World Library  
Flag as Inappropriate
Email this Article

Scalar field

Article Id: WHEBN0000161879
Reproduction Date:

Title: Scalar field  
Author: World Heritage Encyclopedia
Language: English
Subject: Advection, Higgs boson, Vector field, Line integral, Field (physics)
Collection: Articles Containing Video Clips, Multivariable Calculus, Quantum Field Theory
Publisher: World Heritage Encyclopedia

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. The scalar may either be a mathematical number or a physical quantity. Scalar fields are required to be coordinate-independent, 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 1
  • Uses in physics 2
    • Examples in quantum theory and relativity 2.1
  • Other kinds of fields 3
  • See also 4
  • References 5


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 \sin (2\pi(xy+\sigma)) oscillating as \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. 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 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 supposed to cause the accelerated expansion of the universe (inflation [12]), 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.[13]

Other kinds of fields

See also


  1. ^  
  2. ^ Hazewinkel, Michiel, ed. (2001), "Scalar",  
  3. ^ Hazewinkel, Michiel, ed. (2001), "Scalar field",  
  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.  
  6. ^ P. Jordan Schwerkraft und Weltall, Vieweg (Braunschweig) 1955.
  7. ^ C. Brans and R. Dicke; Phys. Rev. 124(3): 925, 1961.
  8. ^ A. Zee; Phys. Rev. Lett. 42(7): 417, 1979.
  9. ^ H. Dehnen et al.; Int. J. of Theor. Phys. 31(1): 109, 1992.
  10. ^ H. Dehnen and H. Frommmert, Int. J. of theor. Phys. 30(7): 987, 1991.
  11. ^ C.H. Brans; "The Roots of scalar-tensor theory", arXiv:gr-qc/0506063v1, June 2005.
  12. ^ A. Guth; Phys. Rev. D23: 347, 1981.
  13. ^ J.L. Cervantes-Cota and H. Dehnen; Phys. Rev. D51, 395, 1995.
This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.

Copyright © World Library Foundation. All rights reserved. eBooks from Project Gutenberg are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.