In a physical context, scalar fields are required to be independent of the choice of reference frame. That is, 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.
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.^{[
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:
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]}
^Technically, pions are actually examples of
pseudoscalar mesons, which fail to be invariant under spatial inversion, but are otherwise invariant under Lorentz transformations.