Scalar field

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Scalar field

In mathematics

Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
and physics

Physics

Physics is the natural science which examines basic concepts such as energy, force, and spacetime and all that derives from these, such as mass, charge, matter and its Motion ....
, a scalar field associates a scalar value, which can be either mathematical

Scalar (mathematics)

In linear algebra, real numbers are called scalars and relate to vectors in a vector space through the operation of scalar multiplication, in which a vector can be multiplied by a number to produce another vector....
in definition, or physical

Scalar (physics)

In physics, a scalar is a simple physical quantity that is not changed by coordinate system rotations or translations , or by Lorentz transformations or space-time translations ....
, to every point in space. Scalar fields

Scalar field theory

In theoretical physics, scalar field theory can refer to a Classical field theory or Quantum field theory of scalar fields.Such a field is distinguished by its invariance under a Lorentz transformation, hence the name "scalar", in contrast to a vector field or tensor field....
are often used in physics, for instance to indicate the temperature

Temperature

In physics, temperature is a physical property of a Physical system that underlies the common notions of hot and cold; something that feels hotter generally has the greater temperature....
distribution throughout space, or the air pressure

Pressure

Pressure is the force per unit area applied to an object in a direction surface normal to the surface. Gauge pressure is the pressure relative to the local atmospheric or ambient pressure....
. In mathematics, or more specifically, differential geometry, the set of function

Function (mathematics)

The mathematical concept of a function expresses dependence between two quantities, one of which is known and the other which is produced. A function associates a single output to each input element drawn from a fixed Set , such as the real numbers , although different inputs may have the same output....
s defined on a manifold

Manifold

In mathematics, more specifically topology, a manifold is a topological space in which every point has a neighborhood which "resembles" Euclidean space....
define the commutative ring

Commutative ring

In ring theory, a branch of abstract algebra, a commutative ring is a Ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra....
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.

>scalar field is a function

Function (mathematics)

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Encyclopedia

In mathematics

Mathematics

Physics

Scalar (mathematics)

Scalar (physics)

Scalar field theory

Temperature

Pressure

Function (mathematics)

Manifold

In mathematics, more specifically topology, a manifold is a topological space in which every point has a neighborhood which "resembles" Euclidean space....
define the commutative ring

Commutative ring

Definition

A scalar field is a function

Function (mathematics)

Dimension

In mathematics, the dimension of a space is roughly defined as the minimum number of coordinates needed to specify every point within it. For example: a point on the unit circle in the plane can be specified by two Cartesian coordinates but one can make do with a single coordinate , so the circle is 1-dimensional even though it exists in...
al Euclidean space

Euclidean space

Around 300 Before Christ, the Ancient Greece mathematician Euclid undertook a study of relationships among distances and angles, first in a plane and then in space....
with real

Real number

In mathematics, the real numbers may be described informally in several different ways. The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of two; or, a real number can be given by an infinite decimal representation, such as 2.4871773339...., where the digits co...
values. Often it is required to be continuous

Continuous function

In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. Otherwise, a function is said to be discontinuous....
, or one or more times differentiable, that is, a function of class C^k

Smooth function

In mathematical analysis, a differentiability class is a classification of function according to the properties of their derivatives. Higher order differentiability classes correspond to the existence of more derivatives....
.

The scalar field can be visualized as a n-dimensional space with a real or complex number

Complex number

In mathematics, the complex numbers are an extension of the real numbers obtained by adjoining an imaginary unit, denoted i, which satisfies:...
attached to each point in the space.

The derivative

Derivative

In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much a quantity is changing at a given point....
of a scalar field results in a vector field

Vector field

In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a Euclidean space.Vector fields are often used in physics to model, for example, the speed and direction of a moving fluid throughout space, or the strength and direction of some force, such as the magnetic field or gravity for...
called the gradient

Gradient

In vector calculus, the gradient of a scalar field is a vector field which points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change....
.

Differential geometry

See main article differential form
Differential form

In the mathematics fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates....
.

A scalar field on a C^k-manifold

Manifold

In mathematics, more specifically topology, a manifold is a topological space in which every point has a neighborhood which "resembles" Euclidean space....
is a C^k function to the real numbers. Taking Rⁿ as manifold gives back the special case of vector calculus

Vector calculus

Vector calculus is a branch of mathematics concerned with derivative and integral of vector fields. The term "vector calculus" is sometimes used as a synonym for the broader subject of multivariable calculus, which includes vector calculus as well as partial derivative and multiple integral....
.

A scalar field is also a 0-form. The set of all scalar fields on a manifold forms a commutative ring

Commutative ring

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.

