DeWitt notation

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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 φ's, 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 are φ^α(x) where x is a point of M.

In the deWitt notation φ^α(x) is written as φⁱ where i is now understood as an index covering both α and x.

So, given a smooth functional A, A_,i stands for the functional derivative

as a functional of φ. In other words, a "1-form" field over the infinite dimensional "functional manifold".

The Einstein summation convention is used. Alternatively,