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].

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

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

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

${\mathrm{Psigma}}_1$

${\mathrm{\sigma }}_1$

${\mathrm{Psigma}}_{1\,\mathrm{matrix}}$

${\mathrm{Psigma}}_1\=\left(\left[\begin{array}{rr}0& 1\\ 1& 0\end{array}\right]\right)$

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

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

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

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

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

$\mathrm{`.`}\left(\left[\begin{array}{rr}0& 1\\ 1& 0\end{array}\right]\,\left[\begin{array}{cc}0& -\mathrm{I}\\ \mathrm{I}& 0\end{array}\right]\right)\+\mathrm{`.`}\left(\left[\begin{array}{cc}0& -\mathrm{I}\\ \mathrm{I}& 0\end{array}\right]\,\left[\begin{array}{rr}0& 1\\ 1& 0\end{array}\right]\right)$

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

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

${\mathrm{Psigma}}_1{\mathrm{Psigma}}_1$

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

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

$2$

${\mathrm{Psigma}}_1{\mathrm{Psigma}}_2$

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

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

$0$

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

$\mathrm{\%Commutator}\left({\mathrm{Psigma}}_{\mathrm{\μ}}\,{\mathrm{Psigma}}_{\mathrm{\ν}}\right)=2\mathrm{I}{\mathrm{\epsilon }}_{4,\mathrm{\mu },\mathrm{\nu }\phantom{\mathrm{\alpha }}}^{\phantom{4}\phantom{,\mathrm{\mu },\mathrm{\nu }}\mathrm{\alpha }}{\mathrm{\sigma }}_{\mathrm{\alpha }},\mathrm{\%AntiCommutator}\left({\mathrm{Psigma}}_{\mathrm{\μ}}\,{\mathrm{Psigma}}_{\mathrm{\ν}}\right)=2{\mathrm{\sigma }}_{\mathrm{\mu }}{\mathrm{g\_}}_{\mathrm{\ν}\,\mathrm{~4}}+2{\mathrm{\sigma }}_{\mathrm{\nu }}{\mathrm{g\_}}_{\mathrm{\μ}\,\mathrm{~4}}-2g_{\mathrm{\mu },\mathrm{\nu }}$

