deWitt notation

{{Short description|Notation used in quantum field theory}}

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

:$A\_\{,i\}[\backslash varphi]\; \backslash \; \backslash stackrel\{\backslash mathrm\{def\}\}\{=\}\backslash \; \backslash frac\{\backslash delta\}\{\backslash delta\; \backslash varphi^\backslash alpha(x)\}A[\backslash varphi]$

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,

:$A^i\; B\_i\; \backslash \; \backslash stackrel\{\backslash mathrm\{def\}\}\{=\}\backslash \; \backslash int\_M\; \backslash sum\_\backslash alpha\; A^\backslash alpha(x)\; B\_\backslash alpha(x)\; d^dx$