13.19.1 Tensor Contraction Dummy Index Removal

Tensor contraction removes dummy indices by summing over them, simplifying tensor expressions in algebraic computations.

Tensor Contraction Dummy Index Removal is the simplification step in which a repeated index appearing once as an upper position and once as a lower position within a tensor expression is eliminated from the written notation by carrying out the implied summation explicitly, replacing the pair of occurrences with the resulting summed quantity and leaving no trace of the original dummy symbol in the simplified expression.

Definition

Given an expression containing a repeated index $a$ , appearing once up and once down under the Einstein summation convention:

T_{a}^{i} U_{a}^{j}

dummy index removal replaces this expression with a newly named tensor whose components are the explicit sum, and the label $a$ no longer appears:

S_{i}^{j} = \sum_{a} T_{a}^{i} U_{a}^{j}

Why Dummy Index Removal Is Valid

Absence of External Meaning

A dummy index carries no meaning outside the specific term in which it is repeated; it exists solely to indicate which pair of slots is summed over. Once that summation has been performed, the symbol itself is no longer needed to interpret the resulting value.

Renaming Invariance

Because a dummy index can be freely renamed to any other unused symbol without changing the value of the expression, the specific letter chosen for it carries no information that needs to survive into the simplified notation:

T_{a}^{i} U_{a}^{j} = T_{b}^{i} U_{b}^{j}

confirming that removing it entirely, once the sum is carried out, does not discard any distinguishing content.

Procedure

Step One: Locate the Repeated Pair

Identify the index symbol occurring exactly twice in a single term, once as an upper index and once as a lower index.

Step Two: Perform the Summation

Carry out the sum over the located index's full range, producing a new set of components indexed only by the remaining free symbols.

Step Three: Assign a New Symbol

Name the resulting object with a fresh tensor symbol, dropping the now-summed dummy index entirely from the notation, leaving only the free indices $i$ and $j$ visible.

Effect on Free Index Count

Reduction by Two Positions

Each dummy index removed accounts for two index positions vanishing from the written expression, one upper and one lower, consistent with the general order reduction behavior associated with a single contraction.

Multiple Removals

An expression containing several independent repeated index pairs undergoes dummy index removal separately for each pair, with the final simplified expression retaining only the indices that never repeated within any single term.

Diagram

Relation to Broader Simplification

Dummy index removal is the elementary operation underlying the more general tensor contraction expression simplification, and it is the mechanism by which patterns such as trace recognition and Kronecker delta elimination achieve their simplified forms: each of those patterns is, at its core, an instance of removing a repeated dummy index once its summation has been recognized and carried out.