The Dgamma[mu] command is used to represent the Dirac ${\mathrm{\gamma }}_{\mathrm{\mu }}$ matrices, where $\mathrm{\mu }$ ranges from 1 to the dimension d of spacetime; these are noncommutative objects satisfying

where the products in the above are noncommutative, constructed by using the `*` operator of the Physics package, and $g_{\mathrm{\mu },\mathrm{\nu }}$ is the metric tensor. The properties of the Dirac matrices are derived from this defining relation (anticommutator algebra) above. The Simplify command simplifies products of Dirac matrices and the Trace command computes traces of these products, taking this defining relation into account.

This defining anticommutator algebra satisfied by the Dirac matrices is invariant under a unitary transformation. Thus these matrices are determined up to a transformation of that kind, and conventions are necessary to construct their representations. The most common representations are the standard (also known as Dirac), the chiral (also known as Weyl or spinor), and the Majorana representations.

When the Physics package is loaded, the default spacetime is of Minkowski type with signature (- - - +). With that signature, the standard representation for Dirac's matrices, uniform in the literature, is:

where $𝕀$ is the 2 x 2 identity matrix, k runs from 1 to 3, ${\mathrm{\sigma }}_{}^k=-\mathrm{\σ\_\_k}$ (due to the signature (---+)), and $\mathrm{\σ\_\_k}$ are the three Pauli matrices. (All of ${\mathrm{\sigma }}_{}^0=\mathrm{\σ\_\_0}\equiv 𝕀$ and $\mathrm{\σ\_\_k}$ are represented by Psigma, and together form the four vector Psigma[mu], displayed as $\mathrm{\σ\_\_\μ}$.) As is the case for all spacetime tensors, when the position of the timelike component is the last one (the case when the signature is (---+)), the value d of a spacetime index in d dimensions is also represented by the number 0.

The conventions for the chiral and Majorana representations are not uniform in the literature. The conventions adopted here are the same ones shown in Wikipedia and in ref.[1-3], so that in the chiral representation, the ${\mathrm{\gamma }}_{}^k$ are the same as in the standard representation, while ${\mathrm{\gamma }}_{}^0$ changes to

The convention implemented for the Majorana representation, that is, a representation where all the nonzero components of the Dirac matrices are imaginary, is

where $\ⅈ$ is the imaginary unit (to represent it with a lowercase letter as in the above, see interface imaginaryunit).

The form of the contravariant Dirac matrices shown above for the three representations does not change with the signature, which could be (- - - +) (default when you load Physics), (+ - - -), (+ + + -) or (- + + +). Correspondingly, for the signatures (+ + + -) or (- + + +) there is an extra minus sign in the right-hand side of the defining algebra rule, that becomes ${\mathrm{\gamma }}_{}^{\mu }{\mathrm{\gamma }}_{}^{\nu }+{\mathrm{\gamma }}_{}^{\nu }{\mathrm{\gamma }}_{}^{\mu }=-2g_{}^{\mu ,\nu }$ To query about the signature, enter Setup(signature).  To change the value of the signature see Setup.

In all of these three representations, the timelike component of the Dirac matrices is Hermitian: ${\mathrm{\gamma }}_{}^0=\left({{\mathrm{\gamma }}_{}^0}^{†}\right)$, and the spacelike components are anti-Hermitian: ${\mathrm{\gamma }}_{}^k=-\left({{\mathrm{\gamma }}_{}^k}^{†}\right)$. In a four dimensional Minkowski spacetime, an Hermitian matrix ${\mathrm{\gamma }}_{}^5={{\mathrm{\gamma }}_{}^5}^{†}$ satisfying

is given by:

Note the minus sign on the right-hand side of this definition, according to [1], [2] and [3], but not Wikipedia and not uniform in the literature. This definition of ${\mathrm{\gamma }}_{}^5$ can also be written as

from where there is no distinction between a covariant or contravariant character for its index, ${\mathrm{\gamma }}_5={\mathrm{\gamma }}_{}^5$.

