Difference between revisions of "Pauli algebra"
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| − | The | + | The $2^3$-dimensional real [[Clifford algebra|Clifford algebra]] generated by the [[Pauli matrices|Pauli matrices]] [[#References|[a1]]] |
| − | + | \begin{equation}\sigma_x=\begin{pmatrix}0&1\\1&0\end{pmatrix},\sigma_y=\begin{pmatrix}0&-i\\i&0\end{pmatrix},\sigma_z=\begin{pmatrix}1&0\\0&-1\end{pmatrix},\end{equation} | |
| − | where | + | where $i$ is the complex unit $\sqrt{-1}$. The matrices $\sigma_x$, $\sigma_y$ and $\sigma_z$ satisfy $\sigma^2_x=\sigma^2_y=\sigma^2_z=1$ and the anti-commutative relations: |
| − | + | \begin{equation}\sigma_i\sigma_j+\sigma_j\sigma_i=0\text{ for }i,j\in\{x,y,z\}.\end{equation} | |
| − | These matrices are used to describe angular momentum, spin- | + | These matrices are used to describe angular momentum, spin-$1/2$ fermions (which include the electron) and to describe isospin for the neutron, proton, mesons and other particles. |
| − | The angular momentum algebra is generated by elements | + | The angular momentum algebra is generated by elements $\{J_1,J_2,J_3\}$ satisfying |
| − | + | \begin{equation}J_1J_2=J_2J_1=iJ_3\end{equation} | |
| − | + | \begin{equation}J_2J_3-J_3J_2=iJ_1J_3J_1-J_1J_3=iJ_2.\end{equation} | |
The Pauli matrices provide a non-trivial representation of the generators of this algebra. The correspondence | The Pauli matrices provide a non-trivial representation of the generators of this algebra. The correspondence | ||
| − | + | \begin{equation}1\leftrightarrow\begin{pmatrix}1&0\\0&1\end{pmatrix},I\leftrightarrow i\sigma_1,J\leftrightarrow i\sigma_2,K\leftrightarrow i\sigma_3\end{equation} | |
leads to a realization of the quaternion division algebra (cf. also [[Quaternion|Quaternion]]) as a subring of the Pauli algebra. See [[#References|[a2]]], [[#References|[a3]]] for algebras with three anti-commuting elements. | leads to a realization of the quaternion division algebra (cf. also [[Quaternion|Quaternion]]) as a subring of the Pauli algebra. See [[#References|[a2]]], [[#References|[a3]]] for algebras with three anti-commuting elements. | ||
Latest revision as of 06:50, 25 December 2020
The $2^3$-dimensional real Clifford algebra generated by the Pauli matrices [a1]
\begin{equation}\sigma_x=\begin{pmatrix}0&1\\1&0\end{pmatrix},\sigma_y=\begin{pmatrix}0&-i\\i&0\end{pmatrix},\sigma_z=\begin{pmatrix}1&0\\0&-1\end{pmatrix},\end{equation}
where $i$ is the complex unit $\sqrt{-1}$. The matrices $\sigma_x$, $\sigma_y$ and $\sigma_z$ satisfy $\sigma^2_x=\sigma^2_y=\sigma^2_z=1$ and the anti-commutative relations:
\begin{equation}\sigma_i\sigma_j+\sigma_j\sigma_i=0\text{ for }i,j\in\{x,y,z\}.\end{equation}
These matrices are used to describe angular momentum, spin-$1/2$ fermions (which include the electron) and to describe isospin for the neutron, proton, mesons and other particles.
The angular momentum algebra is generated by elements $\{J_1,J_2,J_3\}$ satisfying
\begin{equation}J_1J_2=J_2J_1=iJ_3\end{equation}
\begin{equation}J_2J_3-J_3J_2=iJ_1J_3J_1-J_1J_3=iJ_2.\end{equation}
The Pauli matrices provide a non-trivial representation of the generators of this algebra. The correspondence
\begin{equation}1\leftrightarrow\begin{pmatrix}1&0\\0&1\end{pmatrix},I\leftrightarrow i\sigma_1,J\leftrightarrow i\sigma_2,K\leftrightarrow i\sigma_3\end{equation}
leads to a realization of the quaternion division algebra (cf. also Quaternion) as a subring of the Pauli algebra. See [a2], [a3] for algebras with three anti-commuting elements.
References
| [a1] | W. Pauli, "Zur Quantenmechanik des magnetischen Elektrons" Z. f. Phys. , 43 (1927) pp. 601–623 |
| [a2] | Y. Ilamed, N. Salingaros, "Algebras with three anticommuting emements I: spinors and quaternions" J. Math. Phys. , 22 (1981) pp. 2091–2095 |
| [a3] | N. Salingaros, "Algebras with three anticommuting elements II" J. Math. Phys. , 22 (1881) pp. 2096–2100 |
Pauli algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pauli_algebra&oldid=51078