Scalar field

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A scalar field such as temperature or pressure, where intensity of the field is represented by different hues of colors. Scalar field.png
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 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.

Contents

Definition

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.

The scalar field of oscillating as 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 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.

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:

Examples in quantum theory and 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. [8] [9] This field interacts gravitationally and Yukawa-like (short-ranged) with the particles that get mass through it. [10]

Other kinds of fields

See also

Related Research Articles

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No-hair theorem Theorem related to black holes

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Supersymmetry Symmetry between bosons and fermions

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Top quark Type of quark

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Higgs mechanism

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 Z0 bosons actually have relatively large masses of around 80 GeV/c2. 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.

Hierarchy problem

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False vacuum decay A hypothetical vacuum, less stable than true vacuum

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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 Englert Belgian theoretical physicist

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Christopher T. Hill

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.

Higgs boson Elementary particle related to the Higgs field giving particles mass

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.

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