13.21.2 Tensor Contraction Repeated Index Notation

Tensor Contraction Repeated Index Notation is the specific notational device within tensor contraction notation whereby a single index symbol is written twice in the same term, once as a superscript and once as a subscript, to signal that the corresponding pair of tensor slots is to be summed over rather than treated as two independent free positions.

Definition

Repeated index notation asserts that whenever a symbol such as $$a$$ appears exactly twice within a single term, once upper and once lower, the term stands for the sum over all admissible values of that symbol:

where $$n$$ is the dimension of the space the index ranges over.

Rules Governing Repetition

Exactly Two Occurrences

A symbol used in this repeated fashion must occur exactly twice within a single term; a symbol occurring only once is a free index, and a symbol occurring three or more times is not covered by the convention and signals a notational error, as identified by the index balance check.

One Upper and One Lower

The two occurrences of a repeated symbol must be of opposite variance, one upper and one lower; a symbol appearing twice with the same variance does not represent a valid contraction under this notation, matching the requirement enforced by the pair validity check.

Scope Limited to a Single Term

The repetition convention applies within a single multiplicative term; the same symbol may be reused as a repeated pair in a separate, unrelated term elsewhere in a larger expression without conflict, since each term's summation is independent of any other term's.

Distinguishing Repeated Indices From Free Indices

Visual Identification

A repeated index can be identified by scanning a term for any symbol occurring twice; every other symbol occurring exactly once is a free index and is not summed, remaining in the notation for the final resulting tensor.

Renaming Freedom

Because a repeated index is a dummy variable standing only for the act of summation, it may be renamed to any unused symbol without altering the value of the term, provided the new symbol does not collide with another index already in use within that term.

Diagram

Position Within Notation

Repeated index notation is the specific mechanism underlying the Einstein summation convention described within the broader tensor contraction notation, and it is the notational feature that all downstream procedures, including simplification, cost analysis, and verification, must correctly parse before any of their respective operations can be meaningfully applied to a given expression.