9.4.3 Tensor Product Basis Factor Labeling

Tensor Product Basis Factor Labeling is the practice of tagging each position of a tensor product basis element, and each entry of its multi index address, with an explicit marker identifying which factor space that position belongs to, so that positions can be told apart even when several factors are copies of the same underlying space and would otherwise carry indistinguishable indices; it supplies the layer of bookkeeping that keeps the origin of every index attached to it permanently, regardless of how many factors are combined or how similar those factors are to one another.

The Problem Factor Labeling Solves

Repeated Factors Produce Repeated Index Ranges

When a tensor product is formed from several copies of the same space, each factor contributes indices drawn from the identical range, for instance 1 through n in every copy. Without further marking, an index value alone cannot say which copy it came from, since the same numeral could equally well belong to the first factor or the third.

A Bare Multi Index Is Not Always Enough

A plain multi index address distinguishes positions by their order within the tuple, but once that tuple is manipulated, reordered, or referenced outside its original context, the positional information can be lost. Tensor product basis factor labeling attaches an explicit, portable tag to each entry so that the entry continues to announce its factor of origin independently of its position in any particular tuple.

How Factor Labeling Is Carried Out

Superscripted or Subscripted Factor Tags

A common labeling convention attaches a tag directly to the index symbol, such as writing i^{(1)} for an index belonging to the first factor and i^{(2)} for an index belonging to the second, so that the basis element itself carries the labeling explicitly.

Named Factor Slots

An alternative form of factor labeling assigns each factor a distinct name rather than a numeral, for example labeling the factors A and B and writing the corresponding basis element as e_i^{A} ⊗ e_j^{B}, which is useful when the factors play conceptually distinct roles rather than being interchangeable copies of one space.

Distinguishing Factor Labeling From Ordering Alone

Order Encodes Position, Labeling Encodes Identity

Ordering a tensor product basis element by writing its factors left to right encodes only where in the sequence a given index sits. Factor labeling encodes something different and additional: which specific factor space that index is drawn from, a fact that remains true even if the tensor product is later rewritten with its factors permuted.

Both Are Needed Together in General

In practice, a fully specified tensor product basis element uses ordering to fix a definite sequence of positions and factor labeling to fix the identity of the space attached to each position, so that the two conventions operate together rather than substituting for one another.

Diagram of Factor Labeling

Consequences of Consistent Factor Labeling

Contractions Only Act Within a Labeled Factor

When a contraction or index operation is intended to act on one particular factor of a tensor product, factor labeling makes clear which entries of the multi index address are eligible, preventing an operation meant for one factor from being applied inadvertently to an index belonging to a different, identically ranged factor.

Symmetrization Across Identical Factors Becomes Well Defined

When identical factors are permuted to express symmetry or antisymmetry of a tensor under exchange of those factors, factor labeling is what makes the permutation meaningful: the operation is defined as an exchange of the labels attached to two positions, and without those labels there would be nothing distinguishing one arrangement from another to permute in the first place.