13.1.1 Tensor Contraction Operation Scope

Tensor contraction operation scope defines how indices are summed over in tensor algebra, specifying the range and rules for contraction in multi-linear operations.

Tensor Contraction Operation Scope is the delineation of exactly which index pair, tensor, and surrounding expression a single act of contraction applies to, identifying the specific portion of a larger tensor expression that a given summation over repeated indices is understood to affect.

Defining the Extent of a Contraction

The Tensor Subject to Contraction

The scope of a contraction begins with the identification of the particular tensor, or product of tensors, whose indices are being summed, distinguishing that tensor from any other tensors that may appear elsewhere in a larger expression but are not involved in this particular contraction.

T_{i}^{i} = \sum_{i = 1}^{n} T_{i}^{i}

The Specific Index Pair Involved

The scope further narrows to the exact pair of indices, one upper and one lower, that are set equal and summed, since a tensor may possess several indices, only two of which participate in any single contraction operation.

Scope Within Larger Expressions

Isolating a Contraction Within a Product

When a contraction occurs within a larger product of several tensors, the scope of that contraction extends only to the two indices being summed and does not extend to other indices present on the same tensors that remain free and uncontracted.

A_{i} B^{i j} = \sum_{i = 1}^{n} A_{i} B^{i j}

In this expression, the free index $j$ lies outside the scope of the contraction over $i$ , remaining an open index of the resulting tensor.

Multiple Contractions Within One Expression

An expression may contain more than one contraction, each with its own scope confined to its particular pair of repeated indices, so that separate summations proceed independently over their respective index pairs without interference between them.

A_{i} B_{j}^{i} C^{j}

Scope Relative to the Free Indices of a Result

Boundary Between Summed and Free Indices

The scope of contraction draws a clear boundary between indices that are summed away and disappear from the final expression and indices that remain free, appearing unmodified as indices of the resulting tensor, with this boundary determining the order of the tensor produced by the operation.

Determining the Order of the Result

Because indices within the scope of a contraction are removed through summation while indices outside that scope persist, the order of the resulting tensor equals the combined order of the tensors involved in the expression minus twice the number of index pairs falling within the scope of contraction.

Scope and the Repeated Index Convention

Scope Fixed by the Repetition of a Symbol

Under the standard summation convention, the scope of a contraction is fixed entirely by which index symbol is repeated as both an upper and lower index within a single term, so that identifying the scope of a contraction reduces to identifying repeated index symbols within that term.

Restriction to a Single Term

The scope of a contraction governed by the summation convention does not extend across separate terms joined by addition, since a repeated index appearing in two different terms of a sum is understood as two distinct, independently scoped contractions rather than one contraction spanning both terms.

Relationship to Tensor Operation Notation

The operation scope of a contraction is read directly from the arrangement of indices in tensor operation notation, since the placement of a repeated upper and lower index within a single term, as opposed to across separate terms, is precisely what notation uses to signal the boundaries of a particular contraction's scope.