The Psigma[n] command, where n ranges from 1 to 3, represents the three Pauli matrices, displayed on the screen as ${\mathrm{\sigma }}_n$; these are the Hermitian and unitary matrices

where $\mathrm{I}$ is the imaginary unit (to represent it with a lowercase $i$, see interface,imaginaryunit). Psigma[...] can be also be indexed with the letters x, y and z, with the standard correspondence ${\mathrm{\sigma }}_x={\mathrm{\sigma }}_1$ ${\mathrm{\sigma }}_y={\mathrm{\sigma }}_2$ and ${\mathrm{\sigma }}_z={\mathrm{\sigma }}_3$, and also with `+` and `-` (include the ``), representing the ladder operators

To see the matrix form of any of these formulas use Physics:-Library:-RewriteInMatrixForm.

The Pauli matrices satisfy the commutation relations

where ${\mathrm{\delta }}_{a,b}$ and ${\mathrm{\epsilon }}_{a,b,c}$ are respectively the KroneckerDelta and LeviCivita symbols, and $a,b,c$ range from 1 to 3. The Pauli matrices satisfy $\mathrm{Det}\left({\mathrm{\sigma }}_a\right)=\mathrm{-1},\mathrm{Trace}\left({\mathrm{\sigma }}_a\right)=0$, and ${\mathrm{\sigma }}_a^2=1$ (the 2 x 2 identity matrix), where Det represents the determinant, and Trace represents/computes the trace.

When Physics is loaded, the three Pauli matrices Psigma[a], with a ranging from 1 to 3, together with Psigma[4] representing the 2 x 2 identity, are the components of a 4-vector in spacetime, Psigma[mu], with mu ranging from 1 to 4 and displayed as ${\mathrm{\sigma }}_{\mathrm{\mu }}$. As with all spacetime tensors, you can use the value 0 of an index to refer to the position of the timelike component (by default equal to 4). The defining algebra for the 3D Pauli matrices ${\mathrm{\sigma }}_a$ shown above is extended to the 4D ${\mathrm{\sigma }}_{\mathrm{\mu }}$ as follows

Note: The default metric is of Minkowski type with signature (---+), so the space components of the contravariant ${\mathrm{\sigma }}^{\mathrm{\mu }}$ change in sign with respect to the definition of the Pauli matrices shown above.

You can use Setup to change the metric or set a different signature, for example with the timelike component in position 1, as in (+---) or (-+++). That, however, makes the values 1, 2 and 3 of the index $\mathrm{\mu }$ respectively refer to the identity matrix and the Pauli matrices 1, 2. To work with (+---) or (-+++) and avoid the inconvenience of having Psigma[1] referring to the identity matrix instead of ${\mathrm{\sigma }}_x$, set spaceindices or su2indices, for example via $\mathrm{Setup}\left(\mathrm{su2indices}=\mathrm{lowercaselatin\_ah}\right)$, and use Define with its redo option to redefine the type of tensor of Psigma, for example entering $\mathrm{Define}\left(\mathrm{redo}\,{\mathrm{Psigma}}_a\right)$. In that way Psigma[a] becomes a 3D tensor, free of the issue mentioned about the numerical value of the index 1 in Psigma[mu].

$\mathrm{with}\left(\mathrm{Physics}\right)\:$

$\mathrm{Setup}\left(\mathrm{mathematicalnotation}=\mathrm{true}\right)$

$\left[\mathrm{mathematicalnotation}=\mathrm{true}\right]$

$\mathrm{Psigma}\left[1\right]$

${\mathrm{\sigma }}_1$

$\mathrm{Psigma}\left[1,\mathrm{matrix}\right]$

${\mathrm{\sigma }}_1=\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]$

$\mathrm{Library}:-\mathrm{RewriteInMatrixForm}\left(\right)$

$\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]$

$\mathrm{Psigma}\left[1\right]\mathrm{Psigma}\left[2\right]+\mathrm{Psigma}\left[2\right]\mathrm{Psigma}\left[1\right]$

${\mathrm{\sigma }}_1{\mathrm{\sigma }}_2+{\mathrm{\sigma }}_2{\mathrm{\sigma }}_1$

