13.1 Tensor Contraction Scope

Tensor Contraction Scope is the general concept describing the extent to which a contraction affects a tensor expression, encompassing which indices are summed, which tensors and terms are involved, and which parts of a larger expression lie outside the reach of a particular contraction.

The General Notion of Scope in Contraction

Scope as a Boundary of Effect

Scope, in the context of tensor contraction, refers to the boundary separating the portion of an expression altered by a summation over repeated indices from the portion left unaffected, so that describing the scope of a contraction amounts to describing exactly what that contraction reaches and what it does not.

Necessity of Precisely Defined Scope

Because contraction removes indices through summation, an imprecise or ambiguous notion of scope would leave uncertain which indices are summed and which remain, making a precise definition of scope essential to the coherent application of contraction within tensor algebra.

Dimensions Along Which Scope Is Described

Scope With Respect to Index Pairs

One dimension of scope concerns which specific pair of indices, one contravariant and one covariant, is being summed, distinguishing the indices participating in a given contraction from other indices that a tensor may carry but that remain free.

Scope With Respect to Terms in an Expression

Another dimension of scope concerns which term of a larger sum a contraction belongs to, since the summation convention confines a contraction to the single term in which the repeated index symbol appears, rather than spanning across separately added terms.

Scope With Respect to Nested or Sequential Contractions

A further dimension of scope arises when multiple contractions are applied in sequence or are nested within a single expression, in which case the scope of each individual contraction must be tracked separately to determine the cumulative effect of applying them together.

Consequences of Scope for the Resulting Tensor

Effect on Order

The scope of all contractions applied within an expression collectively determines how many indices are removed through summation, and therefore determines the order of the tensor that remains once every contraction within its scope has been carried out.

Effect on Which Indices Remain Free

Indices lying outside the scope of any contraction in the expression persist as free indices of the result, retaining their original variance and dimension, while indices lying within the scope of some contraction are summed away entirely.

Scope as a Category Encompassing Specific Cases

Relation to the Scope of a Single Operation

The scope associated with one particular act of contraction, applied to one specific index pair on one specific tensor or product of tensors, represents a specific instance of the more general notion of contraction scope, narrowed to a single operation rather than considered across an entire expression.

Consistency Across Instances

Regardless of how many contractions appear within an expression or how they are combined, the general notion of scope applies uniformly to each one, ensuring that the same principles used to delineate the reach of a single contraction extend consistently to more complex expressions containing several contractions.

Relationship to Tensor Operation Notation

Contraction scope is made explicit through the placement and repetition of index symbols within tensor operation notation, since it is the pattern of a symbol appearing once as an upper index and once as a lower index within a single term that notation uses to delimit precisely which indices, and which portion of an expression, fall within the scope of a given contraction.