13.16.2 Tensor Contraction Diagram Edge

A Tensor Contraction Diagram Edge visually represents tensor index contraction, simplifying complex algebraic operations through graphical notation.

Tensor Contraction Diagram Edge is a line segment in a contraction diagram that joins two legs, one bearing an upper index and one bearing a lower index, and represents the implicit summation of a single contracted index pair, connecting either two distinct nodes or two legs of the same node.

Definition

An edge in a contraction diagram is a connection between exactly two legs, denoted:

edge = {leg (A, i), leg (B, j)}

where one of the two legs carries an upper index and the other carries a lower index. The edge encodes the algebraic identification of those two index positions, standing for the sum:

\sum_{a} A^{\dots a \dots} B_{\dots a \dots}

Properties of an Edge

Variance Requirement

An edge is only valid between one upper leg and one lower leg. Two legs of the same variance cannot be joined by an edge without introducing a metric tensor to raise or lower one of them first, since contraction as such requires a natural pairing between a vector space and its dual.

Endpoint Types

An edge may connect two legs belonging to different nodes, representing a contraction between two distinct tensors, or two legs belonging to the same node, representing a contraction of a tensor against one of its own index pairs, commonly called a self-loop.

Range of Summation

The implicit summation associated with an edge runs over the full dimension of the underlying vector space, so an edge in a diagram drawn for an $n$ -dimensional space represents a sum of $n$ terms.

Edge Types by Configuration

Direct Edge Between Two Nodes

Self-Loop Edge

A self-loop reduces the order of the node's tensor by two on its own, since both indices involved belong to the same tensor.

Counting Edges in a Diagram

Relation to Contractions

The number of edges in a diagram equals the number of independent contractions applied in the corresponding algebraic expression. A diagram with three edges represents an expression with three separate summed index pairs.

Relation to Order Reduction

Each edge reduces the total order of the diagram's represented tensor by two, consistent with the general order reduction behavior of contraction:

order = (total legs) - 2 \times (number of edges)

Independence of Edges

Distinct edges in a well-formed diagram never share an endpoint: each leg belongs to at most one edge, since a single index cannot be contracted against two different partners simultaneously. This ensures that every edge corresponds to a distinct, independently summed dummy index in the algebraic expression the diagram represents.