Английская Википедия:Generalized Clifford algebra

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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 Шаблон:Mvar-dimensional generalized Clifford algebra is defined as an associative algebra over a field Шаблон:Mvar, generated by

<math>\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>e_j^{N_j} = 1 = \omega_{jk}^{N_j} = \omega_{jk}^{N_k} \,</math>

Шаблон:Math.

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

<math>\omega_{jk} = \omega_{kj}^{-1} = e^{2\pi i \nu_{kj}/N_{kj}}</math>

Шаблон:Math,   and <math>N_{kj} ={}</math>gcd<math> (N_j, N_k)</math>. The field Шаблон:Mvar is usually taken to be the complex numbers C.

More specific definition

Шаблон:Main article

In the more common cases of GCA, the Шаблон:Mvar-dimensional generalized Clifford algebra of order Шаблон:Mvar has the property Шаблон:Math, <math>N_k=p</math>   for all j,k, and <math>\nu_{kj}=1</math>. It follows that

<math>\begin{align}

    e_j e_k &= \omega \, e_k e_j \,\\
 \omega e_l &= e_l \omega \,

\end{align}</math>

and

<math>e_j^{p} = 1 = \omega^{p} \,</math>

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

<math>\omega = \omega^{-1} = e^{2\pi i /p}</math>

is the Шаблон:Mvarth 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 Шаблон:Math.

Matrix representation

Шаблон:Main article The Clock and Shift matrices can be represented by Шаблон:Math matrices in Schwinger's canonical notation as

<math>\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, Шаблон:Math, Шаблон:Math (the Weyl braiding relations), and Шаблон:Math (the discrete Fourier transform). With Шаблон:Math, one has three basis elements which, together with Шаблон:Mvar, fulfil the above conditions of the Generalized Clifford Algebra (GCA).

These matrices, Шаблон:Mvar and Шаблон:Mvar, normally referred to as "shift and clock matrices", were introduced by J. J. Sylvester in the 1880s. (Note that the matrices Шаблон:Mvar 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 Шаблон:Math

In this case, we have Шаблон:Mvar = −1, and

<math>\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>\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 Шаблон:Math

In this case we have Шаблон:Mvar = Шаблон:Mvar, and

<math>\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 Шаблон:Math may be determined accordingly.

See also

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

Further reading

• Шаблон:Cite journal
• Шаблон:Cite arXiv (In The legacy of Alladi Ramakrishnan in the mathematical sciences (pp. 465–489). Springer, New York, NY.)
• Шаблон:Cite journal
• Шаблон:Cite journal
• Шаблон:Cite journal