$\mathrm{Library}:-\mathrm{RewriteInMatrixForm}\left(\right)$

$\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]·\left[\begin{array}{cc}0& \mathrm{-I}\\ \mathrm{I}& 0\end{array}\right]+\left[\begin{array}{cc}0& \mathrm{-I}\\ \mathrm{I}& 0\end{array}\right]·\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]$

$\mathrm{Library}:-\mathrm{PerformMatrixOperations}\left(\right)$

$\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]$

$\mathrm{Psigma}\left[1\right]\mathrm{Psigma}\left[1\right]$

${\mathrm{\sigma }}_1^2$

$\mathrm{Trace}\left(\right)$

$2$

$\mathrm{Psigma}\left[1\right]\mathrm{Psigma}\left[2\right]$

${\mathrm{\sigma }}_1{\mathrm{\sigma }}_2$

$\mathrm{Trace}\left(\right)$

$0$

$\mathrm{Library}:-\mathrm{DefaultAlgebraRules}\left(\mathrm{Psigma}\right)$

${\left[{\mathrm{\sigma }}_{\mathrm{\mu }},{\mathrm{\sigma }}_{\mathrm{\nu }}\right]}_{-}=2\mathrm{I}{\mathrm{\epsilon }}_{4,\mathrm{\mu },\mathrm{\nu }\phantom{\mathrm{\alpha }}}^{\phantom{4}\phantom{,\mathrm{\mu },\mathrm{\nu }}\mathrm{\alpha }}{\mathrm{\sigma }}_{\mathrm{\alpha }},{\left[{\mathrm{\sigma }}_{\mathrm{\mu }},{\mathrm{\sigma }}_{\mathrm{\nu }}\right]}_{+}=2{\mathrm{\sigma }}_{\mathrm{\mu }}{\mathrm{\delta }}_{\mathrm{\nu }\phantom{4}}^{\phantom{\mathrm{\nu }}4}+2{\mathrm{\sigma }}_{\mathrm{\nu }}{\mathrm{\delta }}_{\mathrm{\mu }\phantom{4}}^{\phantom{\mathrm{\mu }}4}-2g_{\mathrm{\mu },\mathrm{\nu }}$

