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.
A ''scalar field'' is a Function from ''n'' to . That is, it is a function defined on the ''n''- Dimension al 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 .
A on a C''k''- Manifold is a C''k'' function to the real numbers. Taking '''R'''''n'' 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.
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.
- In Quantum Field Theory , a Scalar Field is associated with spin 0 particles, such as Meson s or Boson s. 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 .
- 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 P. Jordanm ''Schwerkraft und Weltall'', Vieweg (Braunschweig) 1955. as a generalization of the Kaluza-Klein Theory and the Brans-Dicke Theory C. Brans and R. Dicke; ''Phis. Rev. (3): 925'', 1961..
- Scalar fields like the Higgs field can be found within scalar-tensor theories, using as scalar field the Higgs field of the -like (short-ranged) with the particles that get mass through it H. Dehnen and H. Frommmert, ''Int. J. of theor. Phys. (7): 987'', 1991..
- 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 C.H. Brans; "The Roots of scalar-tensor theory", arXiv:gr-qc/0506063v1, June 2005..
- Scalar fields are supposed to cause the accelerated expansion of the universe (inflation A. Guth; ''Pys. Rev. : 346'', 1981.), 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 Inflaton s. Massive (i.e. short-ranged) scalar fields are proposed, too, using for example Higgs-like fields (e.g. J.L. Cervantes-Cota and H. Dehnen; ''Phys. Rev. '''D51''', 395'', 1995.).
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