Scalar field
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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.
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[edit] 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.
[edit] 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 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.
[edit] 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].
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- 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]).
[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. 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".
[edit] See also
[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.