$\mathrm{TensorArray}\left(\left[\right]\right)$

$\left[\left[\begin{array}{cccc}{\left[{\mathrm{\sigma }}_1,{\mathrm{\sigma }}_1\right]}_{-}=0& {\left[{\mathrm{\sigma }}_1,{\mathrm{\sigma }}_2\right]}_{-}=2\mathrm{I}{\mathrm{\sigma }}_3& {\left[{\mathrm{\sigma }}_1,{\mathrm{\sigma }}_3\right]}_{-}=-2\mathrm{I}{\mathrm{\sigma }}_2& {\left[{\mathrm{\sigma }}_1,{\mathrm{\sigma }}_4\right]}_{-}=0\\ {\left[{\mathrm{\sigma }}_2,{\mathrm{\sigma }}_1\right]}_{-}=-2\mathrm{I}{\mathrm{\sigma }}_3& {\left[{\mathrm{\sigma }}_2,{\mathrm{\sigma }}_2\right]}_{-}=0& {\left[{\mathrm{\sigma }}_2,{\mathrm{\sigma }}_3\right]}_{-}=2\mathrm{I}{\mathrm{\sigma }}_1& {\left[{\mathrm{\sigma }}_2,{\mathrm{\sigma }}_4\right]}_{-}=0\\ {\left[{\mathrm{\sigma }}_3,{\mathrm{\sigma }}_1\right]}_{-}=2\mathrm{I}{\mathrm{\sigma }}_2& {\left[{\mathrm{\sigma }}_3,{\mathrm{\sigma }}_2\right]}_{-}=-2\mathrm{I}{\mathrm{\sigma }}_1& {\left[{\mathrm{\sigma }}_3,{\mathrm{\sigma }}_3\right]}_{-}=0& {\left[{\mathrm{\sigma }}_3,{\mathrm{\sigma }}_4\right]}_{-}=0\\ {\left[{\mathrm{\sigma }}_4,{\mathrm{\sigma }}_1\right]}_{-}=0& {\left[{\mathrm{\sigma }}_4,{\mathrm{\sigma }}_2\right]}_{-}=0& {\left[{\mathrm{\sigma }}_4,{\mathrm{\sigma }}_3\right]}_{-}=0& {\left[{\mathrm{\sigma }}_4,{\mathrm{\sigma }}_4\right]}_{-}=0\end{array}\right]\,\left[\begin{array}{cccc}{\left[{\mathrm{\sigma }}_1,{\mathrm{\sigma }}_1\right]}_{+}=2& {\left[{\mathrm{\sigma }}_1,{\mathrm{\sigma }}_2\right]}_{+}=0& {\left[{\mathrm{\sigma }}_1,{\mathrm{\sigma }}_3\right]}_{+}=0& {\left[{\mathrm{\sigma }}_1,{\mathrm{\sigma }}_4\right]}_{+}=2{\mathrm{\sigma }}_1\\ {\left[{\mathrm{\sigma }}_2,{\mathrm{\sigma }}_1\right]}_{+}=0& {\left[{\mathrm{\sigma }}_2,{\mathrm{\sigma }}_2\right]}_{+}=2& {\left[{\mathrm{\sigma }}_2,{\mathrm{\sigma }}_3\right]}_{+}=0& {\left[{\mathrm{\sigma }}_2,{\mathrm{\sigma }}_4\right]}_{+}=2{\mathrm{\sigma }}_2\\ {\left[{\mathrm{\sigma }}_3,{\mathrm{\sigma }}_1\right]}_{+}=0& {\left[{\mathrm{\sigma }}_3,{\mathrm{\sigma }}_2\right]}_{+}=0& {\left[{\mathrm{\sigma }}_3,{\mathrm{\sigma }}_3\right]}_{+}=2& {\left[{\mathrm{\sigma }}_3,{\mathrm{\sigma }}_4\right]}_{+}=2{\mathrm{\sigma }}_3\\ {\left[{\mathrm{\sigma }}_4,{\mathrm{\sigma }}_1\right]}_{+}=2{\mathrm{\sigma }}_1& {\left[{\mathrm{\sigma }}_4,{\mathrm{\sigma }}_2\right]}_{+}=2{\mathrm{\sigma }}_2& {\left[{\mathrm{\sigma }}_4,{\mathrm{\sigma }}_3\right]}_{+}=2{\mathrm{\sigma }}_3& {\left[{\mathrm{\sigma }}_4,{\mathrm{\sigma }}_4\right]}_{+}=4{\mathrm{\sigma }}_4-2\end{array}\right]\right]$

$\mathrm{TensorArray}\left(\left[\right]\,\mathrm{simplifier}=\mathrm{value}\right)$

$\left[\left[\begin{array}{cccc}0=0& 2\mathrm{I}{\mathrm{\epsilon }}_{1,2,4\phantom{\mathrm{\beta }}}^{\phantom{1}\phantom{,2,4}\mathrm{\beta }}{\mathrm{\sigma }}_{\mathrm{\beta }}=2\mathrm{I}{\mathrm{\sigma }}_3& 2\mathrm{I}{\mathrm{\epsilon }}_{1,3,4\phantom{\mathrm{\beta }}}^{\phantom{1}\phantom{,3,4}\mathrm{\beta }}{\mathrm{\sigma }}_{\mathrm{\beta }}=-2\mathrm{I}{\mathrm{\sigma }}_2& 0=0\\ -2\mathrm{I}{\mathrm{\epsilon }}_{1,2,4\phantom{\mathrm{\beta }}}^{\phantom{1}\phantom{,2,4}\mathrm{\beta }}{\mathrm{\sigma }}_{\mathrm{\beta }}=-2\mathrm{I}{\mathrm{\sigma }}_3& 0=0& 2\mathrm{I}{\mathrm{\epsilon }}_{2,3,4\phantom{\mathrm{\beta }}}^{\phantom{2}\phantom{,3,4}\mathrm{\beta }}{\mathrm{\sigma }}_{\mathrm{\beta }}=2\mathrm{I}{\mathrm{\sigma }}_1& 0=0\\ -2\mathrm{I}{\mathrm{\epsilon }}_{1,3,4\phantom{\mathrm{\beta }}}^{\phantom{1}\phantom{,3,4}\mathrm{\beta }}{\mathrm{\sigma }}_{\mathrm{\beta }}=2\mathrm{I}{\mathrm{\sigma }}_2& -2\mathrm{I}{\mathrm{\epsilon }}_{2,3,4\phantom{\mathrm{\beta }}}^{\phantom{2}\phantom{,3,4}\mathrm{\beta }}{\mathrm{\sigma }}_{\mathrm{\beta }}=-2\mathrm{I}{\mathrm{\sigma }}_1& 0=0& 0=0\\ 0=0& 0=0& 0=0& 0=0\end{array}\right]\,\left[\begin{array}{cccc}2=2& 0=0& 0=0& 2{\mathrm{\sigma }}_1=2{\mathrm{\sigma }}_1\\ 0=0& 2=2& 0=0& 2{\mathrm{\sigma }}_2=2{\mathrm{\sigma }}_2\\ 0=0& 0=0& 2=2& 2{\mathrm{\sigma }}_3=2{\mathrm{\sigma }}_3\\ 2{\mathrm{\sigma }}_1=2{\mathrm{\sigma }}_1& 2{\mathrm{\sigma }}_2=2{\mathrm{\sigma }}_2& 2{\mathrm{\sigma }}_3=2{\mathrm{\sigma }}_3& 4{\mathrm{\sigma }}_4-2=4{\mathrm{\sigma }}_4-2\end{array}\right]\right]$

