mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

and

physics Physics is the scientific study of matter, its Elementary particle, 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 whi ...

, a scalar field is a function associating a single

number A number is a mathematical object used to count, measure, and label. The most basic 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 can ...

to each point in a

region In geography, regions, otherwise referred to as areas, zones, lands or territories, are portions of the Earth's surface that are broadly divided by physical characteristics (physical geography), human impact characteristics (human geography), and ...

space Space is a three-dimensional continuum containing positions and directions. In classical physics, physical space is often conceived in three linear dimensions. Modern physicists usually consider it, with time, to be part of a boundless ...

– 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. That is, 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, also called the space-time continuum, is a mathematical model that fuses the three dimensions of space and the one dimension of time into a single four-dimensional continuum. Spacetime diagrams are useful in visualiz ...

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

temperature Temperature is a physical quantity that quantitatively expresses the attribute of hotness or coldness. Temperature is measurement, measured with a thermometer. It reflects the average kinetic energy of the vibrating and colliding atoms making ...

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 and eve ...

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

''U'' is a real or complex-valued function or distribution on ''U''. The region ''U'' may be a set in some

Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...

, Minkowski space, 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 ratio of a substance's mass to its volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' (or ''d'') can also be u ...

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

units of measurement A unit of measurement, or unit of measure, is a definite magnitude (mathematics), 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 qua ...

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 of an object or system due to the body's position relative to other objects, or the configuration of its particles. The energy is equal to the work done against any restoring forces, such as gravity ...

associated with a particular

force In physics, a force is an influence that can cause an Physical object, object to change its velocity unless counterbalanced by other forces. In mechanics, force makes ideas like 'pushing' or 'pulling' mathematically precise. Because the Magnitu ...

. The force is a vector field, which can be obtained as a factor of the gradient of the potential energy scalar field. Examples include: * Potential fields, such as the Newtonian gravitational potential, or the

electric potential Electric potential (also called the ''electric field potential'', potential drop, the electrostatic potential) is defined as electric potential energy per unit of electric charge. More precisely, electric potential is the amount of work (physic ...

electrostatics Electrostatics is a branch of physics that studies slow-moving or stationary electric charges. Since classical antiquity, classical times, it has been known that some materials, such as amber, attract lightweight particles after triboelectric e ...

, are scalar fields which describe the more familiar forces. * A

humidity Humidity is the concentration of water vapor present in the air. Water vapor, the gaseous state of water, is generally invisible to the human eye. Humidity indicates the likelihood for precipitation (meteorology), precipitation, dew, or fog t ...

, or

field, such as those used in

meteorology Meteorology is the scientific study of the Earth's atmosphere and short-term atmospheric phenomena (i.e. weather), with a focus on weather forecasting. It has applications in the military, aviation, energy production, transport, agricultur ...

Examples in quantum theory and relativity

* In

quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines Field theory (physics), field theory and the principle of relativity with ideas behind quantum mechanics. QFT is used in particle physics to construct phy ...

, 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 Scientific theory, theory describing three of the four known fundamental forces (electromagnetism, electromagnetic, weak interaction, weak and strong interactions – excluding gravity) in the unive ...

, as well as the charged

pions In particle physics, a pion (, ) or pi meson, denoted with the Greek letter pi (), is any of three subatomic particles: , , and . Each pion consists of a quark and an antiquark and is therefore a meson. Pions are the lightest mesons and, mo ...

mediating the strong nuclear interaction. * 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 In the Standard Model of particle physics, the Higgs mechanism is essential to explain the Mass generation, generation mechanism of the property "mass" for gauge bosons. Without the Higgs mechanism, all bosons (one of the two classes of particles ...

. A candidate for the

Higgs boson The Higgs boson, sometimes called the Higgs particle, is an elementary particle in the Standard Model of particle physics produced by the excited state, quantum excitation of the Higgs field, one of the field (physics), fields in particl ...

was first detected at CERN in 2012. * In scalar theories of gravitation 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, officially the Hashemite Kingdom of Jordan, is a country in the Southern Levant region of West Asia. Jordan is bordered by Syria to the north, Iraq to the east, Saudi Arabia to the south, and Israel and the occupied Palestinian ter ...

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-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 hypothesized to have caused the high accelerated expansion of the early universe (

inflation In economics, inflation is an increase in the average price of goods and services in terms of money. This increase is measured using a price index, typically a consumer price index (CPI). When the general price level rises, each unit of curre ...

), 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 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 associated with a vector space. Tensors may map between different objects such as vectors, scalars, and even other ...

to every point in space. For example, in

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

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 Georg Friedrich Bernhard Riemann (; ; 17September 182620July 1866) was a German mathematician who made profound contributions to mathematical analysis, analysis, number theory, and differential geometry. In the field of real analysis, he is mos ...

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, Electrical network, electr ...

for the electromagnetic field, plus an extra scalar field known as the " dilaton". (The dilaton scalar is also found among the massless bosonic fields in string theory.)

References

{{DEFAULTSORT:Scalar Field Multivariable calculus Articles containing video clips Field Functions and mappings

Definition

Uses in physics

Examples in quantum theory and relativity

Other kinds of fields

See also

References