In mathematics and physics , a scalar field or scalar-value function associates a scalar value for every point in space —possibly physical space . A scalar can be either a ( dimensionless ) mathematical number or a physical quantity . In a physical context, the scalar field needs to be independent of the choice of reference frame, meaning that any two observers using the same units will be at the same absolute point in space (or spacetime) on the value of the scalar fields. will agree.) regardless of their respective points of origin. Examples used in physics include temperature distributions throughout space, pressure distributions in liquids , and spin-zero quantum fields, such as the Higgs field . These fields are the subject of scalar fields theory .
Definition
According to mathematics, a scalar fields on the field U is a real or complex value function or distribution on U. [1] [2] The field U may be a set in some Euclidean space , Minkowski space , or more generally a subset of manifolds , and it is typical in mathematics to impose further conditions on the field, such as whether it is continuous or often differentiable continuously for some sequence . A scalar field is a tensor field of zero order , [3]And the term “scalar field” can be used to distinguish such a function with the more general tensor field, density or differential form .
The oscillations increase as the scalar field of Red represents positive values, purple represents negative values, and sky blue represents values close to zero.\sin (2\pi(xy+\sigma))\sigma
Physically, a scalar field is additionally distinguished by having units of measurement associated with it. In this context, a scalar field must 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 point in physical space. Should be. Scalar fields are contrasted with other physical quantities, such as vector fields , which relate a vector to every point on a field, as well as tensor fields and spinner fields . [ citation needed ] More precisely, the scalar field is oftenContrasted with pseudoscalar regions.
Use in physics
In physics, scalar fields often describe the potential energy associated with a particular force . Force is a vector field , which can be obtained as a factor of the gradient of the potential energy scalar field . examples include:
- In electrostatics , scalar fields which are more familiar descriptions of forces, such as Newton’s potential fields, gravitational potentials , or electric potentials .
- A temperature , humidity, or pressure area, such as that used in meteorology.
Examples in Quantum Theory and Relativity
- In quantum field theory, a scalar fields is associated with spin 0 particles. The scalar fields can be a real or a complex value. Complex scalar fields represent charged particles. These include the Higgs field of the Standard Model, as well as the charged pions that mediate the strong nuclear interaction. [4]
- In the Standard Model of elementary particles, a scalar Higgs fields is used to give leptons and giant vector bosons their masses, through a combination of the Yukawa interaction and spontaneous symmetry breaking. This mechanism is known as the Higgs system. [5] A candidate for the Higgs boson was first detected at CERN in 2012.
- In the Scalar Theories of Gravity the Scalar Field Gravitational Field is described.
- Scalar-tensor theories represent gravitational interactions through both a tensor and a scalar one. Such attempts are for example in the form of generalizations of the Jordan principle [6] Kaluza–Klein theory and Brauns–Dicke theory. [7]
- Scalar fields such as the Higgs fields can be found within scalar-tensor theories, using the Higgs fields of the Standard Model as the scalar field. [8] [9] This field exerts gravitational and Yukawa-like (short-range) interactions with the particles that gain mass through it. [10]
- Scalar fields are found within superstring theories such as Deleton fields, breaking the string’s analogous symmetry, though balancing the quantum anomalies of this tensor. [11 1]
- Scalar fields are thought to be the cause of the highly accelerated expansion of the early universe (inflation), [12] helping to solve the horizon problem and giving a hypothetical reason for the non-vanishing cosmological constant of cosmology. In this context, large-scale (i.e. long-range) scalar regions are known as inflatons. Large-scale (i.e. short-range) scalar fields are also proposed, for example using Higgs-like fields. [13]
Other types of fields
- Some examples of vector fields, which link a vector to every point in space, include the electromagnetic field and wind flow (wind) in meteorology.
- Tensor field, which associates a tensor at every point in space. For example, gravity in general relativity is associated with a tensor field called the Einstein tensor. In the Kaluza–Klein theory, space is extended to five dimensions and its Riemann curvature tensor can be separated out into ordinary four-dimensional gravity with an additional set, which is equivalent to Maxwell’s equation for electromagnetic fields, with With an additional scalar field known as ” dilton “. [ citation needed ] (The deleton scalar in string theory is also found in massless bosonic regions.)