$\mathrm{Library}:-\mathrm{RewriteInMatrixForm}\left(\left[1\right]\right)$

$\left[\begin{array}{cccc}{\left[\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right],\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]\right]}_{-}=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]& {\left[\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right],\left[\begin{array}{cc}0& \mathrm{-I}\\ \mathrm{I}& 0\end{array}\right]\right]}_{-}=2\mathrm{I}\left[\begin{array}{cc}1& 0\\ 0& \mathrm{-1}\end{array}\right]& {\left[\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right],\left[\begin{array}{cc}1& 0\\ 0& \mathrm{-1}\end{array}\right]\right]}_{-}=-2\mathrm{I}\left[\begin{array}{cc}0& \mathrm{-I}\\ \mathrm{I}& 0\end{array}\right]& {\left[\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right],\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]\right]}_{-}=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]\\ {\left[\left[\begin{array}{cc}0& \mathrm{-I}\\ \mathrm{I}& 0\end{array}\right],\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]\right]}_{-}=-2\mathrm{I}\left[\begin{array}{cc}1& 0\\ 0& \mathrm{-1}\end{array}\right]& {\left[\left[\begin{array}{cc}0& \mathrm{-I}\\ \mathrm{I}& 0\end{array}\right],\left[\begin{array}{cc}0& \mathrm{-I}\\ \mathrm{I}& 0\end{array}\right]\right]}_{-}=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]& {\left[\left[\begin{array}{cc}0& \mathrm{-I}\\ \mathrm{I}& 0\end{array}\right],\left[\begin{array}{cc}1& 0\\ 0& \mathrm{-1}\end{array}\right]\right]}_{-}=2\mathrm{I}\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]& {\left[\left[\begin{array}{cc}0& \mathrm{-I}\\ \mathrm{I}& 0\end{array}\right],\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]\right]}_{-}=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]\\ {\left[\left[\begin{array}{cc}1& 0\\ 0& \mathrm{-1}\end{array}\right],\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]\right]}_{-}=2\mathrm{I}\left[\begin{array}{cc}0& \mathrm{-I}\\ \mathrm{I}& 0\end{array}\right]& {\left[\left[\begin{array}{cc}1& 0\\ 0& \mathrm{-1}\end{array}\right],\left[\begin{array}{cc}0& \mathrm{-I}\\ \mathrm{I}& 0\end{array}\right]\right]}_{-}=-2\mathrm{I}\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]& {\left[\left[\begin{array}{cc}1& 0\\ 0& \mathrm{-1}\end{array}\right],\left[\begin{array}{cc}1& 0\\ 0& \mathrm{-1}\end{array}\right]\right]}_{-}=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]& {\left[\left[\begin{array}{cc}1& 0\\ 0& \mathrm{-1}\end{array}\right],\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]\right]}_{-}=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]\\ {\left[\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right],\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]\right]}_{-}=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]& {\left[\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right],\left[\begin{array}{cc}0& \mathrm{-I}\\ \mathrm{I}& 0\end{array}\right]\right]}_{-}=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]& {\left[\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right],\left[\begin{array}{cc}1& 0\\ 0& \mathrm{-1}\end{array}\right]\right]}_{-}=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]& {\left[\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right],\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]\right]}_{-}=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]\end{array}\right]$

