13.17 Tensor Contraction Sequence Procedure

Tensor Contraction Sequence Procedure is the ordered set of steps by which multiple contractions are applied one after another to a tensor or product of tensors, specifying at each stage which index pair is contracted next, so that the overall multi-step reduction from an initial tensor expression to a final residual tensor is fully determined and reproducible.

Definition

A contraction sequence procedure on an initial tensor $$T_0$$ is a finite ordered list of contraction operations:

where each $$C_i$$ specifies a distinct index pair to contract within the tensor $$T_{i-1}$$, producing $$T_i$$.

Structural Requirements of a Valid Sequence

Index Availability

At each step $$i$$, the chosen index pair must consist of one upper and one lower index that are both still present in $$T_{i-1}$$. A contraction referencing an index already consumed in an earlier step, or an index of the wrong variance, is not a valid step in the sequence.

Well-Defined Length

A sequence procedure applied to a tensor of initial type $$(p_0,q_0)$$ can contain at most $$\text{min}(p_0,q_0)$$ steps, since each step consumes one upper and one lower index and the sequence cannot continue once either supply is exhausted.

Order-Independence Versus Order-Dependence

Commuting Contractions

When the index pairs chosen at each step are disjoint from one another, meaning no two steps share a common index, the final residual tensor does not depend on the order in which the steps are carried out:

Dependent Steps

When a later step's chosen indices depend on the labeling produced by an earlier step, such as contracting a newly-exposed index that only exists after a prior contraction has been performed, the steps must be executed in the specified order, since the later step is not even defined until the earlier one has been carried out.

Procedure Diagram

Termination and Result

Terminal Tensor

The sequence procedure terminates after its final specified step, yielding a residual tensor $$T_k$$ whose type is determined by subtracting one from each of the upper and lower index counts for every step performed:

Reproducibility

Because each step is fully specified by naming the exact index pair contracted, the entire sequence procedure can be replayed identically on any tensor of matching type, producing consistent, well-defined residual tensors across repeated applications.

Relation to Diagram Representation

A contraction sequence procedure corresponds directly to the successive addition of edges to a contraction diagram, one edge per step, with the final diagram's open legs corresponding exactly to the free indices of the terminal tensor produced by the procedure.