# Scalar field

A scalar field such as temperature or pressure, where intensity of the field is represented by different hues of color.

In mathematics and physics, a scalar field associates a scalar value to every point in a space – possibly physical space. The scalar may either be a (dimensionless) mathematical number or a physical quantity. In a physical context, scalar fields are required to be independent of the choice of reference frame, meaning that any two observers using the same units will agree on the value of the scalar field at the same absolute point in space (or spacetime) regardless of their respective points of origin. Examples used in physics include the temperature distribution throughout space, the pressure distribution in a fluid, and spin-zero quantum fields, such as the Higgs field. These fields are the subject of scalar field theory.

## Definition

Mathematically, a scalar field on a region U is a real or complex-valued function or distribution on U.[1][2] The region U may be a set in some Euclidean space, Minkowski space, or more generally a subset of a manifold, and it is typical in mathematics to impose further conditions on the field, such that it be continuous or often continuously differentiable to some order. A scalar field is a tensor field of order zero,[3] and the term "scalar field" may be used to distinguish a function of this kind with a more general tensor field, density, or differential form.

The scalar field of ${\displaystyle \sin(2\pi (xy+\sigma ))}$  oscillating as ${\displaystyle \sigma }$  increases. Red represents positive values, purple represents negative values, and sky blue represents values close to zero.

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, as well as tensor fields and spinor fields.[citation needed] More subtly, scalar fields are often contrasted with pseudoscalar fields.

## 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 a factor of the gradient of the potential energy scalar field. Examples include:

### Examples in quantum theory and relativity

• Scalar fields like the Higgs field can be found within scalar-tensor theories, using as scalar field the Higgs field of the Standard Model.[8][9] This field interacts gravitationally and Yukawa-like (short-ranged) with the particles that get mass through it.[10]
• 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.[11]
• Scalar fields are hypothesized to have caused the high 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. 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.[13]

## References

1. ^ Apostol, Tom (1969). Calculus. II (2nd ed.). Wiley.
2. ^ Hazewinkel, Michiel, ed. (2001) [1994], "Scalar", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
3. ^ Hazewinkel, Michiel, ed. (2001) [1994], "Scalar field", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
4. ^ Technically, pions are actually examples of pseudoscalar mesons, which fail to be invariant under spatial inversion, but are otherwise invariant under Lorentz transformations.
5. ^ P.W. Higgs (Oct 1964). "Broken Symmetries and the Masses of Gauge Bosons". Phys. Rev. Lett. 13 (16): 508. Bibcode:1964PhRvL..13..508H. doi:10.1103/PhysRevLett.13.508.
6. ^ Jordan, P. (1955). Schwerkraft und Weltall. Braunschweig: Vieweg.
7. ^ Brans, C.; Dicke, R. (1961). "Mach's Principle and a Relativistic Theory of Gravitation". Phys. Rev. 124 (3): 925. Bibcode:1961PhRv..124..925B. doi:10.1103/PhysRev.124.925.
8. ^ Zee, A. (1979). "Broken-Symmetric Theory of Gravity". Phys. Rev. Lett. 42 (7): 417. Bibcode:1979PhRvL..42..417Z. doi:10.1103/PhysRevLett.42.417.
9. ^ Dehnen, H.; Frommert, H.; Ghaboussi, F. (1992). "Higgs field and a new scalar-tensor theory of gravity". Int. J. Theor. Phys. 31 (1): 109. Bibcode:1992IJTP...31..109D. doi:10.1007/BF00674344.
10. ^ Dehnen, H.; Frommmert, H. (1991). "Higgs-field gravity within the standard model". Int. J. Theor. Phys. 30 (7): 985–998 [p. 987]. Bibcode:1991IJTP...30..985D. doi:10.1007/BF00673991.
11. ^ Brans, C. H. (2005). "The Roots of scalar-tensor theory". arXiv:gr-qc/0506063. Bibcode:2005gr.qc.....6063B.
12. ^ Guth, A. (1981). "Inflationary universe: A possible solution to the horizon and flatness problems". Phys. Rev. D. 23: 347. Bibcode:1981PhRvD..23..347G. doi:10.1103/PhysRevD.23.347.
13. ^ Cervantes-Cota, J. L.; Dehnen, H. (1995). "Induced gravity inflation in the SU(5) GUT". Phys. Rev. D. 51: 395. arXiv:astro-ph/9412032. Bibcode:1995PhRvD..51..395C. doi:10.1103/PhysRevD.51.395.