$\mathrm{map}\left(\mathrm{expand}\,\mathrm{value}\left(\right)\right)$

$\left[\begin{array}{cccc}\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]& \left[\begin{array}{cc}2\mathrm{I}& 0\\ 0& -2\mathrm{I}\end{array}\right]=\left[\begin{array}{cc}2\mathrm{I}& 0\\ 0& -2\mathrm{I}\end{array}\right]& \left[\begin{array}{cc}0& \mathrm{-2}\\ 2& 0\end{array}\right]=\left[\begin{array}{cc}0& \mathrm{-2}\\ 2& 0\end{array}\right]& \left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]\\ \left[\begin{array}{cc}-2\mathrm{I}& 0\\ 0& 2\mathrm{I}\end{array}\right]=\left[\begin{array}{cc}-2\mathrm{I}& 0\\ 0& 2\mathrm{I}\end{array}\right]& \left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]& \left[\begin{array}{cc}0& 2\mathrm{I}\\ 2\mathrm{I}& 0\end{array}\right]=\left[\begin{array}{cc}0& 2\mathrm{I}\\ 2\mathrm{I}& 0\end{array}\right]& \left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]\\ \left[\begin{array}{cc}0& 2\\ \mathrm{-2}& 0\end{array}\right]=\left[\begin{array}{cc}0& 2\\ \mathrm{-2}& 0\end{array}\right]& \left[\begin{array}{cc}0& -2\mathrm{I}\\ -2\mathrm{I}& 0\end{array}\right]=\left[\begin{array}{cc}0& -2\mathrm{I}\\ -2\mathrm{I}& 0\end{array}\right]& \left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]& \left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]\\ \left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]& \left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]& \left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]& \left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]\end{array}\right]$

$\mathrm{Setup}\left(\mathrm{signature}=\mathrm{`+---`}\right)$

$\left[\mathrm{signature}=\mathrm{+\; -\; -\; -}\right]$

$\mathrm{Psigma}\left[1\right]=\mathrm{Psigma}\left[0\right]$

${\mathrm{\sigma }}_1={\mathrm{\sigma }}_1$

$\mathrm{Psigma}\left[1,\mathrm{matrix}\right]$

${\mathrm{\sigma }}_1=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$

$\mathrm{Psigma}\left[x\right]=\mathrm{Library}:-\mathrm{RewriteInMatrixForm}\left(\mathrm{Psigma}\left[x\right]\right)$

${\mathrm{\sigma }}_2=\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]$

$\mathrm{Setup}\left(\mathrm{spaceindices}=\mathrm{lowercaselatin\_is}\right)$

$\left[\mathrm{spaceindices}=\mathrm{lowercaselatin\_is}\right]$

$\mathrm{Define}\left(\mathrm{redo}\,\mathrm{Psigma}\left[j\right]\right)$

$\mathrm{Defined\; Pauli\; sigma\; matrices\; (Psigma):}{\mathrm{\sigma }}_1,{\mathrm{\sigma }}_2,{\mathrm{\sigma }}_3$

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}$

$\mathrm{Defined\; objects\; with\; tensor\; properties}$

$\left\{{\mathrm{\gamma }}_{\mathrm{\mu }}\,{\mathrm{\sigma }}_j\,{\partial }_{\mathrm{\mu }}\,g_{\mathrm{\mu },\mathrm{\nu }}\,{\mathrm{\gamma }}_{i,j}\,{\mathrm{\epsilon }}_{\mathrm{\alpha },\mathrm{\beta },\mathrm{\mu },\mathrm{\nu }}\right\}$

