8.1.5 Tensor Indexed Contraction Scope

Tensor Indexed Contraction Scope is the identification, within a tensor carrying several indices, of exactly which one superscript slot and one subscript slot are being paired together and summed in a given contraction operation, as distinct from all the other index slots on the same tensor that remain untouched by that particular contraction. It answers the question of which pair of slots a contraction acts on, since a tensor of high rank generally offers more than one candidate pair, and different choices of pair produce different resulting tensors.

Selecting a Pair Among Several Candidates

A High-Rank Tensor Offers Multiple Contraction Choices

A tensor such as T^{ij}_{kl}, carrying two superscript and two subscript indices, admits more than one possible contraction: the superscript i could be paired with the subscript k, or with the subscript l, and likewise for j. The contraction scope of a specific contraction operation names precisely which of these candidate pairs is being acted upon, since writing simply "contract T" without specifying the pair leaves the operation ambiguous.

Contracting the first superscript with the first subscript, as on the left, generally produces a different resulting tensor than contracting the first superscript with the second subscript, as on the right; the two expressions specify different contraction scopes and are not interchangeable in general.

Indices Outside the Chosen Scope Remain Free

Any index slot on the tensor not included in the designated contracted pair lies outside that contraction's scope and survives, unsummed, as a free index of the result. Contracting T^{ij}_{kl} over i and k leaves j and l outside the contraction scope, so the result is a tensor S^{j}_{l} carrying exactly those two remaining free indices.

Contraction Scope Within a Single Tensor Versus Across a Product

Self-Contraction (Trace-Type Scope)

When both members of the contracted pair belong to the same tensor object, the operation is a self-contraction, and its scope is confined entirely within that one tensor's index slots; the type (p, q) of the tensor is reduced to (p − 1, q − 1) by this single internal pairing.

Contraction Between Factors

When the contracted pair spans two different tensors multiplied together, as in A^{i}_{j} B^{j}_{k}, the contraction scope spans both factors: the summed index j links a subscript slot on A to a superscript slot on B, and the scope of this contraction is the pair of slots taken one from each factor, rather than any single tensor's own internal structure.

Diagram of Contraction Scope Choice

Successive Contractions and Their Individual Scopes

One Scope at a Time

When a tensor is contracted more than once, each contraction has its own scope, applied to whatever tensor results from the previous contraction; a full contraction of T^{ij}_{kl} down to a scalar, for instance, requires two successive contraction operations, each with a clearly identified scope, rather than a single operation acting on all four indices simultaneously.

Order Sensitivity of Scope Selection

Because a tensor with multiple candidate index pairs can, in general, be contracted in more than one order or over more than one distinct pairing, specifying the contraction scope at each step is necessary to obtain a well-defined result; omitting this specification leaves open which of several inequivalent final tensors or scalars is intended.

Contraction Scope in the Summation Convention

Made Explicit by the Repeated Letter

Under standard tensor index notation, the contraction scope of a given operation is communicated implicitly but unambiguously by which specific superscript letter is repeated as which specific subscript letter; writing T^{ij}_{ik} rather than T^{ij}_{ki} already fixes, through the placement of the repeated letter i, exactly which pair of slots forms the contraction scope, without any further verbal specification being required.

Consistency Requirement

For the notation to unambiguously convey contraction scope, a given letter must be repeated exactly once as a superscript and once as a subscript within the relevant term; any deviation, such as the same letter appearing in three or more slots, removes the ability of the notation alone to specify a single, well-defined contraction scope, and must be resolved by introducing additional distinct letters.