Английская Википедия:Generalized Clifford algebra

Материал из Онлайн справочника
Версия от 00:34, 12 марта 2024; EducationBot (обсуждение | вклад) (Новая страница: «{{Английская Википедия/Панель перехода}} {{About|the algebra called generalized Clifford algebra (GCA)|(orthogonal) Clifford algebra|Clifford algebra|symplectic Clifford algebra|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,<ref>{{...»)
(разн.) ← Предыдущая версия | Текущая версия (разн.) | Следующая версия → (разн.)
Перейти к навигацииПерейти к поиску

Шаблон:About

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 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 Шаблон:Mvar-dimensional generalized Clifford algebra is defined as an associative algebra over a field Шаблон:Mvar, generated by[12]

<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,[6] 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.[13]

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[14] 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

References

Шаблон:Reflist

Further reading