DeWitt Notation
Get DeWitt Notation essential facts below. View Videos or join the DeWitt Notation discussion. Add DeWitt Notation to your PopFlock.com topic list for future reference or share this resource on social media.
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

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

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,

${\displaystyle A^{i}B_{i}\ {\stackrel {\mathrm {def} }{=}}\ \int _{M}\sum _{\alpha }A^{\alpha }(x)B_{\alpha }(x)d^{d}x}$

## References

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