In mathematics and physics, a **scalar field** or **scalar-valued function ** associates a scalar value to every point in a space – possibly physical space. The scalar may either be a (dimensionless) mathematical number or a physical quantity. 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) 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-zero quantum fields, such as the Higgs field. These fields are the subject of scalar field theory.

Mathematically, scalar fields on a region *U* is a real or complex-valued function or distribution on *U*.^{ [1] }^{ [2] } 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 or often continuously differentiable to some order. A scalar field is a tensor field of order zero,^{ [3] } and the term "scalar field" may be used to distinguish a function of this kind with a more general tensor field, density, or differential form.

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 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.^{[ citation needed ]} More subtly, scalar fields are often contrasted with pseudoscalar fields.

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.

- 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, as well as the charged pions mediating the strong nuclear interaction.
^{ [4] } - In the Standard Model 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.
^{ [5] }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 theories represent the gravitational interaction through both a tensor and a scalar. Such attempts are for example the Jordan theory
^{ [6] }as a generalization of the Kaluza–Klein theory and the Brans–Dicke theory.^{ [7] }

- Scalar fields like the Higgs field can be found within scalar-tensor theories, using as scalar field the Higgs field of the Standard Model.
^{ [8] }^{ [9] }This field interacts gravitationally and Yukawa-like (short-ranged) with the particles that get mass through it.^{ [10] }

- Scalar fields like the Higgs field can be found within scalar-tensor theories, using as scalar field the Higgs field of the Standard Model.

- 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.
^{ [11] } - Scalar fields are hypothesized to have caused the high accelerated expansion of the early universe (inflation),
^{ [12] }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.^{ [13] }

- Vector fields, which associate a vector to every point in space. Some examples of vector fields include the electromagnetic field and air flow (wind) in meteorology.
- Tensor fields, 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 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 ]}(The dilaton scalar is also found among the massless bosonic fields in string theory.)

The **Standard Model** of particle physics is the theory describing three of the four known fundamental forces in the universe, as well as classifying all known elementary particles. It was developed in stages throughout the latter half of the 20th century, through the work of many scientists around the world, with the current formulation being finalized in the mid-1970s upon experimental confirmation of the existence of quarks. Since then, confirmation of the top quark (1995), the tau neutrino (2000), and the Higgs boson (2012) have added further credence to the Standard Model. In addition, the Standard Model has predicted various properties of weak neutral currents and the W and Z bosons with great accuracy.

The **no-hair theorem** states that all black hole solutions of the Einstein–Maxwell equations of gravitation and electromagnetism in general relativity can be completely characterized by only three *externally* observable classical parameters: mass, electric charge, and angular momentum. All other information about the matter that formed a black hole or is falling into it "disappears" behind the black-hole event horizon and is therefore permanently inaccessible to external observers. Physicist John Archibald Wheeler expressed this idea with the phrase "black holes have no hair", which was the origin of the name. In a later interview, Wheeler said that Jacob Bekenstein coined this phrase.

In particle physics, **supersymmetry** (**SUSY**) is a conjectured relationship between two basic classes of elementary particles: bosons, which have an integer-valued spin, and fermions, which have a half-integer-valued spin. A type of spacetime symmetry, supersymmetry is a possible candidate for undiscovered particle physics, and seen by some physicists as an elegant solution to many current problems in particle physics if confirmed correct, which could resolve various areas where current theories are believed to be incomplete. A supersymmetrical extension to the Standard Model could resolve major hierarchy problems within gauge theory, by guaranteeing that quadratic divergences of all orders will cancel out in perturbation theory.

The **top quark**, sometimes also referred to as the **truth quark**, is the most massive of all observed elementary particles. It derives its mass from its coupling to the Higgs Boson. This coupling is very close to unity; in the Standard Model of particle physics, it is the largest (strongest) coupling at the scale of the weak interactions and above. The top quark was discovered in 1995 by the CDF and DØ experiments at Fermilab.

