nLab
pseudoscalar

Contents

Contents

Idea

Given a symmetry group GG equipped with a homomorphism GϕO(s,t)G \overset{\phi}{\longrightarrow} O(s,t) to an orthogonal group (for instance a Pin group), a pseudoscalar is an element of the 1-dimensional linear representation (over a given ground field kk)

1 sgnRep k(G) \mathbf{1}_{{}_{sgn}} \;\in\; Rep_k(G)

that is given by forming the determinant (sign representation):

(1)1 sgn(G×k k (g,c) det(ϕ(g))). \mathbf{1}_{{}_{sgn}} \;\;\coloneqq\;\; \left( \array{ G \times k &\longrightarrow& k \\ (g, c ) &\mapsto& det\big( \phi(g) \big) } \right) \,.

More generally, given a function with values in 1 sgn\mathbf{1}_{{}_{sgn}}, or yet more generally a section of a fiber bundle with typical fiber 1 sgn\mathbf{1}_{{}_{sgn}}, this whole function/section is often called a pseudoscalar; more precisely: a pseudoscalar field. This as opposed to scalar fields, which take values in the 1-dimensional trivial representation 1\mathbf{1}.

If the fiber bundle in question is a “canonical bundle”/determinant line bundle of a Riemannian manifold of dimension nn, hence the top exterior power of a tangent bundle (i.e. top degree (Kähler) differential form-bundle) then such “pseudoscalar fields” of physics are what are called densities in mathematics.

References

See also

Last revised on April 28, 2020 at 11:12:21. See the history of this page for a list of all contributions to it.