13.16.5 Tensor Contraction Diagram Result Reading

Understanding how tensor contraction diagrams visually represent and simplify complex algebraic operations in tensor algebra.

Tensor Contraction Diagram Result Reading is the procedure by which the tensor type, index labels, and algebraic expression represented by a completed contraction diagram are extracted directly from its visual structure, translating nodes, open legs, and edges back into the equivalent indexed formula without needing to re-derive the expression from the original definitions.

Definition

Given a fully drawn contraction diagram, result reading is the mapping:

diagram \to R_{\dots}^{\dots}

that produces the residual tensor $R$ , with its free indices matching the diagram's open legs and its value equal to the product of all node tensors summed over every internal edge.

Step-by-Step Reading Procedure

Step One: Identify Open Legs

List every open leg in the diagram and record its variance, upper or lower. This list directly gives the type pair $(p, q)$ of the result, with $p$ equal to the count of open upper legs and $q$ equal to the count of open lower legs.

Step Two: Assign Index Labels

Assign a distinct symbol to each open leg; these become the free indices of the resulting expression. Assign a distinct dummy symbol to each internal edge; these become the summed indices.

Step Three: Write the Node Factors

For each node, write down its tensor symbol with all of its attached leg labels placed as superscripts or subscripts according to variance, exactly as recorded in steps one and two.

Step Four: Multiply and Sum

Write the product of all node factors from step three, and apply the summation convention to every dummy index introduced in step two, yielding the final expression:

R_{j}^{i} = \sum_{a} A_{a}^{i} B_{j}^{a}

Reading Order Directly From Structure

Order Formula

The order of the result is read immediately by counting open legs, without needing to write out the expression:

order (R) = number of open legs

Scalar Recognition

If a diagram has zero open legs, result reading immediately identifies the represented quantity as a scalar, without needing to inspect the internal edge pattern further.

Example Reading

Verification of a Reading

A reading is verified by checking two consistency conditions: every leg in the diagram appears exactly once in the derived expression, either as a free index or as one instance of a repeated dummy index, and every repeated dummy index appears exactly twice, once upper and once lower, matching the internal edge it corresponds to.

Purpose

Result reading closes the loop between the diagrammatic and algebraic representations of tensor contraction, allowing a diagram constructed purely by visual composition of nodes and edges to be converted back into a standard indexed formula suitable for direct computation or further algebraic manipulation.