7.6.2 Tensor Component Field Membership

Tensor Component Field Membership is the requirement that every scalar value entry of a tensor's component array belong to a single, fixed field, such as the real or complex numbers, so that the arithmetic operations used to combine components, addition, multiplication, and scalar scaling, are guaranteed to be well defined throughout the tensor's value structure.

Definition and Scope

The Underlying Field of a Tensor

A tensor is defined over a vector space that is itself built on a field (F), and every component of the tensor, once a basis is chosen, is an element of that same field:

with (F) commonly taken to be the real numbers (\mathbb{R}) or the complex numbers (\mathbb{C}), though the general theory of tensors over vector spaces permits any field satisfying the algebraic axioms a field requires.

Uniformity Across the Whole Array

Field membership applies uniformly to every entry: it is not permitted for some components of a tensor to lie in one field and others in a different one, since the operations defining a tensor's algebraic structure, linear combination, contraction, and the tensor product, presuppose that all component values can be added and multiplied together within a common field.

Structural Properties

Consequences of the Choice of Field

The specific field chosen affects which operations and identities are available. Over the complex numbers, every component may carry a nonzero imaginary part, and additional structures such as complex conjugation and Hermitian symmetry, distinct from ordinary symmetry, become meaningful; over the real numbers, such structures collapse to their ordinary counterparts, since every real number equals its own conjugate.

Field Membership and Ordering

Some properties meaningful for real-valued components, such as one entry being greater than another, or a tensor's entries all being nonnegative, depend on the field's order structure and have no direct analogue over fields, like the complex numbers, that lack a compatible total order. Field membership therefore determines in advance which comparative statements about a tensor's value structure are even meaningful to ask.

Compatibility in Combined Operations

Two tensors can only be added, or have one contracted against the other, when their components share the same field membership; a tensor with real-valued components and a tensor with complex-valued components cannot be combined directly without first embedding the real-valued tensor into the complex field, a standard but necessary preliminary step.

Role Within Tensor Algebra

Foundation for Well-Defined Operations

Field membership is what guarantees, at the most basic level, that the sums and products appearing in the definitions of tensor addition, scalar multiplication, and contraction always produce a result that itself lies in the same field, so that repeated application of these operations never leaves the field the tensor was originally defined over.

Choosing a Field for an Application

Practical modeling decisions, such as representing rotations using real-valued rotation tensors versus representing wave phenomena using complex-valued tensors carrying phase information, are, at bottom, decisions about which field the components should have membership in, made based on which arithmetic properties, ordering, conjugation, or none of these, the application requires.