In mathematics and

physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which rel ...

, a scalar field is a function associating a single

number A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual numbers ...

to every point in a

space Space is the boundless three-dimensional extent in which objects and events have relative position and direction. In classical physics, physical space is often conceived in three linear dimensions, although modern physicists usually con ...

– possibly physical space. The scalar may either be a pure mathematical number ( dimensionless) or a scalar physical quantity (with units). In a physical context, scalar fields are required to be independent of the choice of reference frame, meaning that any two observers using the same units will agree on the value of the scalar field at the same absolute point in space (or

spacetime In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why diffe ...

) regardless of their respective points of origin. Examples used in physics include the

temperature Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measured with a thermometer. Thermometers are calibrated in various temperature scales that historically have relied on ...

distribution throughout space, the

pressure Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country a ...

distribution in a fluid, and spin-zero quantum fields, such as the Higgs field. These fields are the subject of scalar field theory.

Definition

Mathematically, a scalar field on a

region In geography, regions, otherwise referred to as zones, lands or territories, are areas that are broadly divided by physical characteristics (physical geography), human impact characteristics (human geography), and the interaction of humanity and t ...

''U'' is a real or complex-valued function or

distribution Distribution may refer to: Mathematics *Distribution (mathematics), generalized functions used to formulate solutions of partial differential equations *Probability distribution, the probability of a particular value or value range of a varia ...

on ''U''. The region ''U'' may be a set in some Euclidean space,

Minkowski space In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the ...

, or more generally a subset of a manifold, and it is typical in mathematics to impose further conditions on the field, such that it be continuous or often continuously differentiable to some order. A scalar field is a tensor field of order zero, and the term "scalar field" may be used to distinguish a function of this kind with a more general tensor field,

density Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematicall ...

, or differential form. Physically, a scalar field is additionally distinguished by having

units of measurement A unit of measurement is a definite magnitude of a quantity, defined and adopted by convention or by law, that is used as a standard for measurement of the same kind of quantity. Any other quantity of that kind can be expressed as a mul ...

associated with it. In this context, a scalar field should also be independent of the coordinate system used to describe the physical system—that is, any two observers using the same units must agree on the numerical value of a scalar field at any given point of physical space. Scalar fields are contrasted with other physical quantities such as vector fields, which associate a vector to every point of a region, as well as tensor fields and spinor fields. More subtly, scalar fields are often contrasted with pseudoscalar fields.

Uses in physics

In physics, scalar fields often describe the

potential energy In physics, potential energy is the energy held by an object because of its position relative to other objects, stresses within itself, its electric charge, or other factors. Common types of potential energy include the gravitational potentia ...

associated with a particular force. The force is a vector field, which can be obtained as a factor of the

gradient In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gr ...

of the potential energy scalar field. Examples include: *Potential fields, such as the Newtonian gravitational potential, or the electric potential in electrostatics, are scalar fields which describe the more familiar forces. * A

, humidity, or

field, such as those used in

meteorology Meteorology is a branch of the atmospheric sciences (which include atmospheric chemistry and physics) with a major focus on weather forecasting. The study of meteorology dates back millennia, though significant progress in meteorology did no ...

Examples in quantum theory and relativity

* In quantum field theory, a scalar field is associated with spin-0 particles. The scalar field may be real or complex valued. Complex scalar fields represent charged particles. These include the Higgs field of the

Standard Model The Standard Model of particle physics is the theory describing three of the four known fundamental forces ( electromagnetic, weak and strong interactions - excluding gravity) in the universe and classifying all known elementary particles. I ...

, as well as the charged pions mediating the

strong nuclear interaction The nuclear force (or nucleon–nucleon interaction, residual strong force, or, historically, strong nuclear force) is a force that acts between the protons and neutrons of atoms. Neutrons and protons, both nucleons, are affected by the nucle ...

. * In the

of elementary particles, a scalar Higgs field is used to give the leptons and massive vector bosons their mass, via a combination of the Yukawa interaction and the spontaneous symmetry breaking. This mechanism is known as the Higgs mechanism. A candidate for the Higgs boson was first detected at CERN in 2012. * In

scalar theories of gravitation Scalar theories of gravitation are field theories of gravitation in which the gravitational field is described using a scalar field, which is required to satisfy some field equation. ''Note:'' This article focuses on relativistic classical field t ...

scalar fields are used to describe the gravitational field. * Scalar–tensor theories represent the gravitational interaction through both a tensor and a scalar. Such attempts are for example the

Jordan Jordan ( ar, الأردن; tr. ' ), officially the Hashemite Kingdom of Jordan,; tr. ' is a country in Western Asia. It is situated at the crossroads of Asia, Africa, and Europe, within the Levant region, on the East Bank of the Jordan Ri ...

theory as a generalization of the Kaluza–Klein theory and the Brans–Dicke theory. :* Scalar fields like the Higgs field can be found within scalar–tensor theories, using as scalar field the Higgs field of the

. This field interacts gravitationally and

Yukawa Yukawa (written: 湯川) is a Japanese surname, but is also applied to proper nouns. People * Diana Yukawa (born 1985), Anglo-Japanese solo violinist. She has had two solo albums with BMG Japan, one of which opened to #1 * Hideki Yukawa (1907–1 ...

-like (short-ranged) with the particles that get mass through it. * Scalar fields are found within superstring theories as

dilaton In particle physics, the hypothetical dilaton particle is a particle of a scalar field \varphi that appears in theories with Dimension (mathematics and physics)#Additional dimensions, extra dimensions when the volume of the compactified dimensions ...

fields, breaking the conformal symmetry of the string, though balancing the quantum anomalies of this tensor. * Scalar fields are hypothesized to have caused the high accelerated expansion of the early universe (

inflation In economics, inflation is an increase in the general price level of goods and services in an economy. When the general price level rises, each unit of currency buys fewer goods and services; consequently, inflation corresponds to a reductio ...

), helping to solve the horizon problem and giving a hypothetical reason for the non-vanishing cosmological constant of cosmology. Massless (i.e. long-ranged) scalar fields in this context are known as inflatons. Massive (i.e. short-ranged) scalar fields are proposed, too, using for example Higgs-like fields.

Other kinds of fields

* Vector fields, which associate a vector to every point in space. Some examples of vector fields include the

electromagnetic field An electromagnetic field (also EM field or EMF) is a classical (i.e. non-quantum) field produced by (stationary or moving) electric charges. It is the field described by classical electrodynamics (a classical field theory) and is the classica ...

and air flow ( wind) in meteorology. * Tensor fields, which associate a

tensor In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tens ...

to every point in space. For example, in

general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics. ...

gravitation is associated with the tensor field called Einstein tensor. In Kaluza–Klein theory, spacetime is extended to five dimensions and its Riemann curvature tensor can be separated out into ordinary four-dimensional gravitation plus an extra set, which is equivalent to

Maxwell's equations Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits. Th ...

for the

, plus an extra scalar field known as the "

". (The

scalar is also found among the massless bosonic fields in string theory.)

References

{{DEFAULTSORT:Scalar Field Multivariable calculus Articles containing video clips Scalar physical quantities

Definition

Uses in physics

Examples in quantum theory and relativity

Other kinds of fields

See also

References