In mathematics and physics, a scalar field associates a scalar to every point in space. Scalar fields are often used in physics, for instance to indicate the temperature distribution throughout space, or the air pressure. Euclid, detail from The School of Athens by Raphael. ... Physics (from the Greek, φυσικός (physikos), natural, and φύσις (physis), nature) is the science of the natural world dealing with the fundamental constituents of the universe, the forces they exert on one another, and the results produced by these forces. ... The term scalar is used in mathematics, physics, and computing basically for quantities that are characterized by a single numeric value and/or do not involve the concept of direction. ... Physics (from the Greek, φυσικός (physikos), natural, and φύσις (physis), nature) is the science of the natural world dealing with the fundamental constituents of the universe, the forces they exert on one another, and the results produced by these forces. ... Temperature is also the name of a song by Sean Paul. ... Pressure (symbol: p) is the force per unit area acting on a surface in a direction perpendicular to that surface. ...

Definition

A scalar field is a function from Rⁿ to R. That is, it is a function defined on the n-dimensional Euclidean space with real values. Often it is required to be continuous, or one or more times differentiable, that is, a function of class C^k. Partial plot of a function f. ... 2-dimensional renderings (ie. ... In mathematics, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ... In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ... In mathematics, a continuous function is a function in which arbitrarily small changes in the input produce arbitrarily small changes in the output. ... In mathematics, a smooth function is one that is infinitely differentiable, i. ...

The scalar field can be visualized as a n-dimensional space with a real or complex number attached to each point in the space. Wikibooks Algebra has more about this subject: Complex numbers In mathematics, a complex number is an expression of the form where a and b are real numbers, and i is a specific imaginary number, called the imaginary unit, with the property i 2 = −1. ...

The derivative of a scalar field results in a vector field called the gradient. In mathematics, the derivative is defined as the instantaneous rate of change of a function. ... Vector field given by vectors of the form (-y, x) In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a Euclidean space. ... In the above two images, the scalar field is in black and white, black representing higher values, and its corresponding gradient is represented by blue arrows. ...

Examples found in physics

• Potential field like the Newtonian one for gravitation.
• In quantum field theory a scalar field is associated with spin 0 particles, like mesons. 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.
• In the Standard Model of elementary particles a scalar field is used to reproduce the mass, through the so-called symmetry breakdown within the Higgs mechanism ^[1]. This supposes the existence of a (still hypothetical) spin 0 particle called Higgs particle.
• 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 ^[2] as a generalization of the Kaluza-Klein theory and the Brans-Dicke theory ^[3].
• Scalar fields like the Higgs field can be found within scalar-tensor theories, using as scalar field the Higgs field of the Standard Model ^[4], ^[5]. This field interacts gravitatively and Yukawa-like (short-ranged) with the particles that get mass through it ^[6].
• Scalar fields are found within superstring theories as dilaton fields, breaking the conformal lsymmetry of the string, though balancing the quantum anomalies of this tensor ^[7].
• Scalar fields are supposed to cause the accelerated expansion of the universe (inflation ^[8]), helping to solve the horizon problem and giving an hypothetical reason for the non-vanishing cosmological constant of cosmology. Massless (i.e. long-ranged) scalar fields in this context are known are inflatons. Massive (i.e. short-ranged) scalar fields are proposed, too, using for example Higgs-like fields (e.g. ^[9]).

