{{Short description|Notation used in quantum field theory}} Physics often deals with classical models where the dynamical variables are a collection of functions {''φ''<sup>''α''</sup>}<sub>''α''</sub> 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''<sup>''α''</sup>(''x'').
In the '''DeWitt notation''' (named after theoretical physicist Bryce DeWitt), ''φ''<sup>''α''</sup>(''x'') is written as ''φ''<sup>''i''</sup> where ''i'' is now understood as an index covering both ''α'' and ''x''.
So, given a smooth functional ''A'', ''A''<sub>,''i''</sub> stands for the functional derivative
:<math>A_{,i}[\varphi] \ \stackrel{\mathrm{def}}{=}\ \frac{\delta}{\delta \varphi^\alpha(x)}A[\varphi]</math>
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,
:<math>A^i B_i \ \stackrel{\mathrm{def}}{=}\ \int_M \sum_\alpha A^\alpha(x) B_\alpha(x) d^dx</math>
==References== * {{cite book | first = Claus | last = Kiefer| authorlink = Claus Kiefer |date=April 2007 | title = Quantum gravity |type= hardcover | edition = 2nd | pages = 361 | publisher = Oxford University Press | isbn=978-0-19-921252-1 }}
Category:Quantum field theory Category:Mathematical notation
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