The form of the Dirac matrices implemented in the case of an Euclidean spacetime, for the standard, chiral, and Majorana representations, is obtained from the formulas above for the contravariant Dirac matrices by performing a Wick rotation, equivalent to multiplying the ${\mathrm{\gamma }}_{}^k$ by $-\mathrm{i}$, while ${\mathrm{\gamma }}_{}^0$ remains unchanged, and ${\mathrm{\gamma }}_{}^5$ is given by

These Euclidean Dirac matrices are all Hermitian, ${\mathrm{\gamma }}_{}^{\mathrm{\mu }}={{\mathrm{\gamma }}_{}^{\mathrm{\mu }}}^{†}$, including ${\mathrm{\gamma }}_{}^5$, and they all satisfy the same defining equations and anticommutation algebra rules stated in the previous paragraphs for a Minkowski spacetime.

The following are some representation-free frequently used identities for the Dirac matrices, valid provided the dimension, $d$, is greater than 1, expressed by using the sum rule for repeated indices:

Some representation-free identities for the traces of products of Dirac matrices in four dimensions are:

and the Trace of any product of an odd number of ${\mathrm{\gamma }}_{}^{\mathrm{\mu }}$ is zero. Note the sign +, not -, in the right-hand side of the last formula, related to the convention used for ${\mathrm{\gamma }}_{}^5$. To compute using these formulas, you can use the Physics commands Trace, g_ for the metric and LeviCivita for the totally antisymmetric symbol $\mathrm{\epsilon }$.

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

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

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

$\mathrm{interface}\left(\mathrm{imaginaryunit}=i\right)$

$I$

$\mathrm{Dgamma}\left[\right]$

${\mathrm{\gamma }}_{\mathrm{\mu }}=\left[\begin{array}{cccc}{\mathrm{\gamma }}_1& {\mathrm{\gamma }}_2& {\mathrm{\gamma }}_3& {\mathrm{\gamma }}_4\end{array}\right]$

$\mathrm{Dgamma}\left[\mathrm{`~`}\right]$

${\mathrm{\gamma }}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}=\left[\begin{array}{cccc}{\mathrm{\gamma }}_{\phantom{}\phantom{1}}^{\phantom{}1}& {\mathrm{\gamma }}_{\phantom{}\phantom{2}}^{\phantom{}2}& {\mathrm{\gamma }}_{\phantom{}\phantom{3}}^{\phantom{}3}& {\mathrm{\gamma }}_{\phantom{}\phantom{4}}^{\phantom{}4}\end{array}\right]$

$\mathrm{Library}:-\mathrm{RewriteInMatrixForm}\left(\mathrm{Dgamma}\left[\mathrm{`~`}\right]\right)$

${\mathrm{\gamma }}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}=\left[\begin{array}{cccc}\left[\begin{array}{cccc}0& 0& 0& 1\\ 0& 0& 1& 0\\ 0& \mathrm{-1}& 0& 0\\ \mathrm{-1}& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc}0& 0& 0& \mathrm{-i}\\ 0& 0& \mathrm{i}& 0\\ 0& \mathrm{i}& 0& 0\\ \mathrm{-i}& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc}0& 0& 1& 0\\ 0& 0& 0& \mathrm{-1}\\ \mathrm{-1}& 0& 0& 0\\ 0& 1& 0& 0\end{array}\right]& \left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& \mathrm{-1}& 0\\ 0& 0& 0& \mathrm{-1}\end{array}\right]\end{array}\right]$

$\mathrm{Dgamma}\left[\mathrm{definition}\right]$

${\left[{\mathrm{\gamma }}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}\,{\mathrm{\gamma }}_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}\right]}_{+}=2g_{\phantom{}\phantom{\mathrm{\mu },\mathrm{\nu }}}^{\phantom{}\mathrm{\mu },\mathrm{\nu }}$

$\mathrm{Setup}\left(\mathrm{signature}\right)$

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

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

$4$

$\mathrm{Dgamma}\left[0\right]$

${\mathrm{\gamma }}_4$

$\mathrm{Dgamma}\left[0,\mathrm{matrix}\right]$

${\mathrm{\gamma }}_4=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& \mathrm{-1}& 0\\ 0& 0& 0& \mathrm{-1}\end{array}\right]$

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

${\mathrm{\gamma }}_{\phantom{}\phantom{1}}^{\phantom{}1}=\left[\begin{array}{cccc}0& 0& 0& 1\\ 0& 0& 1& 0\\ 0& \mathrm{-1}& 0& 0\\ \mathrm{-1}& 0& 0& 0\end{array}\right]$