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

$\left[\left[\begin{array}{cccc}\mathrm{\%Commutator}\left({\mathrm{Psigma}}_1\,{\mathrm{Psigma}}_1\right)\=0& \mathrm{\%Commutator}\left({\mathrm{Psigma}}_1\,{\mathrm{Psigma}}_2\right)\=2\mathrm{I}{\mathrm{Psigma}}_3& \mathrm{\%Commutator}\left({\mathrm{Psigma}}_1\,{\mathrm{Psigma}}_3\right)\=-2\mathrm{I}{\mathrm{Psigma}}_2& \mathrm{\%Commutator}\left({\mathrm{Psigma}}_1\,{\mathrm{Psigma}}_4\right)\=0\\ \mathrm{\%Commutator}\left({\mathrm{Psigma}}_2\,{\mathrm{Psigma}}_1\right)\=-2\mathrm{I}{\mathrm{Psigma}}_3& \mathrm{\%Commutator}\left({\mathrm{Psigma}}_2\,{\mathrm{Psigma}}_2\right)\=0& \mathrm{\%Commutator}\left({\mathrm{Psigma}}_2\,{\mathrm{Psigma}}_3\right)\=2\mathrm{I}{\mathrm{Psigma}}_1& \mathrm{\%Commutator}\left({\mathrm{Psigma}}_2\,{\mathrm{Psigma}}_4\right)\=0\\ \mathrm{\%Commutator}\left({\mathrm{Psigma}}_3\,{\mathrm{Psigma}}_1\right)\=2\mathrm{I}{\mathrm{Psigma}}_2& \mathrm{\%Commutator}\left({\mathrm{Psigma}}_3\,{\mathrm{Psigma}}_2\right)\=-2\mathrm{I}{\mathrm{Psigma}}_1& \mathrm{\%Commutator}\left({\mathrm{Psigma}}_3\,{\mathrm{Psigma}}_3\right)\=0& \mathrm{\%Commutator}\left({\mathrm{Psigma}}_3\,{\mathrm{Psigma}}_4\right)\=0\\ \mathrm{\%Commutator}\left({\mathrm{Psigma}}_4\,{\mathrm{Psigma}}_1\right)\=0& \mathrm{\%Commutator}\left({\mathrm{Psigma}}_4\,{\mathrm{Psigma}}_2\right)\=0& \mathrm{\%Commutator}\left({\mathrm{Psigma}}_4\,{\mathrm{Psigma}}_3\right)\=0& \mathrm{\%Commutator}\left({\mathrm{Psigma}}_4\,{\mathrm{Psigma}}_4\right)\=0\end{array}\right]\,\left[\begin{array}{cccc}\mathrm{\%AntiCommutator}\left({\mathrm{Psigma}}_1\,{\mathrm{Psigma}}_1\right)\=2& \mathrm{\%AntiCommutator}\left({\mathrm{Psigma}}_1\,{\mathrm{Psigma}}_2\right)\=0& \mathrm{\%AntiCommutator}\left({\mathrm{Psigma}}_1\,{\mathrm{Psigma}}_3\right)\=0& \mathrm{\%AntiCommutator}\left({\mathrm{Psigma}}_1\,{\mathrm{Psigma}}_4\right)\=2{\mathrm{Psigma}}_1\\ \mathrm{\%AntiCommutator}\left({\mathrm{Psigma}}_2\,{\mathrm{Psigma}}_1\right)\=0& \mathrm{\%AntiCommutator}\left({\mathrm{Psigma}}_2\,{\mathrm{Psigma}}_2\right)\=2& \mathrm{\%AntiCommutator}\left({\mathrm{Psigma}}_2\,{\mathrm{Psigma}}_3\right)\=0& \mathrm{\%AntiCommutator}\left({\mathrm{Psigma}}_2\,{\mathrm{Psigma}}_4\right)\=2{\mathrm{Psigma}}_2\\ \mathrm{\%AntiCommutator}\left({\mathrm{Psigma}}_3\,{\mathrm{Psigma}}_1\right)\=0& \mathrm{\%AntiCommutator}\left({\mathrm{Psigma}}_3\,{\mathrm{Psigma}}_2\right)\=0& \mathrm{\%AntiCommutator}\left({\mathrm{Psigma}}_3\,{\mathrm{Psigma}}_3\right)\=2& \mathrm{\%AntiCommutator}\left({\mathrm{Psigma}}_3\,{\mathrm{Psigma}}_4\right)\=2{\mathrm{Psigma}}_3\\ \mathrm{\%AntiCommutator}\left({\mathrm{Psigma}}_4\,{\mathrm{Psigma}}_1\right)\=2{\mathrm{Psigma}}_1& \mathrm{\%AntiCommutator}\left({\mathrm{Psigma}}_4\,{\mathrm{Psigma}}_2\right)\=2{\mathrm{Psigma}}_2& \mathrm{\%AntiCommutator}\left({\mathrm{Psigma}}_4\,{\mathrm{Psigma}}_3\right)\=2{\mathrm{Psigma}}_3& \mathrm{\%AntiCommutator}\left({\mathrm{Psigma}}_4\,{\mathrm{Psigma}}_4\right)\=4{\mathrm{Psigma}}_4-2\end{array}\right]\right]$

