DeWitt notation

Physics often deals with classical models where the dynamical variables are a collection of functions {φ^α}_α over a d-dimensional space/spacetime manifold M where α is the "flavor" index. This involves functionals over the φs, functional derivatives, functional integrals, etc. From a functional point of view this is equivalent to working with an infinite-dimensional smooth manifold where its points are an assignment of a function for each α, and the procedure is in analogy with differential geometry where the coordinates for a point x of the manifold M are φ^α(x).

In the DeWitt notation (named after theoretical physicist Bryce DeWitt), φ^α(x) is written as φⁱ where i is now understood as an index covering both α and x.