The Trace command represents and when possible computes the generalized trace of an object. The %Trace command is the inert form of Trace; that is, it represents the same mathematical operation, while displaying the operation unevaluated. To evaluate the operation, use the value command.

When tracing an algebraic expression, Trace considers as scalars everything but the Physics algebraic representation for Dirac (Dgamma) and Pauli (Psigma) matrices as tensors of one index, or objects that are of type matrix or Matrix (Array). This default is convenient in formulations in particle physics. This behavior, however, can be changed so that everything but constants are considered traceable objects - for instance any quantum operator set with Setup - for this purpose either pass the optional argument onlymatrices = false or set this behavior to be the default using Setup with its keyword onlymatrices = false.

When tracing an algebraic expression, if the expression is polynomial in Dirac matrices, Trace considers all the expression as matricial, so that every term not containing Dirac matrices is assumed to have as implicit factor the identity matrix. For example, in $p_{}^{\mathrm{\mu }}{\mathrm{\gamma }}_{\mathrm{\mu }}+m$, the term $m$ is assumed to have the identity matrix as a factor.

The result returned by Trace when onlymatrices = false is built as follows:

If $f$ is a constant, then return $f$.

If $f$ is a single matrix, then return the trace of $f$.

If $f$ is a (noncommutative) product, then

If the operands are Dirac matrices or Pauli matrices, then use standard related formulas (see below).

If all of the operands are anticommutative, then

If the number of operands is even, then return 0.

Otherwise, return the Trace after normalizing the product.

If there are constants in the operands, then return the constants times the Trace of the rest.

If $f$ is a (commutative) product, then return the constants times the Trace of the rest.

If $f$ is a sum, not just of noncommutative objects, distribute Trace according to:$\mathrm{Trace}\left(A+B\right)\to \mathrm{Trace}\left(A\right)+\mathrm{Trace}\left(B\right)$.

For all other cases, return the unevaluated expression, $'\mathrm{Trace}'\left(f\right)$.

The Trace of a product of Dirac matrices is based on their anticommutation relation ${\mathrm{gamma}}_{\mathrm{mu}}{\mathrm{gamma}}_{\mathrm{nu}}+{\mathrm{gamma}}_{\mathrm{nu}}{\mathrm{gamma}}_{\mathrm{mu}}=2\mathrm{g\_}\[\mathrm{mu},\mathrm{nu}\]$, where g_ is the metric, and the formula is valid in 2, 3, and 4 dimensions. Thus, traces of products of an odd number of Dirac matrices are always equal to zero, while traces $\mathrm{Trace}\left({\mathrm{\gamma }}_{\mathrm{\ν1}}\,{\mathrm{\gamma }}_{\mathrm{\ν2}}\,\mathrm{...}\,{\mathrm{\gamma }}_{\mathrm{\ν2}n}\right)$ of an even number ( $2n$ ) of them can be expressed as a sum of terms of the form $4\mathrm{g\_}\[\mathrm{mu1},\mathrm{mu2}\]\mathrm{g\_}\[\mathrm{mu3},\mathrm{mu4}\]\mathrm{...}\mathrm{g\_}\[{\mathrm{mu}}_{\mathrm{2n}-1}$, ${\mathrm{mu}}_{\mathrm{2n}}\]$, with the sign of each term being determined by whether the permutation of indices, from $\[{\mathrm{nu}}_1,{\mathrm{nu}}_2,\mathrm{...},{\mathrm{nu}}_{\mathrm{2n}}\]$ to $\[{\mathrm{mu}}_1,{\mathrm{mu}}_2,\mathrm{...},{\mathrm{mu}}_{\mathrm{2n}}\]$, is odd or even.

