13.16.1 Tensor Contraction Diagram Node

Tensor Contraction Diagram Node is the basic graphical unit within a contraction diagram that represents a single tensor, drawn as a shape from which a fixed number of lines emerge, each line corresponding to one index of that tensor, whether that index is free or destined to be contracted.

Definition

A diagram node associated to a tensor $$T$$ of type $$(p,q)$$ is a labeled point in the plane together with exactly $$p+q$$ lines, called legs, attached to it. The node itself carries no numerical content beyond its label; all quantitative information about $$T$$ is understood to reside in the tensor the node names, not in the drawing.

Anatomy of a Node

Label

Every node is marked with a symbol identifying which tensor it represents. Distinct nodes in the same diagram must carry distinct labels unless they are explicitly intended to denote the same tensor appearing twice.

Legs

The legs attached to a node are partitioned into upper legs, corresponding to contravariant indices, and lower legs, corresponding to covariant indices. A common convention draws upper legs emerging from the top of the node and lower legs from the bottom, though any consistent visual distinction, such as arrow direction or line style, is equally valid.

Valence

The total number of legs attached to a node is called its valence, and it equals the order of the tensor the node represents. A node of valence zero represents a scalar and is drawn with no legs at all.

Node States Within a Diagram

Unconnected Node

A node all of whose legs remain open, unconnected to any other node or to itself, represents a tensor that has not yet participated in any contraction within the diagram.

Partially Connected Node

A node with some legs open and others joined to edges represents a tensor that retains some free indices after one or more of its indices have been contracted against another tensor or against itself.

Fully Internal Node

A node all of whose legs are joined to edges, whether to other nodes or via self-loops, represents a tensor that contributes no free indices to the overall result; its entire content is absorbed into internal summations.

Node Diagram

Composition of Multiple Nodes

A complete contraction diagram consists of one node per tensor factor in the algebraic expression it represents, joined by edges wherever a contraction pairs one node's leg with another node's leg, or with one of its own legs. The node is therefore the atomic building block from which all larger diagram representations are assembled, and every property of the overall diagram, such as total order or number of contractions, can be derived by examining the valence and connection pattern of its constituent nodes.