symmetric monoidal (∞,1)-category of spectra
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.
See also
Last revised on April 6, 2020 at 11:42:57. See the history of this page for a list of all contributions to it.