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

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

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

$\mathrm{Setup}\left(\mathrm{anticommutativeprefix}=Q\,\mathrm{noncommutativeprefix}=Z\right)$

$\left[\mathrm{anticommutativeprefix}=\left\{Q\right\}\,\mathrm{noncommutativeprefix}=\left\{Z\right\}\right]$

$\mathrm{Trace}\left(\mathrm{Q1}\mathrm{Q2}\,\mathrm{onlymatrices}=\mathrm{false}\right)$

$0$

$\mathrm{Trace}\left(\mathrm{Q3}\mathrm{Q1}\mathrm{Q2}\right)$

$\mathrm{Q1}\mathrm{Q2}\mathrm{Q3}$

$\mathrm{Trace}\left(a\mathrm{Q3}b\mathrm{Q1}c\mathrm{Q2}\right)$

$abc\mathrm{Q1}\mathrm{Q2}\mathrm{Q3}$

$\mathrm{Trace}\left(\mathrm{\pi }\mathrm{Q3}\mathrm{\gamma }\mathrm{Q1}\mathrm{I}\mathrm{Q2}\right)$

$\mathrm{I}\mathrm{\pi }\mathrm{\gamma }\mathrm{Q1}\mathrm{Q2}\mathrm{Q3}$

$\mathrm{Trace}\left(A+B+\mathrm{\pi }\mathrm{Q3}\mathrm{\gamma }\mathrm{Q1}\mathrm{I}\mathrm{Q2}\right)$

$A+B+\mathrm{I}\mathrm{\pi }\mathrm{\gamma }\mathrm{Q1}\mathrm{Q2}\mathrm{Q3}$

$\mathrm{Dgamma}\left[3\right]\mathrm{Dgamma}\left[5\right]\mathrm{Dgamma}\left[5\right]$

${\mathrm{\gamma }}_3{\mathrm{\gamma }}_5^2$

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

$0$

$\mathrm{Dgamma}\left[3\right]\mathrm{Dgamma}\left[\mathrm{\xi }\right]$

${\mathrm{\gamma }}_3{\mathrm{\gamma }}_{\mathrm{\xi }}$

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

$4g_{3,\mathrm{\xi }}$

$\mathrm{Dgamma}\left[\mathrm{\alpha }\right]\mathrm{Dgamma}\left[\mathrm{\nu }\right]\mathrm{Dgamma}\left[\mathrm{\xi }\right]$

${\mathrm{\gamma }}_{\mathrm{\alpha }}{\mathrm{\gamma }}_{\mathrm{\nu }}{\mathrm{\gamma }}_{\mathrm{\xi }}$

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

$0$

$\mathrm{Dgamma}\left[\mathrm{\alpha }\right]\mathrm{Dgamma}\left[\mathrm{\nu }\right]\mathrm{Dgamma}\left[\mathrm{\xi }\right]\mathrm{Dgamma}\left[\mathrm{\rho }\right]$

${\mathrm{\gamma }}_{\mathrm{\alpha }}{\mathrm{\gamma }}_{\mathrm{\nu }}{\mathrm{\gamma }}_{\mathrm{\xi }}{\mathrm{\gamma }}_{\mathrm{\rho }}$

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

$4g_{\mathrm{\alpha },\mathrm{\nu }}{g}_{\mathrm{\rho },\mathrm{\xi }}+4g_{\mathrm{\alpha },\mathrm{\rho }}{g}_{\mathrm{\nu },\mathrm{\xi }}-4g_{\mathrm{\alpha },\mathrm{\xi }}{g}_{\mathrm{\nu },\mathrm{\rho }}$

$\mathrm{Psigma}\left[3\right]$

${\mathrm{\sigma }}_3$

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

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

$0$

$5\mathrm{Psigma}\left[2\right]\mathrm{Psigma}\left[2\right]$

$5{\mathrm{\sigma }}_2^2$

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

$5{\left[\right]}^2$

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

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

$10$

$\mathrm{\lambda }\mathrm{Psigma}\left[2\right]\mathrm{Psigma}\left[3\right]$

$\mathrm{\lambda }{\mathrm{\sigma }}_2{\mathrm{\sigma }}_3$

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

$0$

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

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

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

$3\mathrm{Dgamma}\left[1\right]\mathrm{Dgamma}\left[2\right]\mathrm{Psigma}\left[3\right]$

$3{\mathrm{\gamma }}_1{\mathrm{\gamma }}_2{\mathrm{\sigma }}_3$

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

$0$

$\mathrm{Setup}\left(\mathrm{dimension}=4\,\mathrm{signature}=\mathrm{`-`}\,\mathrm{op}=\left\{k\,p\right\}\right)$

