13.1.2 Tensor Index Pairing Scope

Tensor Index Pairing Scope defines how indices pair in tensors, structuring components and transformation rules in multi-linear algebra.

Tensor Index Pairing Scope is the specific form of tensor contraction scope that identifies which single contravariant index and which single covariant index, among possibly many candidates present on the tensors involved, are matched together to form the pair over which a given contraction sums.

The Act of Pairing an Index Pair

Selecting Among Available Candidates

A tensor or product of tensors may present several contravariant and several covariant indices simultaneously, and index pairing scope identifies exactly which one contravariant index and which one covariant index have been selected to form the pair set equal for a particular contraction, leaving all other indices outside that pairing.

T_{j k}^{i l}

In a tensor carrying multiple upper and lower indices such as this, index pairing scope specifies, for instance, that the pairing involves the index labeled $i$ and the index labeled $j$ , distinguishing this choice from other possible pairings such as $i$ with $k$ .

Pairing Fixed by a Shared Symbol

Under the summation convention, index pairing scope is made explicit by assigning the same symbol to the selected upper and lower index while leaving all other indices with distinct symbols, so that the shared symbol itself identifies the extent of the pairing.

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

Distinguishing Pairing Scope from Order Reduction Scope

Focus on Identity Rather Than Count

Where order reduction scope reports only how many indices are removed as a numerical outcome, index pairing scope specifies which particular indices, identified by their position and label, are the ones being removed, providing information at a finer level of detail than a simple count.

Necessity of Pairing Scope for Well-Defined Results

Because a tensor with several available upper and lower indices can generally be contracted in more than one way, each producing a different resulting tensor, index pairing scope must be specified explicitly or unambiguously implied by notation for the outcome of a contraction to be determined.

Multiple Pairings on the Same Tensor

Distinct Results from Distinct Pairings

Contracting different pairs of indices on the same tensor generally produces different resulting tensors, so index pairing scope is essential to distinguishing one legitimate contraction of a tensor from another equally legitimate but distinct contraction of the same tensor.

T_{j}^{i} \neq T_{i}^{j}

when both expressions denote contractions over different index positions of a tensor with more than two indices.

Independent Pairings Within a Single Expression

When an expression involves more than one contraction, each pairing is scoped independently, using a distinct shared symbol for each pair, so that the pairings do not interfere with one another and each summation proceeds over its own designated pair of indices.

Requirements Governing a Valid Pairing

Opposite Variance Requirement

Index pairing scope is only valid when the two indices selected for pairing have opposite variance, one upper and one lower, since pairing two indices of the same variance would not correspond to a defined contraction.

Equal Dimension Requirement

The two indices selected for pairing must range over spaces of the same dimension, since the summation defining the pairing requires both indices to take values over an identical range.

Relationship to Tensor Operation Notation

Index pairing scope is communicated entirely through the choice of index symbols in tensor operation notation, since assigning an identical symbol to one upper and one lower index position, while keeping all other index symbols distinct, is precisely how notation specifies which particular pair among several available candidates is selected for a given contraction.