DeWitt Notation
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DeWitt Notation

Physics often deals with classical models where the dynamical variables are a collection of functions {??}? over a d-dimensional space/spacetime manifold M where ? is the "flavor" index. This involves functionals over the ?'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 ?, and the procedure is in analogy with differential geometry where the coordinates for a point x of the manifold M are ??(x).

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

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

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

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

References

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

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