$\mathrm{Dgamma}\left[5,\mathrm{matrix}\right]$

${\mathrm{\gamma }}_5=\left[\begin{array}{cccc}0& 0& \mathrm{-1}& 0\\ 0& 0& 0& \mathrm{-1}\\ \mathrm{-1}& 0& 0& 0\\ 0& \mathrm{-1}& 0& 0\end{array}\right]$

$\mathrm{Dgamma}\left[5,\mathrm{definition}\right]$

${\mathrm{\gamma }}_5={\mathrm{\gamma }}_{\phantom{}\phantom{5}}^{\phantom{}5},{\mathrm{\gamma }}_{\phantom{}\phantom{5}}^{\phantom{}5}=\mathrm{-i}{\mathrm{\gamma }}_{\phantom{}\phantom{0}}^{\phantom{}0}{\mathrm{\gamma }}_{\phantom{}\phantom{1}}^{\phantom{}1}{\mathrm{\gamma }}_{\phantom{}\phantom{2}}^{\phantom{}2}{\mathrm{\gamma }}_{\phantom{}\phantom{3}}^{\phantom{}3},{\mathrm{\gamma }}_{\phantom{}\phantom{5}}^{\phantom{}5}{\mathrm{\gamma }}_{\phantom{}\phantom{5}}^{\phantom{}5}=1,{\left[{\mathrm{\gamma }}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}\,{\mathrm{\gamma }}_{\phantom{}\phantom{5}}^{\phantom{}5}\right]}_{+}=0,{\left[{\mathrm{\gamma }}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}\,{\mathrm{\gamma }}_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}\right]}_{+}=2g_{\phantom{}\phantom{\mathrm{\mu },\mathrm{\nu }}}^{\phantom{}\mathrm{\mu },\mathrm{\nu }}$

$\mathrm{value}\left(\mathrm{TensorArray}\left(\left[\right]\left[1..3\right]\,\mathrm{performmatrixoperations}\right)\right)$

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

$\mathrm{value}\left(\mathrm{TensorArray}\left(\left[\right]\left[4\right]\,\mathrm{performmatrixoperations}\right)\right)$

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

$\mathrm{Dgamma}\left[1\right]\mathrm{Dgamma}\left[2\right]+\mathrm{Dgamma}\left[0\right]$

${\mathrm{\gamma }}_1{\mathrm{\gamma }}_2+{\mathrm{\gamma }}_4$

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

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

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

$\left[\begin{array}{cccc}1-\mathrm{i}& 0& 0& 0\\ 0& 1+\mathrm{i}& 0& 0\\ 0& 0& \mathrm{-1}-\mathrm{i}& 0\\ 0& 0& 0& \mathrm{-1}+\mathrm{i}\end{array}\right]$

$\mathrm{Dgamma}\left[\mathrm{\mu }\right]\mathrm{Dgamma}\left[\mathrm{\nu }\right]$

${\mathrm{\gamma }}_{\mathrm{\mu }}{\mathrm{\gamma }}_{\mathrm{\nu }}$

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

$4g_{\mathrm{\mu },\mathrm{\nu }}$

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

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

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

$0$

$\mathrm{e0}≔{\mathrm{Dgamma}\left[\mathrm{\mu }\right]}^2$

$\mathrm{e0}≔{\mathrm{\gamma }}_{\mathrm{\mu }}{\mathrm{\gamma }}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}$

$\mathrm{Simplify}\left(\mathrm{e0}\right)$

$4$

$\mathrm{e1}≔\mathrm{Dgamma}\left[\mathrm{\mu }\right]\mathrm{Dgamma}\left[\mathrm{~nu}\right]\mathrm{Dgamma}\left[\mathrm{\mu }\right]$

$\mathrm{e1}≔{\mathrm{\gamma }}_{\mathrm{\mu }}{\mathrm{\gamma }}_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}{\mathrm{\gamma }}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}$

$\mathrm{Simplify}\left(\mathrm{e1}\right)$

$-2{\mathrm{\gamma }}_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}$

$\mathrm{e2}≔\mathrm{Dgamma}\left[\mathrm{\mu }\right]\mathrm{Dgamma}\left[\mathrm{~lambda}\right]\mathrm{Dgamma}\left[\mathrm{~nu}\right]\mathrm{Dgamma}\left[\mathrm{\mu }\right]$

