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Please refer to the picture below. In the first line, I define the angular momentum vector $\vec{L} = \vec{R} \times \vec{P}$ using the Levi-Civita tensor $\epsilon^{i}_{jk}$. The definition relies on repeated indices $j$ and $k$.

In the third line, I try to define the scalar $\vec{L}.\vec{L}$ which throws an error possibly because the resulting expression has 4 '$j$' and 4 '$k$' indices (which is nonsensical).

Next, I try to circumvent this problem by using 'ReplaceDummies' which seems to have solved the problem.

Is there another shorter and more elegant way to go because I may not want to stick 'ReplaceDummies' all the time?

enter image description here

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There are two alternatives if you don't want to add 'ReplaceDummies' all the time.

  1. Add 'ReplaceDummies' in the definition of $L^i$

L[i_] := ReplaceDummies[epsilon[Delta][i, -j, -k] R[j] P[k]] ;

L[i] L[-i] // ContractMetric // ToCanonical

will produce the desired result.

  1. Use 'MakeRule' which is smarter and replaces the dummy indices on its own

LtorcrosspRule = MakeRule[{L[i], epsilon[Delta][i, -j, -k] R[j] P[k]}, MetricOn -> All];

L[i] L[-i] /. LtorcrosspRule // ContractMetric // ToCanonical

will also work.

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