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

$\left[\left[\begin{array}{cccc}0\=0& 2\mathrm{I}{\mathrm{LeviCivita}}_{1\,2\,4\,\mathrm{~beta}}{\mathrm{Psigma}}_{\mathrm{\β}}\=2\mathrm{I}{\mathrm{Psigma}}_3& 2\mathrm{I}{\mathrm{LeviCivita}}_{1\,3\,4\,\mathrm{~beta}}{\mathrm{Psigma}}_{\mathrm{\β}}\=-2\mathrm{I}{\mathrm{Psigma}}_2& 0\=0\\ -2\mathrm{I}{\mathrm{LeviCivita}}_{1\,2\,4\,\mathrm{~beta}}{\mathrm{Psigma}}_{\mathrm{\β}}\=-2\mathrm{I}{\mathrm{Psigma}}_3& 0\=0& 2\mathrm{I}{\mathrm{LeviCivita}}_{2\,3\,4\,\mathrm{~beta}}{\mathrm{Psigma}}_{\mathrm{\β}}\=2\mathrm{I}{\mathrm{Psigma}}_1& 0\=0\\ -2\mathrm{I}{\mathrm{LeviCivita}}_{1\,3\,4\,\mathrm{~beta}}{\mathrm{Psigma}}_{\mathrm{\β}}\=2\mathrm{I}{\mathrm{Psigma}}_2& -2\mathrm{I}{\mathrm{LeviCivita}}_{2\,3\,4\,\mathrm{~beta}}{\mathrm{Psigma}}_{\mathrm{\β}}\=-2\mathrm{I}{\mathrm{Psigma}}_1& 0\=0& 0\=0\\ 0\=0& 0\=0& 0\=0& 0\=0\end{array}\right]\,\left[\begin{array}{cccc}2\=2& 0\=0& 0\=0& 2{\mathrm{Psigma}}_1\=2{\mathrm{Psigma}}_1\\ 0\=0& 2\=2& 0\=0& 2{\mathrm{Psigma}}_2\=2{\mathrm{Psigma}}_2\\ 0\=0& 0\=0& 2\=2& 2{\mathrm{Psigma}}_3\=2{\mathrm{Psigma}}_3\\ 2{\mathrm{Psigma}}_1\=2{\mathrm{Psigma}}_1& 2{\mathrm{Psigma}}_2\=2{\mathrm{Psigma}}_2& 2{\mathrm{Psigma}}_3\=2{\mathrm{Psigma}}_3& 4{\mathrm{Psigma}}_4-2\=4{\mathrm{Psigma}}_4-2\end{array}\right]\right]$

