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

$\left[\mathrm{`*`}\,\mathrm{`.`}\,\mathrm{Annihilation}\,\mathrm{AntiCommutator}\,\mathrm{Antisymmetrize}\,\mathrm{Assume}\,\mathrm{Bra}\,\mathrm{Bracket}\,\mathrm{Check}\,\mathrm{Christoffel}\,\mathrm{Coefficients}\,\mathrm{Commutator}\,\mathrm{CompactDisplay}\,\mathrm{Coordinates}\,\mathrm{Creation}\,\mathrm{D\_}\,\mathrm{Dagger}\,\mathrm{Decompose}\,\mathrm{Define}\,\mathrm{D\γ}\,\mathrm{DiracConjugate}\,\mathrm{Einstein}\,\mathrm{EnergyMomentum}\,\mathrm{Expand}\,\mathrm{ExteriorDerivative}\,\mathrm{Factor}\,\mathrm{FeynmanDiagrams}\,\mathrm{FeynmanIntegral}\,\mathrm{Fundiff}\,\mathrm{Geodesics}\,\mathrm{GrassmannParity}\,\mathrm{Gtaylor}\,\mathrm{Intc}\,\mathrm{Inverse}\,\mathrm{Ket}\,\mathrm{KillingVectors}\,\mathrm{KroneckerDelta}\,\mathrm{LagrangeEquations}\,\mathrm{LeviCivita}\,\mathrm{Library}\,\mathrm{LieBracket}\,\mathrm{LieDerivative}\,\mathrm{Normal}\,\mathrm{NumericalRelativity}\,\mathrm{Parameters}\,\mathrm{PerformOnAnticommutativeSystem}\,\mathrm{Projector}\,\mathrm{Psigma}\,\mathrm{Redefine}\,\mathrm{Ricci}\,\mathrm{Riemann}\,\mathrm{Setup}\,\mathrm{Simplify}\,\mathrm{SortProducts}\,\mathrm{SpaceTimeVector}\,\mathrm{StandardModel}\,\mathrm{Substitute}\,\mathrm{SubstituteTensor}\,\mathrm{SubstituteTensorIndices}\,\mathrm{SumOverRepeatedIndices}\,\mathrm{Symmetrize}\,\mathrm{TensorArray}\,\mathrm{Tetrads}\,\mathrm{ThreePlusOne}\,\mathrm{ToContravariant}\,\mathrm{ToCovariant}\,\mathrm{ToFieldComponents}\,\mathrm{ToSuperfields}\,\mathrm{Trace}\,\mathrm{TransformCoordinates}\,\mathrm{Vectors}\,\mathrm{Weyl}\,\mathrm{`^`}\,\mathrm{dAlembertian}\,\mathrm{d\_}\,\mathrm{diff}\,\mathrm{g\_}\,\mathrm{gamma\_}\right]$

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

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

To check the free and repeated indices found in tensor expressions for correctness, first Define the tensor objects of your problem.

$\mathrm{Define}\left(A\,B\,C\right)$

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

$\left\{A\,B\,C\,{\mathrm{\gamma }}_{\mathrm{\mu }}\,{\mathrm{\sigma }}_{\mathrm{\mu }}\,{\partial }_{\mathrm{\mu }}\,g_{\mathrm{\mu },\mathrm{\nu }}\,{\mathrm{\epsilon }}_{\mathrm{\alpha },\mathrm{\beta },\mathrm{\mu },\mathrm{\nu }}\right\}$

Consider now an expression where the (spacetime) tensors $A$, $B$, and $C$ appear in products and sums.

$\mathrm{expr}≔A\left[\mathrm{\mu },\mathrm{\nu }\right]B\left[\mathrm{\mu },\mathrm{\rho }\right]+C\left[\mathrm{\nu },\mathrm{\rho }\right]$

$\mathrm{expr}≔A_{\mathrm{\mu },\mathrm{\nu }}{B}_{\mathrm{\mu },\mathrm{\rho }}+C_{\mathrm{\nu },\mathrm{\rho }}$

By inspecting this expression, you can see that $\mathrm{\mu }$ is a repeated index, so the sum rule over $\mathrm{\mu }$ ranging from 1 to the dimension of the spacetime is assumed, and $\mathrm{\nu }$ and $\mathrm{\rho }$ are free indices, the same in both terms of the sum. So everything is correct, and the Check command returns NULL (that is, it just displays the message).

