Editing Scalar field

{{Short description|Assignment of numbers to points in space}}
{{About|associating a scalar value with every point in a space|the set whose members are [[Scalar (mathematics)|scalars]]|Field (mathematics){{!}}field}}
{{Use American English|date = March 2019}}

[[File:Scalar field.png|thumb|right|A scalar field such as temperature or pressure, where intensity of the field is represented by different hues of colors.]]

In [[mathematics]] and [[physics]], a '''scalar field''' is a [[function (mathematics)|function]] associating a single{{Dubious|date=May 2025}} [[number]] to each [[point (geometry)|point]] in a [[region (mathematics)|region]] of [[space (mathematics)|space]] – possibly [[physical space]]. The scalar may either be a pure [[Scalar (mathematics)|mathematical number]] ([[dimensionless]]) or a [[scalar (physics)|scalar physical quantity]] (with [[unit of measurement|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]]) regardless of their respective points of origin.  Examples used in physics include the [[temperature]] distribution throughout space, the [[pressure]] distribution in a fluid, and [[Spin (physics)|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 (mathematical analysis)|region]] ''U'' is a [[real-valued function|real]] or [[complex-valued function]] or [[distribution (mathematics)|distribution]] on ''U''.<ref>{{cite book |first=Tom |last=Apostol |author-link=Tom Apostol |title=Calculus  
   |volume=II |publisher=Wiley |year=1969 |edition=2nd }}</ref><ref>{{springer|title=Scalar|id=s/s083240}}</ref> The region ''U'' may be a set in some [[Euclidean space]], [[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 function|continuous]] or often [[continuously differentiable]] to some order. A scalar field is a [[tensor field]] of order zero,<ref>{{springer|id=s/s083260|title=Scalar field}}</ref> and the term "scalar field" may be used to distinguish a function of this kind with a more general tensor field, [[density bundle|density]], or [[differential form]].

[[File:Scalar Field.ogv|thumb|The scalar field of <math>\sin (2\pi(xy+\sigma))</math> oscillating as <math>\sigma</math> increases. Red represents positive values, purple represents negative values, and sky blue represents values close to zero.]]

Physically, a scalar field is additionally distinguished by having [[units of measurement]] 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 [[observer (special relativity)|observer]]s 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 field]]s, which associate a [[Euclidean vector|vector]] to every point of a region, as well as [[tensor field]]s and [[spinor|spinor fields]].{{Citation needed|date=June 2012}} More subtly, scalar fields are often contrasted with [[pseudoscalar]] fields.

== Uses in physics ==
In physics, scalar fields often describe the [[potential energy]] associated with a particular [[force]].  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]] in [[electrostatics]], are scalar fields which describe the more familiar forces.
* A [[temperature]], [[humidity]], or [[pressure]] field, such as those used in [[meteorology]].

