nLab scalar




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.


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

Discussion of scalars in the context of (dagger-, compact-, closed) monoidal categories, as forming the endomorphism ring of the tensor unit (cf. quantum information theory via dagger-compact categories):

Last revised on September 20, 2023 at 08:22:50. See the history of this page for a list of all contributions to it.