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

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):

• Samson Abramsky, Bob Coecke, §6 in: A categorical semantics of quantum protocols, Proceedings of the 19th IEEE conference on Logic in Computer Science (LiCS’04). IEEE Computer Science Press (2004) $[$arXiv:quant-ph/0402130, doi:10.1109/LICS.2004.1319636$]$

• Jamie Vicary, §3 in: Completeness of dagger-categories and the complex numbers, J. Math. Phys. 52 (2011) 082104 [arXiv:0807.2927, doi:10.1063/1.3549117]

• Jamie Vicary, §4 in: Categorical Properties of The Complex Numbers, Electronic Notes in Theoretical Computer Science 270 2 (2011) 163-189 [doi:10.1016/j.entcs.2011.01.030]

• Chris Heunen, Jamie Vicary, §2.1 in: Categories for Quantum Theory, Oxford University Press 2019 [ISBN:9780198739616]

  based on:

  Chris Heunen, Jamie Vicary, §2.1 in: Lectures on categorical quantum mechanics (2012) [pdf, pdf]

• Chris Heunen, Andre Kornell, Axioms for the category of Hilbert spaces, PNAS 119 9 (2022) e2117024119 [arXiv:2109.07418, doi:10.1073/pnas.2117024119]

• Chris Heunen, Andre Kornell, Nesta van der Schaaf, Axioms for the category of Hilbert spaces and linear contractions [arXiv:2211.02688]