$\mathrm{e2}≔{\mathrm{\gamma }}_{\mathrm{\mu }}{\mathrm{\gamma }}_{\phantom{}\phantom{\mathrm{\lambda }}}^{\phantom{}\mathrm{\lambda }}{\mathrm{\gamma }}_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}{\mathrm{\gamma }}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}$

$\mathrm{Simplify}\left(\mathrm{e2}\right)$

$4g_{\phantom{}\phantom{\mathrm{\lambda },\mathrm{\nu }}}^{\phantom{}\mathrm{\lambda },\mathrm{\nu }}$

$\mathrm{e3}≔\mathrm{Dgamma}\left[\mathrm{\mu }\right]\mathrm{Dgamma}\left[\mathrm{~lambda}\right]\mathrm{Dgamma}\left[\mathrm{~nu}\right]\mathrm{Dgamma}\left[\mathrm{~rho}\right]\mathrm{Dgamma}\left[\mathrm{\mu }\right]$

$\mathrm{e3}≔{\mathrm{\gamma }}_{\mathrm{\mu }}{\mathrm{\gamma }}_{\phantom{}\phantom{\mathrm{\lambda }}}^{\phantom{}\mathrm{\lambda }}{\mathrm{\gamma }}_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}{\mathrm{\gamma }}_{\phantom{}\phantom{\mathrm{\rho }}}^{\phantom{}\mathrm{\rho }}{\mathrm{\gamma }}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}$

$\mathrm{Simplify}\left(\mathrm{e3}\right)$

$-2{\mathrm{\gamma }}_{\phantom{}\phantom{\mathrm{\rho }}}^{\phantom{}\mathrm{\rho }}{\mathrm{\gamma }}_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}{\mathrm{\gamma }}_{\phantom{}\phantom{\mathrm{\lambda }}}^{\phantom{}\mathrm{\lambda }}$

$\mathrm{e4}≔\mathrm{Dgamma}\left[\mathrm{\mu }\right]\mathrm{Dgamma}\left[\mathrm{~lambda}\right]\mathrm{Dgamma}\left[\mathrm{~nu}\right]\mathrm{Dgamma}\left[\mathrm{~rho}\right]\mathrm{Dgamma}\left[\mathrm{~sigma}\right]\mathrm{Dgamma}\left[\mathrm{\mu }\right]$

$\mathrm{e4}≔{\mathrm{\gamma }}_{\mathrm{\mu }}{\mathrm{\gamma }}_{\phantom{}\phantom{\mathrm{\lambda }}}^{\phantom{}\mathrm{\lambda }}{\mathrm{\gamma }}_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}{\mathrm{\gamma }}_{\phantom{}\phantom{\mathrm{\rho }}}^{\phantom{}\mathrm{\rho }}{\mathrm{\gamma }}_{\phantom{}\phantom{\mathrm{\sigma }}}^{\phantom{}\mathrm{\sigma }}{\mathrm{\gamma }}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}$

$\mathrm{Simplify}\left(\mathrm{e4}\right)$

$2{\mathrm{\gamma }}_{\phantom{}\phantom{\mathrm{\sigma }}}^{\phantom{}\mathrm{\sigma }}{\mathrm{\gamma }}_{\phantom{}\phantom{\mathrm{\lambda }}}^{\phantom{}\mathrm{\lambda }}{\mathrm{\gamma }}_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}{\mathrm{\gamma }}_{\phantom{}\phantom{\mathrm{\rho }}}^{\phantom{}\mathrm{\rho }}+2{\mathrm{\gamma }}_{\phantom{}\phantom{\mathrm{\rho }}}^{\phantom{}\mathrm{\rho }}{\mathrm{\gamma }}_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}{\mathrm{\gamma }}_{\phantom{}\phantom{\mathrm{\lambda }}}^{\phantom{}\mathrm{\lambda }}{\mathrm{\gamma }}_{\phantom{}\phantom{\mathrm{\sigma }}}^{\phantom{}\mathrm{\sigma }}$

$\mathrm{e1}=$

${\mathrm{\gamma }}_{\mathrm{\mu }}{\mathrm{\gamma }}_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}{\mathrm{\gamma }}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}=-2{\mathrm{\gamma }}_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}$