$\mathrm{*\; Partial\; match\; of\; \text{'}}\mathrm{op}\mathrm{\text{'}\; against\; keyword\; \text{'}}\mathrm{quantumoperators}\text{'}$

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

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

$\left[\mathrm{dimension}=4\,\mathrm{quantumoperators}=\left\{k\,p\right\}\,\mathrm{signature}=\mathrm{-\; -\; -\; +}\right]$

$\mathrm{Define}\left(p\,k\,\mathrm{quiet}\right)$

$\left\{k\,p\,{\mathrm{\gamma }}_{\mathrm{\mu }}\,{\mathrm{\sigma }}_{\mathrm{\mu }}\,{\partial }_{\mathrm{\mu }}\,g_{\mathrm{\mu },\mathrm{\nu }}\,{\mathrm{\epsilon }}_{\mathrm{\alpha },\mathrm{\beta },\mathrm{\mu },\mathrm{\nu }}\right\}$

$\left(p\left[\mathrm{\mu }\right]\mathrm{Dgamma}\left[\mathrm{\mu }\right]+m\right)\left(k\left[\mathrm{\nu }\right]\mathrm{Dgamma}\left[\mathrm{\nu }\right]+m\right)$

$\left(p_{\mathrm{\mu }}{\mathrm{\gamma }}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}+m\right)\left(k_{\mathrm{\nu }}{\mathrm{\gamma }}_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}+m\right)$

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

$4m^2+4p_{\mathrm{\nu }}{k}_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}$

$\left(p\left[\mathrm{\mu }\right]\mathrm{Dgamma}\left[\mathrm{\mu }\right]+m\right)\left(k\left[\mathrm{\nu }\right]\mathrm{Dgamma}\left[\mathrm{\nu }\right]+m\right)\left(k\left[\mathrm{\rho }\right]\mathrm{Dgamma}\left[\mathrm{\rho }\right]+m\right)\left(k\left[\mathrm{\sigma }\right]\mathrm{Dgamma}\left[\mathrm{\sigma }\right]+m\right)$

$\left(p_{\mathrm{\mu }}{\mathrm{\gamma }}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}+m\right)\left(k_{\mathrm{\nu }}{\mathrm{\gamma }}_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}+m\right)\left(k_{\mathrm{\rho }}{\mathrm{\gamma }}_{\phantom{}\phantom{\mathrm{\rho }}}^{\phantom{}\mathrm{\rho }}+m\right)\left(k_{\mathrm{\sigma }}{\mathrm{\gamma }}_{\phantom{}\phantom{\mathrm{\sigma }}}^{\phantom{}\mathrm{\sigma }}+m\right)$

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

$4m^4+12m^2{k}_{\mathrm{\lambda }}{k}_{\phantom{}\phantom{\mathrm{\lambda }}}^{\phantom{}\mathrm{\lambda }}+12m^2{p}_{\mathrm{\lambda }}{k}_{\phantom{}\phantom{\mathrm{\lambda }}}^{\phantom{}\mathrm{\lambda }}+4p_{\mathrm{\lambda }}{k}_{\mathrm{\tau }}{k}_{\phantom{}\phantom{\mathrm{\tau }}}^{\phantom{}\mathrm{\tau }}{k}_{\phantom{}\phantom{\mathrm{\lambda }}}^{\phantom{}\mathrm{\lambda }}-4p_{\mathrm{\lambda }}{k}_{\mathrm{\tau }}{k}_{\phantom{}\phantom{\mathrm{\lambda }}}^{\phantom{}\mathrm{\lambda }}{k}_{\phantom{}\phantom{\mathrm{\tau }}}^{\phantom{}\mathrm{\tau }}+4p_{\mathrm{\tau }}{k}_{\phantom{}\phantom{\mathrm{\tau }}}^{\phantom{}\mathrm{\tau }}{k}_{\mathrm{\lambda }}{k}_{\phantom{}\phantom{\mathrm{\lambda }}}^{\phantom{}\mathrm{\lambda }}$