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

$\left[\begin{array}{cccc}\mathrm{\%Commutator}\left(\left[\begin{array}{rr}0& 1\\ 1& 0\end{array}\right]\,\left[\begin{array}{rr}0& 1\\ 1& 0\end{array}\right]\right)\=\left(\left[\begin{array}{rr}0& 0\\ 0& 0\end{array}\right]\right)& \mathrm{\%Commutator}\left(\left[\begin{array}{rr}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(\left[\begin{array}{rr}1& 0\\ 0& -1\end{array}\right]\right)& \mathrm{\%Commutator}\left(\left[\begin{array}{rr}0& 1\\ 1& 0\end{array}\right]\,\left[\begin{array}{rr}1& 0\\ 0& -1\end{array}\right]\right)\=-2\mathrm{I}\left(\left[\begin{array}{cc}0& -\mathrm{I}\\ \mathrm{I}& 0\end{array}\right]\right)& \mathrm{\%Commutator}\left(\left[\begin{array}{rr}0& 1\\ 1& 0\end{array}\right]\,\left[\begin{array}{rr}1& 0\\ 0& 1\end{array}\right]\right)\=\left(\left[\begin{array}{rr}0& 0\\ 0& 0\end{array}\right]\right)\\ \mathrm{\%Commutator}\left(\left[\begin{array}{cc}0& -\mathrm{I}\\ \mathrm{I}& 0\end{array}\right]\,\left[\begin{array}{rr}0& 1\\ 1& 0\end{array}\right]\right)\=-2\mathrm{I}\left(\left[\begin{array}{rr}1& 0\\ 0& -1\end{array}\right]\right)& \mathrm{\%Commutator}\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(\left[\begin{array}{rr}0& 0\\ 0& 0\end{array}\right]\right)& \mathrm{\%Commutator}\left(\left[\begin{array}{cc}0& -\mathrm{I}\\ \mathrm{I}& 0\end{array}\right]\,\left[\begin{array}{rr}1& 0\\ 0& -1\end{array}\right]\right)\=2\mathrm{I}\left(\left[\begin{array}{rr}0& 1\\ 1& 0\end{array}\right]\right)& \mathrm{\%Commutator}\left(\left[\begin{array}{cc}0& -\mathrm{I}\\ \mathrm{I}& 0\end{array}\right]\,\left[\begin{array}{rr}1& 0\\ 0& 1\end{array}\right]\right)\=\left(\left[\begin{array}{rr}0& 0\\ 0& 0\end{array}\right]\right)\\ \mathrm{\%Commutator}\left(\left[\begin{array}{rr}1& 0\\ 0& -1\end{array}\right]\,\left[\begin{array}{rr}0& 1\\ 1& 0\end{array}\right]\right)\=2\mathrm{I}\left(\left[\begin{array}{cc}0& -\mathrm{I}\\ \mathrm{I}& 0\end{array}\right]\right)& \mathrm{\%Commutator}\left(\left[\begin{array}{rr}1& 0\\ 0& -1\end{array}\right]\,\left[\begin{array}{cc}0& -\mathrm{I}\\ \mathrm{I}& 0\end{array}\right]\right)\=-2\mathrm{I}\left(\left[\begin{array}{rr}0& 1\\ 1& 0\end{array}\right]\right)& \mathrm{\%Commutator}\left(\left[\begin{array}{rr}1& 0\\ 0& -1\end{array}\right]\,\left[\begin{array}{rr}1& 0\\ 0& -1\end{array}\right]\right)\=\left(\left[\begin{array}{rr}0& 0\\ 0& 0\end{array}\right]\right)& \mathrm{\%Commutator}\left(\left[\begin{array}{rr}1& 0\\ 0& -1\end{array}\right]\,\left[\begin{array}{rr}1& 0\\ 0& 1\end{array}\right]\right)\=\left(\left[\begin{array}{rr}0& 0\\ 0& 0\end{array}\right]\right)\\ \mathrm{\%Commutator}\left(\left[\begin{array}{rr}1& 0\\ 0& 1\end{array}\right]\,\left[\begin{array}{rr}0& 1\\ 1& 0\end{array}\right]\right)\=\left(\left[\begin{array}{rr}0& 0\\ 0& 0\end{array}\right]\right)& \mathrm{\%Commutator}\left(\left[\begin{array}{rr}1& 0\\ 0& 1\end{array}\right]\,\left[\begin{array}{cc}0& -\mathrm{I}\\ \mathrm{I}& 0\end{array}\right]\right)\=\left(\left[\begin{array}{rr}0& 0\\ 0& 0\end{array}\right]\right)& \mathrm{\%Commutator}\left(\left[\begin{array}{rr}1& 0\\ 0& 1\end{array}\right]\,\left[\begin{array}{rr}1& 0\\ 0& -1\end{array}\right]\right)\=\left(\left[\begin{array}{rr}0& 0\\ 0& 0\end{array}\right]\right)& \mathrm{\%Commutator}\left(\left[\begin{array}{rr}1& 0\\ 0& 1\end{array}\right]\,\left[\begin{array}{rr}1& 0\\ 0& 1\end{array}\right]\right)\=\left(\left[\begin{array}{rr}0& 0\\ 0& 0\end{array}\right]\right)\end{array}\right]$

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

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

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

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

${\mathrm{Psigma}}_1\={\mathrm{Psigma}}_0$

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

${\mathrm{Psigma}}_{1\,\mathrm{matrix}}$

${\mathrm{Psigma}}_1\=\left(\left[\begin{array}{rr}1& 0\\ 0& 1\end{array}\right]\right)$