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

${\mathrm{\gamma }}_1{\mathrm{\gamma }}_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}{\mathrm{\gamma }}_{\phantom{}\phantom{1}}^{\phantom{}1}+{\mathrm{\gamma }}_2{\mathrm{\gamma }}_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}{\mathrm{\gamma }}_{\phantom{}\phantom{2}}^{\phantom{}2}+{\mathrm{\gamma }}_3{\mathrm{\gamma }}_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}{\mathrm{\gamma }}_{\phantom{}\phantom{3}}^{\phantom{}3}+{\mathrm{\gamma }}_4{\mathrm{\gamma }}_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}{\mathrm{\gamma }}_{\phantom{}\phantom{4}}^{\phantom{}4}=-2{\mathrm{\gamma }}_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}$

$T≔\mathrm{TensorArray}\left(\right)$

$T≔\left[\begin{array}{cccc}{\mathrm{\gamma }}_{\phantom{}\phantom{1}}^{\phantom{}1}+{\mathrm{\gamma }}_2{\mathrm{\gamma }}_{\phantom{}\phantom{1}}^{\phantom{}1}{\mathrm{\gamma }}_{\phantom{}\phantom{2}}^{\phantom{}2}+{\mathrm{\gamma }}_3{\mathrm{\gamma }}_{\phantom{}\phantom{1}}^{\phantom{}1}{\mathrm{\gamma }}_{\phantom{}\phantom{3}}^{\phantom{}3}+{\mathrm{\gamma }}_4{\mathrm{\gamma }}_{\phantom{}\phantom{1}}^{\phantom{}1}{\mathrm{\gamma }}_{\phantom{}\phantom{4}}^{\phantom{}4}=-2{\mathrm{\gamma }}_{\phantom{}\phantom{1}}^{\phantom{}1}& {\mathrm{\gamma }}_3{\mathrm{\gamma }}_{\phantom{}\phantom{2}}^{\phantom{}2}{\mathrm{\gamma }}_{\phantom{}\phantom{3}}^{\phantom{}3}+{\mathrm{\gamma }}_4{\mathrm{\gamma }}_{\phantom{}\phantom{2}}^{\phantom{}2}{\mathrm{\gamma }}_{\phantom{}\phantom{4}}^{\phantom{}4}=-2{\mathrm{\gamma }}_{\phantom{}\phantom{2}}^{\phantom{}2}& -{\mathrm{\gamma }}_{\phantom{}\phantom{3}}^{\phantom{}3}+{\mathrm{\gamma }}_4{\mathrm{\gamma }}_{\phantom{}\phantom{3}}^{\phantom{}3}{\mathrm{\gamma }}_{\phantom{}\phantom{4}}^{\phantom{}4}=-2{\mathrm{\gamma }}_{\phantom{}\phantom{3}}^{\phantom{}3}& -2{\mathrm{\gamma }}_{\phantom{}\phantom{4}}^{\phantom{}4}=-2{\mathrm{\gamma }}_{\phantom{}\phantom{4}}^{\phantom{}4}\end{array}\right]$

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

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

$\mathrm{Setup}\left(\mathrm{dimension}=3\,\mathrm{signature}\right)$

$\mathit{Warning,\; unable\; to\; define\; the\; Pauli\; sigma\; matrices\; (Psigma)\; as\; a\; tensor\; in\; a\; spacetime\; with\; dimension\; =}\mathbf{3}\mathit{where\; the\; metric\; is\; not\; Euclidean.\; You\; can\; still\; refer\; to\; the\; Pauli\; matrices\; using}{\mathit{Psigma}}_{\mathit{x}}\mathit{,}{\mathit{Psigma}}_{\mathit{y}}\mathit{and}{\mathit{Psigma}}_{\mathit{z}}$

$\mathrm{The\; dimension\; and\; signature\; of\; the\; tensor\; space\; are\; set\; to}\left[3\,\left(\mathrm{-\; -\; +}\right)\right]$

$\left[\mathrm{dimension}=3\,\mathrm{signature}=\mathrm{-\; -\; +}\right]$

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

${\mathrm{\gamma }}_{\phantom{}\phantom{1}}^{\phantom{}1}=\left[\begin{array}{cc}0& \mathrm{i}\\ \mathrm{i}& 0\end{array}\right]$