$\mathrm{Check}\left(\mathrm{expr}\right)$

$\mathrm{The\; repeated\; and\; free\; indices\; in\; the\; given\; expression\; check\; ok.}$

To avoid displaying the message, use the optional keyword quiet, as in Check(expr, quiet). To actually return the free and repeated indices, use the optional parameter kind_of_indices.

$\mathrm{Check}\left(\mathrm{expr}\,\mathrm{all}\right)$

$\mathrm{The\; repeated\; indices\; per\; term\; are:}\left[\left\{\mathrm{...}\right\}\,\left\{\mathrm{...}\right\}\,\mathrm{...}\right]\mathrm{,\; the\; free\; indices\; are:}\left\{\mathrm{...}\right\}$

$\left[\left\{\mathrm{\mu }\right\}\,\varnothing \right],\left\{\mathrm{\nu }\,\mathrm{\rho }\right\}$

$\mathrm{Check}\left(\mathrm{expr}\,\mathrm{free}\,\mathrm{quiet}\right)$

$\left\{\mathrm{\nu }\,\mathrm{\rho }\right\}$

$\mathrm{Check}\left(\mathrm{expr}\,\mathrm{repeated}\,\mathrm{quiet}\right)$

$\left[\left\{\mathrm{\mu }\right\}\,\varnothing \right]$

When the free or repeated indices are incorrect, an error interruption points to the problem found.

$A\left[\mathrm{\mu },\mathrm{\nu }\right]B\left[\mathrm{\mu },\mathrm{\rho }\right]+C\left[\mathrm{\mu },\mathrm{\rho }\right]$

$A_{\mathrm{\mu },\mathrm{\nu }}{B}_{\mathrm{\mu },\mathrm{\rho }}+C_{\mathrm{\mu },\mathrm{\rho }}$

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

$\mathrm{Found\; different\; free\; indices\; in\; different\; operands\; of\; a\; sum;}\mathrm{in\; operand}1:\left[\mathrm{\nu }\,\mathrm{\rho }\right],\mathrm{in\; operand}2:\left[\mathrm{\mu }\,\mathrm{\rho }\right],\mathrm{in}{A}_{\mathrm{\mu },\mathrm{\nu }}{B}_{\mathrm{\mu },\mathrm{\rho }}+C_{\mathrm{\mu },\mathrm{\rho }}$

Error, (in Physics:-Check) found different free indices in different operands of a sum

When checking indices in equations (more generally, relations), the free indices of the left-hand and right-hand sides are expected to be the same, or an error message interrupting the computation is displayed, and the output is presented also as an equation (relation)

$\mathrm{Check}\left(A\left[\mathrm{\mu },\mathrm{\nu }\right]B\left[\mathrm{\mu },\mathrm{\rho }\right]=-C\left[\mathrm{\mu },\mathrm{\rho }\right]\,\mathrm{all}\right)$

$\mathrm{Free\; indices\; on\; both\; sides\; of\; the\; equation\; are\; different:\; found}\left\{\mathrm{\nu }\,\mathrm{\rho }\right\}\mathrm{on\; the\; left-hand\; side\; and}\left\{\mathrm{\mu }\,\mathrm{\rho }\right\}\mathrm{on\; the\; right-hand\; side}$

Error, (in Physics:-Check) free indices on both sides of the equation are different

$\mathrm{Check}\left(A\left[\mathrm{\mu },\mathrm{\nu }\right]B\left[\mathrm{\mu },\mathrm{\rho }\right]=-C\left[\mathrm{\nu },\mathrm{\rho }\right]\,\mathrm{all}\right)$

$\mathrm{The\; repeated\; indices\; per\; term\; are:}\left[\left\{\mathrm{...}\right\}\,\left\{\mathrm{...}\right\}\,\mathrm{...}\right]\mathrm{,\; the\; free\; indices\; are:}\left\{\mathrm{...}\right\}$

$\left(\left[\left\{\mathrm{\mu }\right\}\right]\,\left\{\mathrm{\nu }\,\mathrm{\rho }\right\}\right)=\left(\left[\varnothing \right]\,\left\{\mathrm{\nu }\,\mathrm{\rho }\right\}\right)$

For practical reasons, this request of having the same free indices on both sides of an equation (more generally, a relation) is disregarded when one of these two sides is equal to 0, in which case the equation is first transformed into the left-hand minus right-hand sides, then processed, and the output is returned accordingly, not as a relation. So, for example, the output for the following equation is the same as the one you obtain for Check(A[mu,nu]*B[mu,rho] + C[nu,rho], all).

