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 φs, 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 φ i 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}[\varphi] \ \stackrel{\mathrm{def}}{=}\ \frac{\delta}{\delta \varphi^\alpha(x)}A[\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 \ \stackrel{\mathrm{def}}{=}\ \int_M \sum_\alpha A^\alpha(x) B_\alpha(x) d^dx

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 φs, 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 φ i 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}[\varphi] \ \stackrel{\mathrm{def}}{=}\ \frac{\delta}{\delta \varphi^\alpha(x)}A[\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 \ \stackrel{\mathrm{def}}{=}\ \int_M \sum_\alpha A^\alpha(x) B_\alpha(x) d^dx ==References==

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