$\mathrm{Setup}\left(\mathrm{dimension}=2\right)$

$\mathit{Warning,\; unable\; to\; define\; the\; Pauli\; sigma\; matrices\; (Psigma)\; as\; a\; tensor\; in\; a\; space\; with\; dimension\; =}\mathbf{2}\mathit{<\; 3.\; You\; can\; still\; refer\; to\; the\; Pauli\; matrices\; using}{\mathit{Psigma}}_{\mathit{x}}\mathit{,}{\mathit{Psigma}}_{\mathit{y}}\mathit{and}{\mathit{Psigma}}_{\mathit{z}}$

$\mathrm{The\; dimension\; and\; signature\; of\; the\; tensor\; space\; are\; set\; to}\left[2\&comma;\left(\mathrm{-\; +}\right)\right]$

$\left[\mathrm{dimension}=2\right]$

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

${\mathrm{\gamma }}_{\phantom{}\phantom{1}}^{\phantom{}1}=\left[\begin{array}{cc}0& 1\\ \mathrm{-1}& 0\end{array}\right]$

$\mathrm{Setup}\left(\mathrm{dimension}=4\right)$

$\mathrm{Defined\; Pauli\; sigma\; matrices\; (Psigma):}{\mathrm{\sigma }}_1,{\mathrm{\sigma }}_2,{\mathrm{\sigma }}_3,{\mathrm{\sigma }}_0$

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

$\mathrm{The\; dimension\; and\; signature\; of\; the\; tensor\; space\; are\; set\; to}\left[4\&comma;\left(\mathrm{-\; -\; -\; +}\right)\right]$

$\left[\mathrm{dimension}=4\right]$

$\mathrm{g\_}\left[\right]$

$g_{\mathrm{\mu },\mathrm{\nu }}=\left[\begin{array}{cccc}\mathrm{-1}& 0& 0& 0\\ 0& \mathrm{-1}& 0& 0\\ 0& 0& \mathrm{-1}& 0\\ 0& 0& 0& 1\end{array}\right]$

$\mathrm{Setup}\left(\mathrm{Dgamma}=\mathrm{standard}\&comma;\mathrm{quiet}\right)$

$\left[\mathrm{Dgammarepresentation}=\mathrm{standard}\right]$

$\mathrm{TensorArray}\left(\mathrm{\%Dgamma}\left[\mathrm{~mu}\right]=\mathrm{Dgamma}\left[\mathrm{~mu}\right]\&comma;\mathrm{performmatrixoperations}\right)$

$\left[\begin{array}{cccc}{\mathrm{\gamma }}_{\phantom{}\phantom{1}}^{\phantom{}1}=\left[\begin{array}{cccc}0& 0& 0& 1\\ 0& 0& 1& 0\\ 0& \mathrm{-1}& 0& 0\\ \mathrm{-1}& 0& 0& 0\end{array}\right]& {\mathrm{\gamma }}_{\phantom{}\phantom{2}}^{\phantom{}2}=\left[\begin{array}{cccc}0& 0& 0& \mathrm{-i}\\ 0& 0& \mathrm{i}& 0\\ 0& \mathrm{i}& 0& 0\\ \mathrm{-i}& 0& 0& 0\end{array}\right]& {\mathrm{\gamma }}_{\phantom{}\phantom{3}}^{\phantom{}3}=\left[\begin{array}{cccc}0& 0& 1& 0\\ 0& 0& 0& \mathrm{-1}\\ \mathrm{-1}& 0& 0& 0\\ 0& 1& 0& 0\end{array}\right]& {\mathrm{\gamma }}_{\phantom{}\phantom{4}}^{\phantom{}4}=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& \mathrm{-1}& 0\\ 0& 0& 0& \mathrm{-1}\end{array}\right]\end{array}\right]$

$\mathrm{Setup}\left(\mathrm{Dgamma}=\mathrm{chiral}\&comma;\mathrm{quiet}\right)$

$\left[\mathrm{Dgammarepresentation}=\mathrm{chiral}\right]$

$\mathrm{TensorArray}\left(\mathrm{\%Dgamma}\left[\mathrm{~mu}\right]=\mathrm{Dgamma}\left[\mathrm{~mu}\right]\&comma;\mathrm{performmatrixoperations}\right)$

