# Scalar field

In mathematics and physics, a scalar field associates a scalar value, which can be either mathematical in definition, or physical, to every point in space. Scalar fields are often used in physics, for instance to indicate the temperature distribution throughout space, or the air pressure. In mathematics, or more specifically, differential geometry, the set of functions defined on a manifold define the commutative ring of functions.

Just as the concept of a scalar in mathematics is identical to the concept of a scalar in physics, so also the scalar field defined in differential geometry is identical to, in the abstract, to the (unquantized) scalar fields of physics.

## Definition

A scalar field is a function from Rn to R. That is, it is a function defined on the n-dimensional Euclidean space with real values. Often it is required to be continuous, or one or more times differentiable, that is, a function of class Ck.

The scalar field can be visualized as a n-dimensional space with a real or complex number attached to each point in the space.

The derivative of a scalar field results in a vector field called the gradient.

## Differential geometry

See main article differential form.

A scalar field on a Ck-manifold is a Ck function to the real numbers. Taking Rn as manifold gives back the special case of vector calculus.

A scalar field is also a 0-form. The set of all scalar fields on a manifold forms a commutative ring, under the natural operations of multiplication and addition, point by point.

## Uses in physics

In physics, scalar fields can be used to ascribe forces (which are usually vector fields) to a more general scalar field, the gradient of which describes the force.

### Examples in quantum theory and relativity

• Scalar fields like the Higgs field can be found within scalar-tensor theories, using as scalar field the Higgs field of the Standard Model , . This field interacts gravitatively and Yukawa-like (short-ranged) with the particles that get mass through it .
• Scalar fields are found within superstring theories as dilaton fields, breaking the conformal symmetry of the string, though balancing the quantum anomalies of this tensor .
• Scalar fields are supposed to cause the accelerated expansion of the universe (inflation ), helping to solve the horizon problem and giving an hypothetical reason for the non-vanishing cosmological constant of cosmology. Massless (i.e. long-ranged) scalar fields in this context are known are inflatons. Massive (i.e. short-ranged) scalar fields are proposed, too, using for example Higgs-like fields (e.g. ).

## Other kinds of fields 