1.2.8 Scalar Field Definition

Scalar Field Definition is the characterization of a scalar field, in the algebraic sense relevant to tensor theory, as a set of elements called scalars, equipped with addition and multiplication operations satisfying the field axioms, over which a vector space is defined and from which the coefficients used in linear combinations, bases, and tensor components are drawn. It fixes the algebraic setting that underlies every vector space, dual space, and tensor construction built on top of it, since the properties of the scalar field determine what kinds of coefficients and components are available throughout tensor algebra.

The Field Axioms

A field is a set equipped with two operations, conventionally called addition and multiplication, satisfying a specific list of axioms: addition and multiplication are each associative and commutative, multiplication distributes over addition, there exist distinct identity elements for addition and multiplication, every element has an additive inverse, and every nonzero element has a multiplicative inverse. These axioms guarantee that the four basic arithmetic operations — addition, subtraction, multiplication, and division by a nonzero element — are all well defined and behave in the familiar way within the field.

The expression above states the distributive law, one of the defining axioms that a set of scalars, together with its two operations, must satisfy to qualify as a field.

Role of the Scalar Field in Vector Spaces

A vector space is always defined over a specific scalar field: the operation of scalar multiplication, which combines a vector with a scalar to produce another vector, draws its scalars from this field, and the field's arithmetic determines how linear combinations of vectors behave. Changing the scalar field over which a vector space is defined, even while keeping the underlying set of vectors superficially similar, can produce a mathematically distinct structure with different properties, since the available coefficients for linear combinations change accordingly.

Because bases, coordinates, linear independence, and linear maps are all defined relative to a chosen scalar field, every one of these foundational concepts inherits its algebraic character directly from the properties of that field.

Common Choices of Scalar Field

The Real Numbers

The field of real numbers is the most common choice of scalar field in applied mathematics and physics, since it supports the notions of order, magnitude, and continuity needed to model physical quantities such as length, time, and mass. Vector spaces over the real numbers underlie most treatments of tensor algebra as applied to classical and relativistic physics.

The Complex Numbers

The field of complex numbers is essential wherever a theory requires solutions to polynomial equations that have no real roots, or where the mathematical structure of quantum mechanics is involved, since the state spaces of quantum systems are vector spaces over the complex numbers, and tensors built from these spaces inherit complex-valued components.

Finite Fields

Finite fields, containing only a finite number of elements, arise in algebra, coding theory, and cryptography. Vector spaces and tensor constructions over finite fields obey the same general definitions as those over the real or complex numbers, since the field axioms alone, rather than any assumption of infinitude, are what the theory requires.

Why the Choice of Scalar Field Matters for Tensor Algebra

Every tensor, at the most basic level, is built from a vector space defined over some scalar field, and the components of a tensor, once a basis is chosen, are themselves elements of that field. The tensor transformation law involves multiplying and adding these components using the field's operations, so the validity of every construction in tensor algebra — the tensor product, contraction, the classification of tensors by type — depends on the scalar field satisfying the field axioms precisely as stated. This is why the scalar field is treated as one of the first pieces of foundational apparatus specified before any of the more elaborate machinery of tensor algebra can be developed.