13.16.3 Tensor Contraction Diagram Open Leg

Tensor Contraction Diagram Open Leg is a line attached to a node in a contraction diagram that does not connect to any other leg, and instead terminates freely, representing an index of the corresponding tensor that remains uncontracted and therefore appears as a free index in the algebraic expression the diagram represents.

Definition

If a node representing a tensor $$T$$ has a leg corresponding to index $$i$$ that is not joined by an edge to any other leg, that leg is an open leg, and $$i$$ is a free index of the expression represented by the whole diagram:

Role of Open Legs

Determining Result Order

The total number of open legs across an entire diagram, summed over all nodes, equals the order of the tensor represented by the diagram as a whole. A diagram with three open legs represents a rank-3 tensor, regardless of how many internal edges or nodes it contains.

Determining Result Variance

Each open leg retains the variance, upper or lower, of the index it represents. The collection of open legs therefore determines both the order and the type pair $$(p,q)$$ of the resulting tensor, with $$p$$ equal to the number of open upper legs and $$q$$ equal to the number of open lower legs.

Naming Convention

Open legs are typically labeled with the same index symbol used in the corresponding algebraic expression, so that the diagram and the indexed notation can be read interchangeably. Internal edges, by contrast, are often left unlabeled or labeled only for clarity, since their associated index is a dummy variable summed over and not part of the final expression's free index list.

Distinguishing Open Legs From Internal Edges

Structural Test

A leg is open if and only if it has exactly one endpoint attached to a node; an internal edge has two such endpoints, one at each end of the connecting line, possibly both on the same node in the case of a self-loop.

Consequence for Summation

No summation convention applies to an open leg, since summation under the Einstein convention is only implied by a repeated index appearing once up and once down within the same term, which corresponds precisely to an internal edge, not an open one.

Example Diagram

In this diagram, the single connecting edge is an internal edge representing one contraction, while the three lines labeled $$i$$, $$j$$, and $$k$$ are open legs, so the diagram represents a rank-3 tensor with free indices $$i$$, $$j$$, and $$k$$.

Special Case: No Open Legs

A diagram with zero open legs, in which every node's every leg is joined by an internal edge, represents a scalar quantity, since a tensor of order zero has no free indices to display as legs.