DeWitt notation

In physics, we often deal 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. We then deal with functionals over the φ's, functional derivatives, functional integrals, etc. If we choose to take a functional point of view, it's as if we are working with an infinite-dimensional smooth manifold where its points are an assignment of a function for each α and we can proceed in analogy with differential geometry where the coordinates are φ^α(x) where x is a point of M.

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

So, if we have a smooth functional A, A_,i stands for the functional derivative

<math>\frac{\delta}{\delta \phi^\alpha(x)}A[\phi]<math>

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

The Einstein summation convention is used. In other words,

<math>A^i B_i\equiv \int_M d^dx \sum_\alpha A^\alpha(x) B_\alpha(x)<math>