8.7.5 Tensor Dummy Index Disappearance Rule

The Tensor Dummy Index Disappearance Rule governs how indices vanish in tensor contractions, simplifying expressions through implicit summation.

Tensor Dummy Index Disappearance Rule is the rule stating that once the implicit summation associated with a dummy index has been carried out, that index no longer appears in the resulting expression at all, neither as an upper nor as a lower index, having contributed only to the numerical or symbolic value of the sum rather than to the structure of the output.

Statement of the Rule

Complete Removal After Summation

The disappearance rule specifies that a dummy index, having served its role of pairing an upper and a lower slot for summation, is entirely absent from the final result: it does not appear as a free index, it does not appear with any residual variance, and it leaves behind only its numerical contribution folded into the value of the expression.

C = A_{i} B^{i}

Although the right-hand side is written using the index (i), the left-hand side (C) carries no index at all, since (i) has fully disappeared through summation, leaving a plain scalar.

Rank Reduction as the Visible Consequence

The disappearance of a dummy index is precisely what causes the rank of a contracted expression to be lower than the combined rank of its original factors, since each disappearing dummy index removes one upper slot and one lower slot from the count of indices that survive into the output.

R_{k} = A_{i k} B^{i}

The combined factors (A) and (B) together carry three index slots before contraction, but the disappearance of (i) removes two of them, leaving only the free index (k) in the result (R).

Disappearance Across Multiple Dummy Indices

Each Dummy Index Disappears Independently

When an expression contains several distinct dummy indices, each one disappears independently upon summation, and the cumulative effect on rank is the sum of the reductions contributed by each dummy index individually.

s = A_{i j} B^{i j}

Here both (i) and (j) satisfy the paired-occurrence pattern and both disappear upon summation, reducing a pair of rank-two tensors to a single scalar (s).

Full Disappearance Producing a Scalar

When every index present in an expression is a dummy index, meaning none remain free, the disappearance rule implies that the entire result is a scalar, carrying no indices whatsoever, regardless of how many tensor factors or how many dummy indices were involved in reaching that result.

Why Disappearance Does Not Mean Loss of Information

Value Retained, Symbol Removed

The disappearance rule concerns only the notational bookkeeping of indices, not the informational content of the computation: the numerical contribution of every value the dummy index ranged over is fully retained in the resulting sum, even though the symbol used to track that summation vanishes from the final notation.

Irreversibility of the Symbol

Because the specific values summed over are absorbed into a single resulting number or tensor, the disappearance of a dummy index is a one-way notational operation; the original individual terms of the sum cannot be recovered from the result alone without returning to the pre-summation expression.

Practical Illustration

The disappearance rule is the natural counterpart to the free index equation preservation rule: while a free index must persist unchanged through every manipulation of an expression, a dummy index is guaranteed to vanish entirely once its summation has been performed, and tracking which indices disappear versus which persist is what allows the rank and structure of any final tensor result to be predicted directly from the original expression.