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

${\mathrm{Psigma}}_2\=\left(\left[\begin{array}{rr}0& 1\\ 1& 0\end{array}\right]\right)$

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

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

$\mathrm{Define}\left(\mathrm{redo}\,{\mathrm{Psigma}}_j\right)$

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

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

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

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

${\mathrm{Psigma}}_{1\,\mathrm{matrix}}$

${\mathrm{Psigma}}_1\=\left(\left[\begin{array}{rr}0& 1\\ 1& 0\end{array}\right]\right)$

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

$\mathrm{\%Commutator}\left({\mathrm{Psigma}}_i\,{\mathrm{Psigma}}_j\right)=2\mathrm{I}{\mathrm{\epsilon }}_{i,j,k}{\mathrm{\sigma }}_k,\mathrm{\%AntiCommutator}\left({\mathrm{Psigma}}_i\,{\mathrm{Psigma}}_j\right)=2{\mathrm{\gamma }}_{i,j}$

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

$\left[\left[\begin{array}{ccc}\mathrm{\%Commutator}\left({\mathrm{Psigma}}_1\,{\mathrm{Psigma}}_1\right)\=0& \mathrm{\%Commutator}\left({\mathrm{Psigma}}_1\,{\mathrm{Psigma}}_2\right)\=2\mathrm{I}{\mathrm{Psigma}}_3& \mathrm{\%Commutator}\left({\mathrm{Psigma}}_1\,{\mathrm{Psigma}}_3\right)\=-2\mathrm{I}{\mathrm{Psigma}}_2\\ \mathrm{\%Commutator}\left({\mathrm{Psigma}}_2\,{\mathrm{Psigma}}_1\right)\=-2\mathrm{I}{\mathrm{Psigma}}_3& \mathrm{\%Commutator}\left({\mathrm{Psigma}}_2\,{\mathrm{Psigma}}_2\right)\=0& \mathrm{\%Commutator}\left({\mathrm{Psigma}}_2\,{\mathrm{Psigma}}_3\right)\=2\mathrm{I}{\mathrm{Psigma}}_1\\ \mathrm{\%Commutator}\left({\mathrm{Psigma}}_3\,{\mathrm{Psigma}}_1\right)\=2\mathrm{I}{\mathrm{Psigma}}_2& \mathrm{\%Commutator}\left({\mathrm{Psigma}}_3\,{\mathrm{Psigma}}_2\right)\=-2\mathrm{I}{\mathrm{Psigma}}_1& \mathrm{\%Commutator}\left({\mathrm{Psigma}}_3\,{\mathrm{Psigma}}_3\right)\=0\end{array}\right]\,\left[\begin{array}{ccc}\mathrm{\%AntiCommutator}\left({\mathrm{Psigma}}_1\,{\mathrm{Psigma}}_1\right)\=2& \mathrm{\%AntiCommutator}\left({\mathrm{Psigma}}_1\,{\mathrm{Psigma}}_2\right)\=0& \mathrm{\%AntiCommutator}\left({\mathrm{Psigma}}_1\,{\mathrm{Psigma}}_3\right)\=0\\ \mathrm{\%AntiCommutator}\left({\mathrm{Psigma}}_2\,{\mathrm{Psigma}}_1\right)\=0& \mathrm{\%AntiCommutator}\left({\mathrm{Psigma}}_2\,{\mathrm{Psigma}}_2\right)\=2& \mathrm{\%AntiCommutator}\left({\mathrm{Psigma}}_2\,{\mathrm{Psigma}}_3\right)\=0\\ \mathrm{\%AntiCommutator}\left({\mathrm{Psigma}}_3\,{\mathrm{Psigma}}_1\right)\=0& \mathrm{\%AntiCommutator}\left({\mathrm{Psigma}}_3\,{\mathrm{Psigma}}_2\right)\=0& \mathrm{\%AntiCommutator}\left({\mathrm{Psigma}}_3\,{\mathrm{Psigma}}_3\right)\=2\end{array}\right]\right]$

