Scalar field

From Free net encyclopedia

This article is about scalar values fields in the sense of quantum field theory. For the field of scalars of a vector space, see scalar (mathematics).

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.

1 Definition
2 Examples found in physics
3 Other kinds of fields
4 Differential geometry
5 References

[edit]

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.

The scalar field can be visualized as a n-dimensional space with a real or complex number attached to each point in the space.

The derivative of a scalar field results in a vector field called the gradient.

[edit]

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)).

[edit]

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".

[edit]

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.

A scalar field is also a 0-form. See differential forms.

[edit]

References

P.W. Higgs; Phys. Rev. Lett. 13(16): 508, Oct. 1964.
P. Jordanm 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.de:Skalarfeld