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 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 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.

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

A_{,i}[\phi] \ \stackrel{\mathrm{def}}{=}\ \frac{\delta}{\delta \phi^\alpha(x)}A[\phi]

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

In integrals, the Einstein summation convention is used. Alternatively,

A^i B_i \ \stackrel{\mathrm{def}}{=}\ \int_M \sum_\alpha A^\alpha(x) B_\alpha(x) d^dx

References

Kiefer, Claus (April 2007). Quantum gravity (hardcover) (2nd ed.). Oxford University Press. p. 361. ISBN 978-0-19-921252-1.