13.16.4 Tensor Contraction Diagram Closed Loop

A closed loop in tensor contraction diagrams visually represents the cyclic summation of tensor indices, simplifying complex algebraic operations in tensor calculus.

Tensor Contraction Diagram Closed Loop is a cycle of edges within a contraction diagram in which a sequence of nodes is connected end to end so that the last node's edge returns to the first node, leaving no open legs anywhere along the cycle, and representing an expression whose value is a scalar produced entirely by internal summation with no free indices remaining.

Definition

A closed loop is a sequence of nodes $T_{1}, T_{2}, \dots, T_{m}$ together with edges joining $T_{1}$ to $T_{2}$ , $T_{2}$ to $T_{3}$ , and so forth, up to an edge joining $T_{m}$ back to $T_{1}$ , such that every leg used in the cycle is consumed by exactly one of these edges:

T_{1}^{a} T_{2}^{b} \dots T_{m}^{a}

with the pattern of index repetitions matching a full cycle through all the nodes involved.

Simplest Case: One-Node Loop

Self-Loop as Minimal Closed Loop

The smallest possible closed loop involves a single node with two of its own legs joined directly to one another, corresponding to a trace-like contraction such as $T_{a}^{a}$ . This is the base case from which longer closed loops are built by inserting additional nodes into the cycle.

General Closed Loop Behavior

No Free Indices Within the Loop

Every leg belonging to a node that lies entirely within a closed loop is consumed by an edge of that loop. Consequently, a diagram consisting purely of one closed loop, with no additional open legs anywhere, represents a scalar quantity.

result order = 0

Loop Length

The number of nodes participating in a closed loop is called its length. A closed loop of length $m$ contains exactly $m$ edges, since each node contributes one outgoing edge to the next node in the cycle.

Coexistence With Open Legs

A diagram may contain a closed loop as one component while other nodes, not part of the loop, retain open legs. In that case, the closed loop contributes a scalar multiplicative factor to the tensor represented by the remaining open legs.

Diagram of a Three-Node Closed Loop

This diagram represents an expression of the form $A_{i}^{j} B_{j}^{k} C_{k}^{i}$ , equivalent to the trace of the product of three matrices.

Significance

Closed loops identify, purely from diagram topology, which portions of a larger contraction expression collapse to scalar factors, without needing to track the algebraic index pattern by hand. Recognizing a closed loop within a larger diagram is equivalent to recognizing a trace-like sub-expression embedded inside a longer chain of tensor contractions.