Potential fields, such as the Newtonian gravitational potential field for gravitation, or the electric potential
Electric potential

At a point in space, the electric potential is the potential energy per unit of electric charge that is associated with a static electric field....
in electrostatics
Electrostatics

Electrostatics is the branch of science that deals with the phenomena arising from stationary or slowly moving electric charges.Since classical antiquity it was known that some materials such as amber attract light particles after Triboelectric effect....
, are scalar fields which describes the more familiar forces.
A temperature
Temperature

In physics, temperature is a physical property of a Physical system that underlies the common notions of hot and cold; something that feels hotter generally has the greater temperature....
, humidity
Humidity

Humidity is the amount of water vapor in the air. In daily language the term "humidity" is normally taken to mean relative humidity. Relative humidity is defined as the ratio of the partial pressure of water vapor in a Air parcel of air to the saturated vapor pressure of water vapor at a prescribed temperature....
or pressure
Pressure

Pressure is the force per unit area applied to an object in a direction surface normal to the surface. Gauge pressure is the pressure relative to the local atmospheric or ambient pressure....
field, such as those used in meteorology
Meteorology

Meteorology is the interdisciplinary scientific study of the Earth's atmosphere that focuses on weather processes and forecasting . Studies in the field stretch back millennia, though significant progress in meteorology did not occur until the eighteenth century....
. Note that when modeling weather on a global basis, the surface of the Earth is not flat, and thus the general language of curvature in differential geometry plays a role. Dopplerized weather radar
Weather radar

A weather radar is a type of radar used to locate precipitation , calculate its motion, estimate its type , and weather forecasting its future position and intensity....
generates a projection of a vector field onto a scalar field.

Examples in quantum theory and relativity

In quantum field theory
Quantum field theory

Quantum field theory or QFT provides a theoretical framework for constructing quantum mechanics models of systems classically described by field or of Many-body problem....
, a scalar field is associated with spin 0 particles, such as meson
Meson

In particle physics, mesons are subatomic particles composed of one quark and one antiquark. They are part of the hadron particle family ? particles made of quarks....
s or boson
Boson

In particle physics, bosons are subatomic particle which obey Bose-Einstein statistics; they are named after Satyendra Nath Bose and Albert Einstein....
s. The scalar field may be real or complex valued (depending on whether it will associate a real or complex number to every point of space-time). Complex scalar fields represent charged particles. These include the Higgs field of the Standard Model
Standard Model

The Standard Model of particle physics is a theory of three of the four known fundamental interactions and the elementary particles that take part in these interactions....
, as well as the pion
Pion

In particle physics, a pion is any of three subatomic particles: , and . Pions are the lightest mesons and play an important role in explaining low-energy properties of the strong nuclear force....
field mediating the strong nuclear interaction.
In the Standard Model
Standard Model

The Standard Model of particle physics is a theory of three of the four known fundamental interactions and the elementary particles that take part in these interactions....
of elementary particles, a scalar field is used to give the lepton
Lepton

Leptons are a family of elementary particles, alongside quarks and gauge bosons . Like quarks, leptons are fermions and are subject to the electromagnetic force, the gravitational force, and weak interaction....
s their mass, via a combination of the Yukawa interaction
Yukawa interaction

In particle physics, Yukawa's interaction, named after Hideki Yukawa, is an interaction between a scalar field and a Dirac field of the type...
and the spontaneous symmetry breaking
Spontaneous symmetry breaking

In physics, spontaneous symmetry breaking occurs when a system that is symmetry in physics with respect to some symmetry group goes into a vacuum state that is not symmetric....
. This mechanism is known as the Higgs mechanism
Higgs mechanism

In quantum field theory, the Higgs mechanism is a way that the massless gauge bosons in a gauge theory get a mass by interacting with a background Higgs field....
. This supposes the existence of a (still hypothetical) spin 0 particle called Higgs boson
Higgs boson

In particle physics, the Higgs boson is a massive Scalar field theory elementary particle predicted to exist by the Standard Model.The Higgs boson is the only Standard Model particle that has not yet been observed....
.
In scalar theories of gravitation
Scalar theories of gravitation

Scalar theories of gravitation are field theories of gravitation in which the gravitational field is described using a Scalar field theory, which is required to satisfy some field equation....
scalar fields are used to describe the gravitational field.
scalar-tensor theories
Scalar-tensor theory

Scalar-tensor theories are theories that include a Scalar field theory as well as a tensor field to represent an interaction, especially the gravitational one....
represent the gravitational interaction through both a tensor and a scalar. Such attempts are for example the Jordan
Jordan

