8.17.2 Tensor Term Free Index Matching

Tensor Term Free Index Matching is the requirement, applied internally within a single term formed from a product of several tensor factors, that the free indices contributed by each individual factor combine consistently into one well-defined set of free indices for the term as a whole, with no factor's free index left unaccounted for and no unintended interaction occurring between the free indices of different factors. It is the local, single-term counterpart to the equation-wide free index matching rule, addressing how the free indices of several multiplied tensors are meant to merge before any comparison against other terms in a larger equation is even possible.

The Local Matching Problem

Free Indices From Multiple Factors Must Coexist Without Conflict

When several tensor factors are multiplied together to form one term, each factor may contribute its own free indices, and term-level matching requires that these contributed indices remain distinct from one another unless a deliberate contraction is intended. In the term

three separate factors each contribute one free index — $i$, $j$, and $k$ — and because all three letters are distinct, the term is internally consistent, resulting in a single object of type $(2,1)$ carrying all three as free indices.

Identifying the Free-Index Structure of a Composite Term

Term-level matching requires correctly identifying, for a term built from multiple factors, exactly which index letters remain free in the assembled term and which have been consumed by contraction between two of the factors. In $A^{i}B_{i}C^{j}$, the letter $i$ is contracted between the first two factors and does not survive as free, while $j$, appearing only once across the whole term, remains free; term-level matching confirms that the term's overall free-index set is ${j}$ alone, correctly excluding $i$.

Sources of Internal Mismatch Within a Term

Unintended Contraction Between Factors

A common failure of term-level matching occurs when two factors, each independently free of any collision on its own, happen to share a letter once multiplied together, causing an unintended contraction that removes a free index the author intended to retain. Multiplying $A^{i}$ by $B_{i}$ when both were meant to contribute independent free indices collapses what should have been a rank-2 result into a contracted scalar, violating the intended free-index structure of the term.

Genuinely Free Indices Colliding Under the Same Letter

A separate failure occurs when two factors each contribute a free index using the same letter without any intention of contraction, since two same-position or ambiguous-position occurrences of the same letter cannot both remain distinct free indices of the assembled term; this is the term-level manifestation of a free–free index collision, discovered specifically at the moment separate factors are multiplied together.

Verifying Term-Level Matching

Enumerating Contributions Factor by Factor

The direct method for verifying term-level free index matching is to list, for each factor in the term, the index letters it carries along with their positions, then check across the combined list whether every letter appears either exactly once (a legitimate free index of the term) or exactly twice in opposite positions (a legitimate internal contraction), with no letter appearing in any other configuration.

Comparing Intended Against Actual Free-Index Sets

Because a term is typically constructed with a specific intended set of free indices in mind — often dictated by what the term is meant to contribute to a larger equation — term-level matching also involves comparing the actual free-index set obtained after accounting for internal contractions against this intended set, catching cases where an unplanned contraction has silently reduced the term's rank below what was intended.

Consequences for the Larger Equation

A Term Failing Internal Matching Cannot Participate Correctly

Before a term's free indices can be compared against the free indices of other terms in a full equation, as required by the equation-level free index matching rule, the term's own internal free-index structure must first be correctly and unambiguously established; a term suffering from an internal collision has no well-defined free-index set to offer for that comparison, making internal, term-level matching a logical prerequisite to equation-level matching rather than a separate, independent concern.

Correcting Internal Mismatches Before Assembling an Equation

Because internal term mismatches are typically resolved by renaming one of the colliding dummy occurrences, or by recognizing and fixing an unintended contraction, this correction is best performed while a term is still being constructed, before it is placed alongside other terms in a full equation where the ambiguity would otherwise be harder to trace back to its source.

Role Within the Index Matching Rule

Term-level free index matching addresses the assembly of a single term from multiple tensor factors, ensuring that the free indices those factors contribute combine into one coherent, unambiguous set before that term is ever measured against the type, position, or letter identity required by the equation-level matching rule. It functions as the foundation on which the broader index matching rule depends, since an equation's terms cannot be meaningfully compared to one another unless each one, individually, already has a well-defined free-index structure of its own.