$\mathrm{Psigma}\left[1,\mathrm{matrix}\right]$

${\mathrm{\sigma }}_1=\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]$

$\mathrm{Library}:-\mathrm{DefaultAlgebraRules}\left(\mathrm{Psigma}\right)$

${\left[{\mathrm{\sigma }}_i,{\mathrm{\sigma }}_j\right]}_{-}=2\mathrm{I}{\mathrm{\epsilon }}_{i,j,k}{\mathrm{\sigma }}_k,{\left[{\mathrm{\sigma }}_i,{\mathrm{\sigma }}_j\right]}_{+}=2{\mathrm{\gamma }}_{i,j}$

$\mathrm{TensorArray}\left(\left[\right]\right)$

$\left[\left[\begin{array}{ccc}{\left[{\mathrm{\sigma }}_1,{\mathrm{\sigma }}_1\right]}_{-}=0& {\left[{\mathrm{\sigma }}_1,{\mathrm{\sigma }}_2\right]}_{-}=2\mathrm{I}{\mathrm{\sigma }}_3& {\left[{\mathrm{\sigma }}_1,{\mathrm{\sigma }}_3\right]}_{-}=-2\mathrm{I}{\mathrm{\sigma }}_2\\ {\left[{\mathrm{\sigma }}_2,{\mathrm{\sigma }}_1\right]}_{-}=-2\mathrm{I}{\mathrm{\sigma }}_3& {\left[{\mathrm{\sigma }}_2,{\mathrm{\sigma }}_2\right]}_{-}=0& {\left[{\mathrm{\sigma }}_2,{\mathrm{\sigma }}_3\right]}_{-}=2\mathrm{I}{\mathrm{\sigma }}_1\\ {\left[{\mathrm{\sigma }}_3,{\mathrm{\sigma }}_1\right]}_{-}=2\mathrm{I}{\mathrm{\sigma }}_2& {\left[{\mathrm{\sigma }}_3,{\mathrm{\sigma }}_2\right]}_{-}=-2\mathrm{I}{\mathrm{\sigma }}_1& {\left[{\mathrm{\sigma }}_3,{\mathrm{\sigma }}_3\right]}_{-}=0\end{array}\right]\,\left[\begin{array}{ccc}{\left[{\mathrm{\sigma }}_1,{\mathrm{\sigma }}_1\right]}_{+}=2& {\left[{\mathrm{\sigma }}_1,{\mathrm{\sigma }}_2\right]}_{+}=0& {\left[{\mathrm{\sigma }}_1,{\mathrm{\sigma }}_3\right]}_{+}=0\\ {\left[{\mathrm{\sigma }}_2,{\mathrm{\sigma }}_1\right]}_{+}=0& {\left[{\mathrm{\sigma }}_2,{\mathrm{\sigma }}_2\right]}_{+}=2& {\left[{\mathrm{\sigma }}_2,{\mathrm{\sigma }}_3\right]}_{+}=0\\ {\left[{\mathrm{\sigma }}_3,{\mathrm{\sigma }}_1\right]}_{+}=0& {\left[{\mathrm{\sigma }}_3,{\mathrm{\sigma }}_2\right]}_{+}=0& {\left[{\mathrm{\sigma }}_3,{\mathrm{\sigma }}_3\right]}_{+}=2\end{array}\right]\right]$

$\mathrm{Setup}\left(\mathrm{su2indices}=\mathrm{lowercaselatin\_ah}\right)$

$\left[\mathrm{su2indices}=\mathrm{lowercaselatin\_ah}\right]$

$\mathrm{Define}\left(\mathrm{redo}\,\mathrm{Psigma}\left[a\right]\right)$

$\mathrm{Defined\; objects\; with\; tensor\; properties}$

$\left\{{\mathrm{\gamma }}_{\mathrm{\mu }}\,{\mathrm{\sigma }}_a\,{\partial }_{\mathrm{\mu }}\,g_{\mathrm{\mu },\mathrm{\nu }}\,{\mathrm{\gamma }}_{i,j}\,{\mathrm{\epsilon }}_{\mathrm{\alpha },\mathrm{\beta },\mathrm{\mu },\mathrm{\nu }}\right\}$

