6.18.4 Tensor Scalar Field Membership

Tensor Scalar Field Membership is the recognition that a type (0,0) tensor is, in its concrete numerical realization, simply an element of the underlying field over which the vector space V is defined, such as the real numbers or the complex numbers, and that this field membership is what equips the space of scalars with its own algebraic structure of addition, multiplication, and multiplicative inverses, independent of any tensorial considerations. This membership is the algebraic starting point from which the entire tensor algebra is built, since every vector space is, by definition, a structure organized over a chosen field, and every tensor space is, in turn, a vector space over that same field.

The Field Underlying the Vector Space

Fields as the Scalars of Linear Algebra

A vector space V is always defined relative to a field F, whose elements are called scalars in the ordinary sense of linear algebra, and which supply the coefficients used in linear combinations of vectors, v = c_1 e_1 + ... + c_n e_n. The type (0,0) tensors on V are precisely the elements of this same field F, so scalar field membership identifies the tensorial notion of a scalar with the pre-existing algebraic notion of a scalar from linear algebra.

Field Axioms Satisfied by Type Zero Zero Tensors

Because type (0,0) tensors are literally elements of F, they automatically satisfy all of the field axioms: closure and associativity of addition and multiplication, existence of additive and multiplicative identities 0 and 1, existence of additive inverses for every element and multiplicative inverses for every nonzero element, and distributivity of multiplication over addition. No additional verification is required beyond confirming that F itself is a field, since the type (0,0) tensors do not form a separate structure layered on top of F, but coincide with F exactly.

Field Membership and the Vector Space of Type Zero Zero Tensors

A One-Dimensional Vector Space Over Itself

Viewed as a tensor space, the collection of all type (0,0) tensors forms a one-dimensional vector space over F, with F itself serving simultaneously as the space of scalars and as the space being scaled, since scaling a type (0,0) tensor c by a field element a produces ac, another type (0,0) tensor. This self-referential structure is unique to the zero order case; no other tensor type coincides with its own scalar field in this way.

Compatibility with Scalar Multiplication of All Tensor Types

Scalar field membership guarantees that the operation of multiplying any type (p, q) tensor by a type (0,0) scalar is always well defined and satisfies the usual distributive and associative laws expected of scalar multiplication, precisely because the scalars performing the multiplication are drawn from the same field F that already governs linear combinations within V and within every tensor product space built from V and V*.

Field Membership Across Different Choices of F

Real Versus Complex Scalars

When F is the field of real numbers, type (0,0) tensors are ordinary real numbers, and every tensor construction, transformation law, and invariant discussed throughout tensor algebra takes values in the reals. When F is instead the field of complex numbers, type (0,0) tensors are complex numbers, and constructions such as Hermitian bilinear forms, which require complex conjugation, become available in a way that has no direct analogue over the reals.

Consequences for Tensor Constructions

The choice of field affects which operations make sense on tensors of every type, not just scalars: a positive-definite inner product, for instance, requires an ordering on the field to make sense of the statement T(v,v) > 0, a condition available over the reals but not meaningful over the complex numbers without first taking a modulus, illustrating how the algebraic properties of the field to which the scalars belong ripple outward into every higher tensor construction.

Diagram of the Field as the Base of the Tensor Hierarchy

Practical Implications of Field Membership

Ensuring Well-Defined Arithmetic on Invariants

Recognizing that scalar tensors belong to a field guarantees that any arithmetic combination of scalar invariants, such as adding two traces or dividing one determinant by another nonzero determinant, produces another valid element of the same field, and therefore another valid type (0,0) tensor, without any risk of leaving the space of legitimate scalar values.

Field Membership as a Prerequisite for Tensor Algebra

Since every definition in tensor algebra, from the transformation law to the tensor product, assumes an ambient field F supplying the scalars, confirming scalar field membership is an implicit prerequisite underlying the entire subject: without a well-defined field, none of the vector space or tensor space constructions built on top of it would be meaningful.