13.4.3 Tensor Single Contraction Dimension Requirement

Tensor Single Contraction Dimension Requirement is the condition that the two index slots forming a single contraction slot pair must range over vector spaces of identical dimension, a requirement that must hold before the associated summation index can be assigned a well-defined range over which to sum.

Statement of the Requirement

Equal Dimension Between the Two Selected Slots

For a single index contraction operation to be defined, the contravariant slot and the covariant slot chosen for the slot pair must each be associated with a vector space of the same dimension, since the summation linking them must run over a single, shared range of values.

The Shared Dimension as the Summation Range

Once the requirement is satisfied, the common dimension supplies the upper bound of the summation defining the contraction, with the summation index running from one up to that shared value.

Why the Requirement Is Necessary

Ensuring a Well-Defined Pairing of Values

Because the covariant and contravariant slots of the pair are set equal to a common index and summed together, each value taken by that index must correspond to a genuine position in both slots, a correspondence that fails to exist consistently if the two slots range over spaces of different size.

Consequence of an Unequal Dimension

If the two selected slots were associated with differing dimensions, the summation would either omit valid positions of the larger space or attempt to reference positions beyond the range of the smaller space, leaving the intended operation without a coherent, well-defined outcome.

Typical Circumstances in Which the Requirement Is Automatically Met

Contraction of a Tensor Against Itself

When both slots of a contraction pair belong to a single tensor built from one common vector space and its dual, the dimension requirement is automatically satisfied, since every contravariant and covariant index of such a tensor shares the same underlying dimension by construction.

Contraction Across a Product of Tensors from the Same Space

When a contraction links an index from one factor of a tensor product to an index from another factor, the dimension requirement is satisfied provided both factors are built from vector spaces of matching dimension, a condition that must be confirmed rather than assumed when the two factors originate from possibly different sources.

Verification of the Requirement

Position Within the Verification Procedure

The dimension requirement for a single contraction is checked during the input verification stage, specifically as part of the dimension verification applied to the slot pair identified through slot pair selection, before the index summation step is permitted to proceed.

Independence from Variance and Slot Existence Checks

The dimension requirement is verified separately from the checks confirming that the two slots exist and carry opposite variance, since a slot pair can satisfy those other requirements while still failing to satisfy the dimension requirement, and all such checks must succeed together for the contraction to be valid.

Relationship to Tensor Operation Notation

The dimension requirement is not itself visible directly within tensor operation notation, since notation records the labels and placement of indices rather than the dimensions of the spaces they range over, but the requirement governs which notated expressions, featuring a repeated index symbol, correspond to a contraction that is actually well defined once the associated dimensions are examined.