$\mathrm{Psigma}\left[1,\mathrm{matrix}\right]$

${\mathrm{\sigma }}_1=\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]$

$\mathrm{Library}:-\mathrm{DefaultAlgebraRules}\left(\mathrm{Psigma}\right)$

${\left[{\mathrm{\sigma }}_a,{\mathrm{\sigma }}_b\right]}_{-}=2\mathrm{I}{\mathrm{\epsilon }}_{a,b,c}{\mathrm{\sigma }}_c,{\left[{\mathrm{\sigma }}_a,{\mathrm{\sigma }}_b\right]}_{+}=2{\mathrm{\delta }}_{a,b}$

$\mathrm{TensorArray}\left(\left[\right]\right)$

$\left[\left[\begin{array}{ccc}{\left[{\mathrm{\sigma }}_1,{\mathrm{\sigma }}_1\right]}_{-}=0& {\left[{\mathrm{\sigma }}_1,{\mathrm{\sigma }}_2\right]}_{-}=2\mathrm{I}{\mathrm{\sigma }}_3& {\left[{\mathrm{\sigma }}_1,{\mathrm{\sigma }}_3\right]}_{-}=-2\mathrm{I}{\mathrm{\sigma }}_2\\ {\left[{\mathrm{\sigma }}_2,{\mathrm{\sigma }}_1\right]}_{-}=-2\mathrm{I}{\mathrm{\sigma }}_3& {\left[{\mathrm{\sigma }}_2,{\mathrm{\sigma }}_2\right]}_{-}=0& {\left[{\mathrm{\sigma }}_2,{\mathrm{\sigma }}_3\right]}_{-}=2\mathrm{I}{\mathrm{\sigma }}_1\\ {\left[{\mathrm{\sigma }}_3,{\mathrm{\sigma }}_1\right]}_{-}=2\mathrm{I}{\mathrm{\sigma }}_2& {\left[{\mathrm{\sigma }}_3,{\mathrm{\sigma }}_2\right]}_{-}=-2\mathrm{I}{\mathrm{\sigma }}_1& {\left[{\mathrm{\sigma }}_3,{\mathrm{\sigma }}_3\right]}_{-}=0\end{array}\right]\,\left[\begin{array}{ccc}{\left[{\mathrm{\sigma }}_1,{\mathrm{\sigma }}_1\right]}_{+}=2& {\left[{\mathrm{\sigma }}_1,{\mathrm{\sigma }}_2\right]}_{+}=0& {\left[{\mathrm{\sigma }}_1,{\mathrm{\sigma }}_3\right]}_{+}=0\\ {\left[{\mathrm{\sigma }}_2,{\mathrm{\sigma }}_1\right]}_{+}=0& {\left[{\mathrm{\sigma }}_2,{\mathrm{\sigma }}_2\right]}_{+}=2& {\left[{\mathrm{\sigma }}_2,{\mathrm{\sigma }}_3\right]}_{+}=0\\ {\left[{\mathrm{\sigma }}_3,{\mathrm{\sigma }}_1\right]}_{+}=0& {\left[{\mathrm{\sigma }}_3,{\mathrm{\sigma }}_2\right]}_{+}=0& {\left[{\mathrm{\sigma }}_3,{\mathrm{\sigma }}_3\right]}_{+}=2\end{array}\right]\right]$

$\mathrm{value}\left(\right)$

$\left[\left[\begin{array}{ccc}0=0& 2\mathrm{I}{\mathrm{\sigma }}_3=2\mathrm{I}{\mathrm{\sigma }}_3& -2\mathrm{I}{\mathrm{\sigma }}_2=-2\mathrm{I}{\mathrm{\sigma }}_2\\ -2\mathrm{I}{\mathrm{\sigma }}_3=-2\mathrm{I}{\mathrm{\sigma }}_3& 0=0& 2\mathrm{I}{\mathrm{\sigma }}_1=2\mathrm{I}{\mathrm{\sigma }}_1\\ 2\mathrm{I}{\mathrm{\sigma }}_2=2\mathrm{I}{\mathrm{\sigma }}_2& -2\mathrm{I}{\mathrm{\sigma }}_1=-2\mathrm{I}{\mathrm{\sigma }}_1& 0=0\end{array}\right]\,\left[\begin{array}{ccc}2=2& 0=0& 0=0\\ 0=0& 2=2& 0=0\\ 0=0& 0=0& 2=2\end{array}\right]\right]$