8.11.2 Tensor Explicit Index Range

Tensor Explicit Index Range specifies the exact index positions in tensors, clarifying component access in multi-dimensional algebra.

Tensor Explicit Index Range is the pair of stated bounds, a starting value and an ending value, attached directly to an explicit summation symbol, specifying precisely which values a summation index sweeps through, in direct contrast to the implicit range assumed silently under the Einstein summation convention.

Components of an Explicit Range

Lower and Upper Bound

An explicit index range consists of two values written directly beneath and above the summation symbol: the lower bound, giving the first value the index takes, and the upper bound, giving the last value the index takes, with the index assumed to step through every integer value in between.

\sum_{i = 1}^{n} A_{i}

Fully Specified, No Inference Required

Because both bounds are stated directly, an explicit index range requires no external context to interpret; a reader does not need to know the dimension of the ambient space or infer any convention, since the range is entirely self-contained within the notation itself.

Non-Standard Ranges

Ranges Not Matching the Ambient Dimension

An explicit index range is particularly valuable when the summation is not intended to span the full dimension of the space, such as when only a subset of components should be included in the sum, a situation the implicit Einstein convention cannot express on its own.

\sum_{i = 2}^{3} A_{i} B^{i}

This range restricts the sum to only the second and third components, deliberately excluding the first, a restriction that would be impossible to express using the standard implicit convention alone.

Ranges Starting from Zero

Some conventions, particularly those involving spacetime indices, use an explicit range beginning at zero rather than one, and stating this starting point explicitly avoids any ambiguity about whether a temporal component is included at the start of the range.

\sum_{μ = 0}^{3} A_{μ} B^{μ}

Range Consistency Requirements

Matching Ranges for the Same Index

If the same summation index appears in more than one place within a longer expression, its explicit range must be stated consistently everywhere it occurs, since a mismatch in stated bounds for what is meant to be the same index would create an internal contradiction.

Range Independence Between Separate Indices

Different summation indices within the same expression may carry entirely different explicit ranges from one another, each governed independently by its own stated bounds, without any requirement that the ranges coincide.

\sum_{i = 1}^{3} \sum_{μ = 0}^{3} A_{i μ}

Practical Illustration

Stating an explicit index range removes any dependence on assumed context or dimensional convention, making it the notation of choice whenever a summation must be restricted to a specific subset of values, begin at a nonstandard starting point, or otherwise deviate from the range that would be automatically inferred under the implicit Einstein convention.