Jordan , officially the Hashemite Kingdom of Jordan, is an Arab country in Southwest Asia spanning the southern part of the Syrian Desert down to the Gulf of Aqaba....
theory as a generalization of the Kaluza-Klein theory and the Brans-Dicke theory
Brans-Dicke theory

In theoretical physics, the Brans-Dicke theory of gravitation is a theoretical framework to explain gravitation. It is a well-known competitor of Albert Einstein's more popular theory of general 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
Standard Model

The Standard Model of particle physics is a theory of three of the four known fundamental interactions and the elementary particles that take part in these interactions....
, . This field interacts gravitatively and Yukawa
Yukawa interaction

In particle physics, Yukawa's interaction, named after Hideki Yukawa, is an interaction between a scalar field and a Dirac field of the type...
-like (short-ranged) with the particles that get mass through it .

Scalar fields are found within superstring theories as dilaton
Dilaton

Dilaton is a hypothetical particle that appears in Kaluza-Klein theory and string theory....
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
Horizon problem

The horizon problem is a problem with the Friedmann-Lema?tre-Robertson-Walker of the Big Bang which was identified in the 1970s. It points out that different regions of the universe have not "contacted" each other due to the great distances between them, but nevertheless they have the same temperature and other physical properties....
and giving an hypothetical reason for the non-vanishing cosmological constant
Cosmological constant

In physical cosmology, the cosmological constant was proposed by Albert Einstein as a modification of his original theory of general relativity to achieve a Einstein's universe....
of cosmology. Massless (i.e. long-ranged) scalar fields in this context are known are inflaton
Inflaton

The inflaton is the generic name of the unidentified Scalar field theory that may be responsible for an episode of cosmic inflation in the very early universe....
s. Massive (i.e. short-ranged) scalar fields are proposed, too, using for example Higgs-like fields (e.g. ).

Other kinds of fields

Vector field
Vector field

In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a Euclidean space.Vector fields are often used in physics to model, for example, the speed and direction of a moving fluid throughout space, or the strength and direction of some force, such as the magnetic field or gravity for...
s, which associate a vector to every point in space. Some examples of vector field
Vector field

In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a Euclidean space.Vector fields are often used in physics to model, for example, the speed and direction of a moving fluid throughout space, or the strength and direction of some force, such as the magnetic field or gravity for...
s include the electromagnetic field
Electromagnetic field

The electromagnetic field is a physical field produced by electric charge. It affects the behavior of charged objects in the vicinity of the field....
and the Newtonian gravitational field.
Tensor field
Tensor field

In mathematics, physics and engineering, a tensor field is a very general concept of variable geometric quantity. It is used in differential geometry and the theory of manifolds, in algebraic geometry, in general relativity, in the analysis of stress and strain tensor in materials, and in numerous applications in the physical sciences and en...
s, which associate a tensor
Tensor

A tensor is an object which extends the notion of Scalar , Vector , and Matrix . The term has slightly different meanings in mathematics and physics....
to every point in space. For example, in general relativity
General relativity

General relativity or the general theory of relativity is the Geometry Theoretical physics of gravitation published by Albert Einstein in 1916....
gravitation is associated with a tensor field (in particular, with the Riemann curvature tensor
Riemann curvature tensor

In the mathematical field of differential geometry, the Riemann curvature tensor or Riemann?Christoffel tensor is the most standard way to express curvature of Riemannian manifolds....
). In Kaluza-Klein theory, spacetime is extended to five dimensions and its Riemann curvature tensor can be separated out into ordinary four-dimensional
Dimension

In mathematics, the dimension of a space is roughly defined as the minimum number of coordinates needed to specify every point within it. For example: a point on the unit circle in the plane can be specified by two Cartesian coordinates but one can make do with a single coordinate , so the circle is 1-dimensional even though it exists in...
gravitation plus an extra set, which is equivalent to Maxwell's equations
Maxwell's equations

In electromagnetism, James Clerk Maxwell equations are a set of four partial differential equations that describe the properties of the electric field and magnetic field fields and relate them to their sources, charge density and current density....
for the electromagnetic field
Electromagnetic field

The electromagnetic field is a physical field produced by electric charge. It affects the behavior of charged objects in the vicinity of the field....
, plus an extra scalar field known as the "dilaton
Dilaton

Dilaton is a hypothetical particle that appears in Kaluza-Klein theory and string theory....
".

Scalar field

Definition

Differential geometry

Uses in physics

Examples in quantum theory and relativity

Other kinds of fields

See also

External links