It has been suggested that this article or section be merged with Scalar potential. ... Quantum field theory (QFT) is the application of quantum mechanics to fields. ... It has been suggested that this article or section be merged with scalar field. ... In particle physics, a meson is a strongly interacting boson, that is, it is a hadron with integral spin. ... The Standard Model of Fundamental Particles and Interactions The Standard Model of particle physics is a theory which describes the strong, weak, and electromagnetic fundamental forces, as well as the fundamental particles that make up all matter. ... The Higgs mechanism, originally discovered by the British physicist Peter Higgs (building on a previous suggestion by Philip Anderson in condensed matter physics), is the mechanism that gives masses to all elementary particles in particle physics. ... Scalar theories of gravitation are models of gravitation in which the gravitational field is modelled as arising out of a single scalar value. ... This article is in need of attention from an expert on the subject. ... In physics, Kaluza-Klein theory (or KK theory, for short) is a model which sought to unify the two fundamental forces of gravitation and electromagnetism. ... In mathematical physics, the Brans-Dicke theory of gravitation (sometimes called the Jordan/Brans/Dicke theory) is a well-known competitor of Einsteins theory of general relativity. ... The Standard Model of Fundamental Particles and Interactions The Standard Model of particle physics is a theory which describes the strong, weak, and electromagnetic fundamental forces, as well as the fundamental particles that make up all matter. ... In particle physics, Yukawa interaction, named after Hideki Yukawa, is an interaction between a scalar field and a Dirac field of the type . The Yukawa interaction can be used to describe the strong nuclear force between nucleons (which are fermions), mediated by pions (which are scalar mesons). ... In theoretical physics, dilaton originally referred to a theoretical scalar field; as a photon refers in one sense to the electromagnetic field. ... When we look at the CMB it comes from 15 billion light years away. ... The cosmological constant (usually denoted by the Greek capital letter lambda: Λ) occurs in Einsteins theory of general relativity. ... The inflaton is the generic name of the unidentified scalar field (and its associated particle), that may be responsible for an episode of inflation in the very early universe. ...

Other kinds of fields

• Vector fields, which associate a vector to every point in space. Some examples of vector fields include the electromagnetic field and the Newtonian gravitational field.
• Tensor fields, which associate a tensor to every point in space. In general relativity, gravity is associated with a tensor field. In particular, with the Riemann curvature 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".

Vector field given by vectors of the form (-y, x) In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a Euclidean space. ... In physics and in vector calculus, a spatial vector is a concept characterized by a magnitude, which is a scalar, and a direction (which can be defined in a 3-dimensional space by the Euler angles). ... Vector field given by vectors of the form (-y, x) In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a Euclidean space. ... In mathematics, physics and engineering, a tensor field is a very general concept of variable geometric quantity. ... In mathematics, a tensor is (in an informal sense) a generalized linear quantity or geometrical entity that can be expressed as a multi-dimensional array relative to a choice of basis; however, as an object in and of itself, a tensor is independent of any chosen frame of reference. ... It has been suggested that Einsteins theory of gravitation be merged into this article or section. ... In differential geometry, the Riemann curvature tensor is the most standard way to express curvature of Riemannian manifolds, or more generally, any manifold with an affine connection, torsionless or with torsion. ... In physics, Kaluza-Klein theory (or KK theory, for short) is a model which sought to unify the two fundamental forces of gravitation and electromagnetism. ... 2-dimensional renderings (ie. ... Maxwells equations (sometimes called the Maxwell equations) are the set of four equations, attributed to James Clerk Maxwell, that describe the behavior of both the electric and magnetic fields, as well as their interactions with matter. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... In theoretical physics, dilaton originally referred to a theoretical scalar field; as a photon refers in one sense to the electromagnetic field. ...

Differential geometry

A scalar field on a C^k-manifold is a C^k function to the real numbers. Taking Rⁿ as manifold gives back the special case of vector calculus. On a sphere, the sum of the angles of a triangle is not equal to 180°. A sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. ... Vector calculus is a field of mathematics concerned with multivariate real analysis of vectors in two or more dimensions. ...

A scalar field is also a 0-form. See differential forms. A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. ... A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. ...

References

1. ↑  P.W. Higgs; Phys. Rev. Lett. 13(16): 508, Oct. 1964.
2. ↑  P. Jordanm Schwerkraft und Weltall, Vieweg (Braunschweig) 1955.
3. ↑  C. Brans and R. Dicke; Phis. Rev. 124(3): 925, 1961.
4. ↑  A. Zee; Phys. Rev. Lett. 42(7): 417, 1979.
5. ↑  H. Dehnen et al.; Int. J. of Theor. Phys. 31(1): 109, 1992.
6. ↑  H. Dehnen and H. Frommmert, Int. J. of theor. Phys. 30(7): 987, 1991.
7. ↑  C.H. Brans; "The Roots of scalar-tensor theory", arXiv:gr-qc/0506063v1, June 2005.
8. ↑  A. Guth; Pys. Rev. D23: 346, 1981.
9. ↑  J.L. Cervantes-Cota and H. Dehnen; Phys. Rev. D51, 395, 1995.