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Dewitt NotationIn physics, we often deal with classical models where the dynamical variables are a collection of functions {φα}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 φα(x) where x is a point of M. In the deWitt notation we write φα(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 -
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, -
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