In particle physics, the **baryon number** is a strictly conserved additive quantum number of a system. It is defined as

In particle physics, the hypothetical **dilaton** particle is a particle of a scalar field that appears in theories with extra dimensions when the volume of the compactified dimensions varies. It appears as a radion in Kaluza–Klein theory's compactifications of extra dimensions. In Brans–Dicke theory of gravity, Newton's constant is not presumed to be constant but instead 1/*G* is replaced by a scalar field and the associated particle is the dilaton.

In the Standard Model of particle physics, the **Higgs mechanism** is essential to explain the generation mechanism of the property "mass" for gauge bosons. Without the Higgs mechanism, all bosons (one of the two classes of particles, the other being fermions) would be considered massless, but measurements show that the W^{+}, W^{−}, and Z^{0} bosons actually have relatively large masses of around 80 GeV/c^{2}. The Higgs field resolves this conundrum. The simplest description of the mechanism adds a quantum field (the Higgs field) that permeates all space to the Standard Model. Below some extremely high temperature, the field causes spontaneous symmetry breaking during interactions. The breaking of symmetry triggers the Higgs mechanism, causing the bosons it interacts with to have mass. In the Standard Model, the phrase "Higgs mechanism" refers specifically to the generation of masses for the W^{±}, and Z weak gauge bosons through electroweak symmetry breaking. The Large Hadron Collider at CERN announced results consistent with the Higgs particle on 14 March 2013, making it extremely likely that the field, or one like it, exists, and explaining how the Higgs mechanism takes place in nature.

In theoretical physics, the **hierarchy problem** is the large discrepancy between aspects of the weak force and gravity. There is no scientific consensus on why, for example, the weak force is 10^{24} times stronger than gravity.

In quantum field theory, a **false vacuum** is a hypothetical vacuum that is not actively decaying, but somewhat yet not entirely stable ("metastable"). It may last for a very long time in that state, and might eventually move to a more stable state, an event known as **false vacuum decay** or **vacuum metastability event**. The most common suggestion of how such a change might happen is called bubble nucleation – if a small region of the universe by chance reached a more stable vacuum, this "bubble" would spread.

The **Alternative models to the Standard Higgs Model** are models which are considered by many particle physicists to solve some of the Higgs boson's existing problems. Two of the most currently researched models are quantum triviality, and Higgs hierarchy problem.

In theoretical physics, a **scalar–tensor theory** is a field theory that includes both a scalar field and a tensor field to represent a certain interaction. For example, the Brans–Dicke theory of gravitation uses both a scalar field and a tensor field to mediate the gravitational interaction.

In theoretical physics, the **unitarity gauge** or **unitary gauge** is a particular choice of a gauge fixing in a gauge theory with a spontaneous symmetry breaking. In this gauge, the scalar fields responsible for the Higgs mechanism are transformed into a basis in which their Goldstone boson components are set to zero. In other words, the unitarity gauge makes the manifest number of scalar degrees of freedom minimal.

**François, Baron Englert** is a Belgian theoretical physicist and 2013 Nobel prize laureate.

**Christopher T. Hill** is an American theoretical physicist at the Fermi National Accelerator Laboratory who did undergraduate work in physics at M.I.T., and graduate work at Caltech. Hill's Ph.D. thesis, "Higgs Scalars and the Nonleptonic Weak Interactions" (1977) contains one of the first discussions of the two-Higgs-doublet model.

The **Higgs boson** is an elementary particle in the Standard Model of particle physics produced by the quantum excitation of the Higgs field, one of the fields in particle physics theory. In the Standard Model, the Higgs particle is a massive scalar boson with zero spin, no electric charge, and no colour charge. It is also very unstable, decaying into other particles almost immediately.

The **1964 PRL symmetry breaking papers** were written by three teams who proposed related but different approaches to explain how mass could arise in local gauge theories. These three papers were written by

**Bumblebee models** are effective field theories describing a vector field with a vacuum expectation value that spontaneously breaks Lorentz symmetry. A bumblebee model is the simplest case of a theory with spontaneous Lorentz symmetry breaking.

In theoretical physics, a **mass generation** mechanism is a theory that describes the origin of mass from the most fundamental laws of physics. Physicists have proposed a number of models that advocate different views of the origin of mass. The problem is complicated because the primary role of mass is to mediate gravitational interaction between bodies, and no theory of gravitational interaction reconciles with the currently popular Standard Model of particle physics.

