}1&0&0&0\\0&i&0&0\\0&0&-1&0\\0&0&0&-i\end{pmatrix}},&W&={\begin{pmatrix}1&1&1&1\\1&i&-1&-i\\1&-1&1&-1\\1&-i&-1&i\end{pmatrix}}\end{aligned}}}$

and e₁, e₂, e₃ may be determined accordingly.

See also

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