Scalar field

In mathematics and physics, a scalar field associates a scalar value, which can be either mathematical in definition, or physical, 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. In mathematics, or more specifically, differential geometry, the set of functions defined on a manifold define the commutative ring of functions.

Just as the concept of a scalar in mathematics is identical to the concept of a scalar in physics, so also the scalar field defined in differential geometry is identical to, in the abstract, to the (unquantized) scalar fields of physics.

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.

Differential geometry

See main article differential form.

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. The set of all scalar fields on a manifold forms a commutative ring, under the natural operations of multiplication and addition, point by point.

Uses in physics

In physics, scalar fields can be used to ascribe forces (which are usually vector fields) to a more general scalar field, the gradient of which describes the force.

Potential fields, such as the Newtonian gravitational potential field for gravitation, or the electric potential in electrostatics, are scalar fields which describes 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 are 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. 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.