$\left[\begin{array}{cccc}{\mathrm{\gamma }}_{\phantom{}\phantom{1}}^{\phantom{}1}=\left[\begin{array}{cccc}0& 0& 0& 1\\ 0& 0& 1& 0\\ 0& \mathrm{-1}& 0& 0\\ \mathrm{-1}& 0& 0& 0\end{array}\right]& {\mathrm{\gamma }}_{\phantom{}\phantom{2}}^{\phantom{}2}=\left[\begin{array}{cccc}0& 0& 0& \mathrm{-i}\\ 0& 0& \mathrm{i}& 0\\ 0& \mathrm{i}& 0& 0\\ \mathrm{-i}& 0& 0& 0\end{array}\right]& {\mathrm{\gamma }}_{\phantom{}\phantom{3}}^{\phantom{}3}=\left[\begin{array}{cccc}0& 0& 1& 0\\ 0& 0& 0& \mathrm{-1}\\ \mathrm{-1}& 0& 0& 0\\ 0& 1& 0& 0\end{array}\right]& {\mathrm{\gamma }}_{\phantom{}\phantom{4}}^{\phantom{}4}=\left[\begin{array}{cccc}0& 0& 1& 0\\ 0& 0& 0& 1\\ 1& 0& 0& 0\\ 0& 1& 0& 0\end{array}\right]\end{array}\right]$

$\mathrm{Dgamma}\left[\mathrm{~5},\mathrm{definition}\right]$

${\mathrm{\gamma }}_{\phantom{}\phantom{5}}^{\phantom{}5}=\mathrm{-i}{\mathrm{\gamma }}_{\phantom{}\phantom{0}}^{\phantom{}0}{\mathrm{\gamma }}_{\phantom{}\phantom{1}}^{\phantom{}1}{\mathrm{\gamma }}_{\phantom{}\phantom{2}}^{\phantom{}2}{\mathrm{\gamma }}_{\phantom{}\phantom{3}}^{\phantom{}3},{\mathrm{\gamma }}_{\phantom{}\phantom{5}}^{\phantom{}5}{\mathrm{\gamma }}_{\phantom{}\phantom{5}}^{\phantom{}5}=1,{\left[{\mathrm{\gamma }}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}\&comma;{\mathrm{\gamma }}_{\phantom{}\phantom{5}}^{\phantom{}5}\right]}_{+}=0,{\left[{\mathrm{\gamma }}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}\&comma;{\mathrm{\gamma }}_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}\right]}_{+}=2g_{\phantom{}\phantom{\mathrm{\mu },\mathrm{\nu }}}^{\phantom{}\mathrm{\mu },\mathrm{\nu }}$

$\mathrm{TensorArray}\left(\left[1\right]\&comma;\mathrm{performmatrixoperations}\right)$

$\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& \mathrm{-1}& 0\\ 0& 0& 0& \mathrm{-1}\end{array}\right]=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& \mathrm{-1}& 0\\ 0& 0& 0& \mathrm{-1}\end{array}\right]$

$\mathrm{Setup}\left(\mathrm{Dgamma}=\mathrm{Majorana}\&comma;\mathrm{quiet}\right)$

$\left[\mathrm{Dgammarepresentation}=\mathrm{Majorana}\right]$

$\mathrm{TensorArray}\left(\mathrm{\%Dgamma}\left[\mathrm{~mu}\right]=\mathrm{Dgamma}\left[\mathrm{~mu}\right]\&comma;\mathrm{performmatrixoperations}\right)$

$\left[\begin{array}{cccc}{\mathrm{\gamma }}_{\phantom{}\phantom{1}}^{\phantom{}1}=\left[\begin{array}{cccc}\mathrm{i}& 0& 0& 0\\ 0& \mathrm{-i}& 0& 0\\ 0& 0& \mathrm{i}& 0\\ 0& 0& 0& \mathrm{-i}\end{array}\right]& {\mathrm{\gamma }}_{\phantom{}\phantom{2}}^{\phantom{}2}=\left[\begin{array}{cccc}0& 0& 0& \mathrm{i}\\ 0& 0& \mathrm{-i}& 0\\ 0& \mathrm{-i}& 0& 0\\ \mathrm{i}& 0& 0& 0\end{array}\right]& {\mathrm{\gamma }}_{\phantom{}\phantom{3}}^{\phantom{}3}=\left[\begin{array}{cccc}0& \mathrm{-i}& 0& 0\\ \mathrm{-i}& 0& 0& 0\\ 0& 0& 0& \mathrm{-i}\\ 0& 0& \mathrm{-i}& 0\end{array}\right]& {\mathrm{\gamma }}_{\phantom{}\phantom{4}}^{\phantom{}4}=\left[\begin{array}{cccc}0& 0& 0& \mathrm{-i}\\ 0& 0& \mathrm{i}& 0\\ 0& \mathrm{-i}& 0& 0\\ \mathrm{i}& 0& 0& 0\end{array}\right]\end{array}\right]$