=== Examples in quantum theory and relativity ===
* In [[quantum field theory]], a [[Bosonic field|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]], as well as the charged [[pions]] mediating the [[strong nuclear interaction]].<ref>Technically, pions are actually examples of [[pseudoscalar meson]]s, which fail to be invariant under spatial inversion, but are otherwise invariant under Lorentz transformations.</ref>
* In the [[Standard Model]] of elementary particles, a scalar [[Higgs field]] is used to give the [[lepton]]s and [[W and Z bosons|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]].<ref>{{cite journal|author=P.W. Higgs|journal=Phys. Rev. Lett.|volume=13|issue=16|pages=508–509|date=Oct 1964|title=Broken Symmetries and the Masses of Gauge Bosons|doi=10.1103/PhysRevLett.13.508|bibcode = 1964PhRvL..13..508H |doi-access=free}}</ref> A candidate for the [[Higgs boson]] was first detected at CERN in 2012.
* In [[scalar theories of gravitation]] scalar fields are used to describe the gravitational field.
* [[Scalar–tensor theory|Scalar–tensor theories]] represent the gravitational interaction through both a tensor and a scalar. Such attempts are for example the [[Pascual Jordan|Jordan]] theory<ref>{{cite book |first=P. |last=Jordan |title=Schwerkraft und Weltall |publisher=Vieweg |location=Braunschweig |year=1955 |url=https://books.google.com/books?id=snJTcgAACAAJ }}</ref> as a generalization of the [[Kaluza–Klein theory]] and the [[Brans–Dicke theory]].<ref>{{cite journal |first1=C. |last1=Brans |first2=R. |last2=Dicke |title=Mach's Principle and a Relativistic Theory of Gravitation |journal=Phys. Rev. |volume=124 |issue=3 |pages=925 |year=1961 |doi=10.1103/PhysRev.124.925 |bibcode=1961PhRv..124..925B }}</ref>
** Scalar fields like the Higgs field can be found within scalar–tensor theories, using as scalar field the Higgs field of the [[Standard Model]].<ref>{{cite journal |first=A. |last=Zee |title=Broken-Symmetric Theory of Gravity |journal=Phys. Rev. Lett. |volume=42 |issue=7 |pages=417–421 |year=1979 |doi=10.1103/PhysRevLett.42.417 |bibcode=1979PhRvL..42..417Z }}</ref><ref>{{cite journal |first1=H. |last1=Dehnen |first2=H. |last2=Frommert |first3=F. |last3=Ghaboussi |title=Higgs field and a new scalar–tensor theory of gravity |journal=Int. J. Theor. Phys. |volume=31 |issue=1 |pages=109 |year=1992 |doi=10.1007/BF00674344 |bibcode=1992IJTP...31..109D |s2cid=121308053 }}</ref> This field interacts gravitationally and [[Yukawa interaction|Yukawa]]-like (short-ranged) with the particles that get mass through it.<ref>{{cite journal |first1=H. |last1=Dehnen |first2=H. |last2=Frommmert |title=Higgs-field gravity within the standard model |journal=Int. J. Theor. Phys. |volume=30 |issue=7 |pages=985–998 [p. 987] |year=1991 |doi=10.1007/BF00673991 |bibcode=1991IJTP...30..985D |s2cid=120164928 }}</ref>
* 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.<ref>{{cite arXiv |first=C. H. |last=Brans |title=The Roots of scalar–tensor theory |eprint=gr-qc/0506063 |year=2005 }}</ref>
* Scalar fields are hypothesized to have caused the high accelerated expansion of the early universe ([[Inflation (cosmology)|inflation]]),<ref>{{cite journal |first=A. |last=Guth |title=Inflationary universe: A possible solution to the horizon and flatness problems |journal=Phys. Rev. D |volume=23 |pages=347–356 |year=1981 |issue=2 |doi=10.1103/PhysRevD.23.347 |bibcode=1981PhRvD..23..347G |doi-access=free }}</ref> 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 [[inflaton]]s. Massive (i.e. short-ranged) scalar fields are proposed, too, using for example Higgs-like fields.<ref>{{cite journal |first1=J. L. |last1=Cervantes-Cota |first2=H. |last2=Dehnen |title=Induced gravity inflation in the SU(5) GUT |journal=Phys. Rev. D |volume=51 |pages=395–404 |year=1995 |issue=2 |doi=10.1103/PhysRevD.51.395 |pmid=10018493 |arxiv=astro-ph/9412032 |bibcode=1995PhRvD..51..395C |s2cid=11077875 }}</ref>

== Other kinds of fields ==
* [[Vector field]]s, which associate a [[vector (geometry)|vector]] to every point in space. Some examples of [[vector field]]s include the air flow ([[wind]]) in meteorology.<!-- the Newtonian gravitational field.{{dubious|date=May 2020|reason=Newtonian gravitation described as scalar in prior §; the writer may have intended the [[Post-Newtonian expansion]]s developed from General Relativity.}} -->
* [[Tensor field]]s, which associate a [[tensor]] to every point in space. For example, in [[general relativity]] 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 [[dimension|four-dimensional]] gravitation plus an extra set, which is equivalent to [[Maxwell's equations]] for the [[electromagnetic field]], plus an extra scalar field known as the "[[dilaton]]".{{citation needed|date=June 2012}} (The [[dilaton]] scalar is also found among the massless bosonic fields in [[string theory]].)

== See also ==
* [[Scalar field theory]]
* [[Vector boson]]
* [[Vector-valued function]]

== References ==
{{reflist|30em}}

{{Authority control}}

{{DEFAULTSORT:Scalar Field}}
[[Category:Multivariable calculus]]
[[Category:Articles containing video clips]]
[[Category:Scalar physical quantities|Field]]
[[Category:Functions and mappings]]