Scalar field: Definition from Answers.com

In mathematics and physics, a scalar field associates a scalar value to every point in a space. The scalar may either be a mathematical number, or a physical quantity. Scalar fields are often used in physics, for instance to indicate the temperature distribution throughout space, or the air pressure.

1 Definition
2 Uses in physics
- 2.1 Examples in quantum theory and relativity
3 Other kinds of fields
4 See also
5 References
6 External links

Definition

Mathematically, a scalar field on a region U is a real or complex-valued function on U. The region U may be a set in some Euclidean space, or more generally a subset of a manifold, and it is typical to impose further conditions on the field, such that it be continuous or often continuously differentiable to some order. In a mathematical context, 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, and tensor fields. 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 the gradient of the potential energy scalar field. Examples include:

Potential fields, such as the Newtonian gravitational potential field for gravitation, 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. 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 generates a projection of a vector field onto a scalar field.

Examples in quantum theory and relativity

In quantum field theory, a scalar field is associated with spin 0 particles, such as mesons or bosons. 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, as well as the pion field mediating the strong nuclear interaction.
In the Standard Model of elementary particles, a scalar field is used to give the leptons their mass, via a combination of the Yukawa interaction and the spontaneous symmetry breaking. This mechanism is known as the Higgs mechanism ^[1]. This supposes the existence of a (still hypothetical) spin 0 particle called Higgs boson.
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 symmetry 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 as inflatons. Massive (i.e. short-ranged) scalar fields are proposed, too, using for example Higgs-like fields (e.g. ^[9]).

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. For example, in general relativity gravitation 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".

References

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

External links

Optical X wave communications, Jian-yu Lu and Shiping He

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