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

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

$\mathrm{Define}\left(\mathrm{redo}\,{\mathrm{Psigma}}_a\right)$

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

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

${\mathrm{Psigma}}_{1\,\mathrm{matrix}}$

${\mathrm{Psigma}}_1\=\left(\left[\begin{array}{rr}0& 1\\ 1& 0\end{array}\right]\right)$

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

$\mathrm{\%Commutator}\left({\mathrm{Psigma}}_a\,{\mathrm{Psigma}}_b\right)=2\mathrm{I}{\mathrm{\epsilon }}_{a,b,c}{\mathrm{\sigma }}_c,\mathrm{\%AntiCommutator}\left({\mathrm{Psigma}}_a\,{\mathrm{Psigma}}_b\right)=2{\mathrm{\delta }}_{a,b}$

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

$\left[\left[\begin{array}{ccc}\mathrm{\%Commutator}\left({\mathrm{Psigma}}_1\,{\mathrm{Psigma}}_1\right)\=0& \mathrm{\%Commutator}\left({\mathrm{Psigma}}_1\,{\mathrm{Psigma}}_2\right)\=2\mathrm{I}{\mathrm{Psigma}}_3& \mathrm{\%Commutator}\left({\mathrm{Psigma}}_1\,{\mathrm{Psigma}}_3\right)\=-2\mathrm{I}{\mathrm{Psigma}}_2\\ \mathrm{\%Commutator}\left({\mathrm{Psigma}}_2\,{\mathrm{Psigma}}_1\right)\=-2\mathrm{I}{\mathrm{Psigma}}_3& \mathrm{\%Commutator}\left({\mathrm{Psigma}}_2\,{\mathrm{Psigma}}_2\right)\=0& \mathrm{\%Commutator}\left({\mathrm{Psigma}}_2\,{\mathrm{Psigma}}_3\right)\=2\mathrm{I}{\mathrm{Psigma}}_1\\ \mathrm{\%Commutator}\left({\mathrm{Psigma}}_3\,{\mathrm{Psigma}}_1\right)\=2\mathrm{I}{\mathrm{Psigma}}_2& \mathrm{\%Commutator}\left({\mathrm{Psigma}}_3\,{\mathrm{Psigma}}_2\right)\=-2\mathrm{I}{\mathrm{Psigma}}_1& \mathrm{\%Commutator}\left({\mathrm{Psigma}}_3\,{\mathrm{Psigma}}_3\right)\=0\end{array}\right]\,\left[\begin{array}{ccc}\mathrm{\%AntiCommutator}\left({\mathrm{Psigma}}_1\,{\mathrm{Psigma}}_1\right)\=2& \mathrm{\%AntiCommutator}\left({\mathrm{Psigma}}_1\,{\mathrm{Psigma}}_2\right)\=0& \mathrm{\%AntiCommutator}\left({\mathrm{Psigma}}_1\,{\mathrm{Psigma}}_3\right)\=0\\ \mathrm{\%AntiCommutator}\left({\mathrm{Psigma}}_2\,{\mathrm{Psigma}}_1\right)\=0& \mathrm{\%AntiCommutator}\left({\mathrm{Psigma}}_2\,{\mathrm{Psigma}}_2\right)\=2& \mathrm{\%AntiCommutator}\left({\mathrm{Psigma}}_2\,{\mathrm{Psigma}}_3\right)\=0\\ \mathrm{\%AntiCommutator}\left({\mathrm{Psigma}}_3\,{\mathrm{Psigma}}_1\right)\=0& \mathrm{\%AntiCommutator}\left({\mathrm{Psigma}}_3\,{\mathrm{Psigma}}_2\right)\=0& \mathrm{\%AntiCommutator}\left({\mathrm{Psigma}}_3\,{\mathrm{Psigma}}_3\right)\=2\end{array}\right]\right]$

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

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

The Physics[Psigma] command was updated in Maple 2020.