$\mathrm{Check}\left(A\left[\mathrm{\mu },\mathrm{\nu }\right]B\left[\mathrm{\mu },\mathrm{\rho }\right]+C\left[\mathrm{\nu },\mathrm{\rho }\right]=0\,\mathrm{all}\right)$

$\mathrm{The\; repeated\; indices\; per\; term\; are:}\left[\left\{\mathrm{...}\right\}\,\left\{\mathrm{...}\right\}\,\mathrm{...}\right]\mathrm{,\; the\; free\; indices\; are:}\left\{\mathrm{...}\right\}$

$\left[\left\{\mathrm{\mu }\right\}\,\varnothing \right],\left\{\mathrm{\nu }\,\mathrm{\rho }\right\}$

Note that when loading Physics at the beginning of this Examples section we have loaded the `*` operator that comes with the package, designed to handle anticommutative and noncommutative operands in products. For the purpose of illustrating Check with noncommutative variables, unload that command, and set a macro to refer to it instead:

$\mathrm{unwith}\left(\mathrm{Physics}\,\mathrm{`*`}\right)$

$\mathrm{macro}\left(\mathrm{`\&*`}=\mathrm{Physics}:-\mathrm{`*`}\right)\:$

Now set prefixes for identifying anticommutative and noncommutative variables.

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

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

So in what follows, variables prefixed by $Q$ or $Z$, followed by a positive integer, are considered anticommutative and noncommutative, respectively.

Now consider the following product constructed by the generalized product operator of the Physics package (in what follows, `&*` represents this operator, due to the macro used), and also using the global product operator `*`

$P≔\left(\mathrm{Q1}\&*\left(\mathrm{Q1}+\mathrm{Q3}\&*\left(\mathrm{Q4}\&*1+\mathrm{Q5}\mathrm{Q7}\right)\right)\right)\&*\mathrm{Q6}$

$P≔\mathrm{Q1}\left(\mathrm{Q1}+\mathrm{Q3}\left(\mathrm{Q5}\mathrm{Q7}+\mathrm{Q4}\right)\right)\mathrm{Q6}$

Although perhaps it is not evident on inspection, this product is ill-formed.

$\mathrm{Check}\left(P\right)$

Error, (in Physics:-Check) found more than 1 anticommutative object in the commutative product: Q5*Q7

Analogously, products can be ill-formed regarding other types of problems.

$\left(\mathrm{Q1}\&*\left(\mathrm{Q1}+\mathrm{Q3}\&*\left(\mathrm{Q4}\&*1+\mathrm{Z5}\mathrm{Z7}\right)\right)\right)\&*\mathrm{Q6}$

$\mathrm{Q1}\left(\mathrm{Q1}+\mathrm{Q3}\left(\mathrm{Z5}\mathrm{Z7}+\mathrm{Q4}\right)\right)\mathrm{Q6}$

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

Error, (in Physics:-Check) found more than 1 noncommutative object in the commutative product: Z5*Z7

$\left(\mathrm{Q1}\&*\left(\mathrm{Q1}+\mathrm{Q3}\&*\left(\mathrm{Q4}\&*1+\mathrm{Q5}\mathrm{Z7}\right)\right)\right)\&*\mathrm{Q6}$

$\mathrm{Q1}\left(\mathrm{Q1}+\mathrm{Q3}\left(\mathrm{Q5}\mathrm{Z7}+\mathrm{Q4}\right)\right)\mathrm{Q6}$

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

Error, (in Physics:-Check) found anticommutative and noncommutative objects in the commutative product Q5*Z7

When the products Check okay, a message is displayed.

$\left(\mathrm{Q1}\&*\left(\mathrm{Q1}+\mathrm{Q3}\&*\left(\mathrm{Q4}\&*1+\mathrm{A5}\mathrm{Q7}\right)\right)\right)\&*\mathrm{Q6}$

$\mathrm{Q1}\left(\mathrm{Q1}+\mathrm{Q3}\left(\mathrm{A5}\mathrm{Q7}+\mathrm{Q4}\right)\right)\mathrm{Q6}$

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

$\mathrm{The\; products\; in\; the\; given\; expression\; check\; ok.}$

To avoid displaying the message, use the optional keyword quiet, as in Check(%, quiet).