The **pressuron** is a hypothetical scalar particle which couples to both gravity and matter theorised in 2013. Although originally postulated without self-interaction potential, the pressuron is also a dark energy candidate when it has such a potential. The pressuron takes its name from the fact that it decouples from matter in pressure-less regimes, allowing the scalar-tensor theory of gravity involving it to pass solar system tests, as well as tests on the equivalence principle, even though it is fundamentally coupled to matter. Such a decoupling mechanism could explain why gravitation seems to be well described by general relativity at present epoch, while it could actually be more complex than that. Because of the way it couples to matter, the pressuron is a special case of the hypothetical string dilaton. Therefore, it is one of the possible solutions to the present non-observation of various signals coming from massless or light scalar fields that are generically predicted in string theory.

**Horndeski's theory** is the most general theory of gravity in four dimensions whose Lagrangian is constructed out of the metric tensor and a scalar field and leads to second order equations of motion. The theory was first proposed by Gregory Horndeski in 1974 and has found numerous applications, particularly in the construction of cosmological models of Inflation and dark energy. Horndeski's theory contains many theories of gravity, including General relativity, Brans-Dicke theory, Quintessence, Dilaton, Chameleon and covariant Galileon as special cases.

- ↑ Apostol, Tom (1969).
*Calculus*.**II**(2nd ed.). Wiley. - ↑ "Scalar",
*Encyclopedia of Mathematics*, EMS Press, 2001 [1994] - ↑ "Scalar field",
*Encyclopedia of Mathematics*, EMS Press, 2001 [1994] - ↑ Technically, pions are actually examples of pseudoscalar mesons, which fail to be invariant under spatial inversion, but are otherwise invariant under Lorentz transformations.
- ↑ P.W. Higgs (Oct 1964). "Broken Symmetries and the Masses of Gauge Bosons".
*Phys. Rev. Lett*.**13**(16): 508–509. Bibcode:1964PhRvL..13..508H. doi: 10.1103/PhysRevLett.13.508 . - ↑ Jordan, P. (1955).
*Schwerkraft und Weltall*. Braunschweig: Vieweg. - ↑ Brans, C.; Dicke, R. (1961). "Mach's Principle and a Relativistic Theory of Gravitation".
*Phys. Rev*.**124**(3): 925. Bibcode:1961PhRv..124..925B. doi:10.1103/PhysRev.124.925. - ↑ Zee, A. (1979). "Broken-Symmetric Theory of Gravity".
*Phys. Rev. Lett*.**42**(7): 417–421. Bibcode:1979PhRvL..42..417Z. doi:10.1103/PhysRevLett.42.417. - ↑ Dehnen, H.; Frommert, H.; Ghaboussi, F. (1992). "Higgs field and a new scalar-tensor theory of gravity".
*Int. J. Theor. Phys*.**31**(1): 109. Bibcode:1992IJTP...31..109D. doi:10.1007/BF00674344. S2CID 121308053. - ↑ Dehnen, H.; Frommmert, H. (1991). "Higgs-field gravity within the standard model".
*Int. J. Theor. Phys*.**30**(7): 985–998 [p. 987]. Bibcode:1991IJTP...30..985D. doi:10.1007/BF00673991. S2CID 120164928. - ↑ Brans, C. H. (2005). "The Roots of scalar-tensor theory". arXiv: gr-qc/0506063 . Bibcode:2005gr.qc.....6063B.Cite journal requires
`|journal=`

(help) - ↑ Guth, A. (1981). "Inflationary universe: A possible solution to the horizon and flatness problems".
*Phys. Rev. D*.**23**(2): 347–356. Bibcode:1981PhRvD..23..347G. doi: 10.1103/PhysRevD.23.347 . - ↑ Cervantes-Cota, J. L.; Dehnen, H. (1995). "Induced gravity inflation in the SU(5) GUT".
*Phys. Rev. D*.**51**(2): 395–404. arXiv: astro-ph/9412032 . Bibcode:1995PhRvD..51..395C. doi:10.1103/PhysRevD.51.395. PMID 10018493. S2CID 11077875.

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