13.3.2 Tensor Contraction Slot Pair Selection

Tensor Contraction Slot Pair Selection is the first step of the tensor contraction operation in which one specific contravariant index slot and one specific covariant index slot, drawn from among all the index positions present on the tensor or tensors involved, are chosen to be linked together for the purpose of summation.

The Act of Selection

Choosing Among Available Slots

When a tensor carries more than one contravariant index or more than one covariant index, slot pair selection identifies exactly which one of each is to be paired for a particular contraction, since the tensor's structure alone does not determine this choice without an explicit decision.

In this tensor, slot pair selection determines, for instance, whether the contravariant slot labeled $i$ is paired with the covariant slot labeled $j$, or instead with the covariant slot labeled $k$, since either pairing is potentially available.

Marking the Selection Through a Shared Symbol

Once a pair is selected, the selection is recorded by assigning the same index symbol to both chosen slots, distinguishing them from the remaining slots, which retain distinct symbols and are thereby marked as excluded from this particular selection.

Requirements the Selection Must Satisfy

Existence of Both Slots

Slot pair selection requires that both chosen positions genuinely exist among the index slots of the tensor, since a selection referring to a slot beyond the tensor's actual order does not correspond to any valid contraction.

Opposite Variance Between the Chosen Slots

Slot pair selection requires that one chosen slot be contravariant and the other be covariant, since selecting two slots of the same variance would not produce a summation consistent with the transformation behavior required of a tensor.

Equal Dimension Between the Chosen Slots

Slot pair selection requires that the two chosen slots range over vector spaces of the same dimension, since the summation to follow pairs values of the two slots term by term and requires an identical range for both.

Consequences of the Selection for Later Steps

Determining Which Values Are Summed

The pair selected determines exactly which components of the tensor are added together during the index summation step, since only components sharing equal values in the two selected slot positions are combined by that summation.

Determining Which Indices Remain Free

Every slot not chosen during selection remains a free index of the resulting tensor, so the selection made at this step directly determines which indices persist, with their original variance and dimension, into the tensor produced by the operation.

Selection When Multiple Contractions Are Intended

Sequential or Simultaneous Selections

When more than one contraction is to be applied to the same tensor or expression, slot pair selection is performed once for each intended contraction, with each selection using its own distinct shared symbol so that the several selections remain independent of one another.

Avoiding Conflicting Selections

A valid set of selections must not reuse a single slot in more than one pairing, since a slot already committed to one selected pair cannot simultaneously be committed to a different pair without producing an ambiguous or undefined summation.

Relationship to Tensor Operation Notation

Slot pair selection is communicated in tensor operation notation through the assignment of a common index symbol to exactly one upper and one lower slot, with the specific letter chosen for that symbol serving no meaning beyond identifying which two slots, among those available on the tensor, have been selected for the contraction.