nLab scalar

Idea

Broadly, a scalar quantity is a “basic form of quantity”, in terms of which more sophisticated objects of algebra are defined; and/or a “plain form of quantity”, not subject to non-trivial transformation laws.

Specifically:

• in mathematics, by a scalar one typically means an element of a ground ring or ground field (see also: number);

  this usage appears in concepts such as scalar product, extension of scalars, restriction of scalars, …

• in physics, by a scalar one typically means an element of a trivial representation of a given symmetry group;

  this usage appears in concepts such as scalar field, scalar meson, …, and in partial negation in concepts such as pseudoscalar, …

These two usages do overlap: The 1-dimensional trivial representation $\mathbf{1} \in Rep_k(G)$ of any symmetry group $G$ over any ground field $k$ has as underlying set that ground field itself: $\mathbf{1} \,\simeq_k\, k$.

For instance, the scalar curvature in Riemannian geometry is a scalar(-valued function) in both senses of the word.

Related concepts

• unit

References

See also

• Wikipedia, Scalar