$\mathrm{Setup}\left(\mathrm{metric}=\mathrm{Euclidean}\&comma;\mathrm{Dgamma}=\mathrm{standard}\&comma;\mathrm{quiet}\right)$

$\left[\mathrm{Dgammarepresentation}=\mathrm{standard}\&comma;\mathrm{metric}=\left\{\left(1\&comma;1\right)=1\&comma;\left(2\&comma;2\right)=1\&comma;\left(3\&comma;3\right)=1\&comma;\left(4\&comma;4\right)=1\right\}\right]$

$\mathrm{g\_}\left[\right]$

$g_{\mathrm{\mu },\mathrm{\nu }}=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$

$\mathrm{TensorArray}\left(\mathrm{\%Dgamma}\left[\mathrm{\mu }\right]=\mathrm{Dgamma}\left[\mathrm{\mu }\right]\&comma;\mathrm{performmatrixoperations}\right)$

$\left[\begin{array}{cccc}{\mathrm{\gamma }}_1=\left[\begin{array}{cccc}0& 0& 0& \mathrm{-i}\\ 0& 0& \mathrm{-i}& 0\\ 0& \mathrm{i}& 0& 0\\ \mathrm{i}& 0& 0& 0\end{array}\right]& {\mathrm{\gamma }}_2=\left[\begin{array}{cccc}0& 0& 0& \mathrm{-1}\\ 0& 0& 1& 0\\ 0& 1& 0& 0\\ \mathrm{-1}& 0& 0& 0\end{array}\right]& {\mathrm{\gamma }}_3=\left[\begin{array}{cccc}0& 0& \mathrm{-i}& 0\\ 0& 0& 0& \mathrm{i}\\ \mathrm{i}& 0& 0& 0\\ 0& \mathrm{-i}& 0& 0\end{array}\right]& {\mathrm{\gamma }}_4=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& \mathrm{-1}& 0\\ 0& 0& 0& \mathrm{-1}\end{array}\right]\end{array}\right]$

$\mathrm{Setup}\left(\mathrm{Dgamma}=\mathrm{chiral}\&comma;\mathrm{quiet}\right)$

$\left[\mathrm{Dgammarepresentation}=\mathrm{chiral}\right]$

$\mathrm{TensorArray}\left(\mathrm{\%Dgamma}\left[\mathrm{\mu }\right]=\mathrm{Dgamma}\left[\mathrm{\mu }\right]\&comma;\mathrm{performmatrixoperations}\right)$

$\left[\begin{array}{cccc}{\mathrm{\gamma }}_1=\left[\begin{array}{cccc}0& 0& 0& \mathrm{-i}\\ 0& 0& \mathrm{-i}& 0\\ 0& \mathrm{i}& 0& 0\\ \mathrm{i}& 0& 0& 0\end{array}\right]& {\mathrm{\gamma }}_2=\left[\begin{array}{cccc}0& 0& 0& \mathrm{-1}\\ 0& 0& 1& 0\\ 0& 1& 0& 0\\ \mathrm{-1}& 0& 0& 0\end{array}\right]& {\mathrm{\gamma }}_3=\left[\begin{array}{cccc}0& 0& \mathrm{-i}& 0\\ 0& 0& 0& \mathrm{i}\\ \mathrm{i}& 0& 0& 0\\ 0& \mathrm{-i}& 0& 0\end{array}\right]& {\mathrm{\gamma }}_4=\left[\begin{array}{cccc}0& 0& 1& 0\\ 0& 0& 0& 1\\ 1& 0& 0& 0\\ 0& 1& 0& 0\end{array}\right]\end{array}\right]$