Editing 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''<sup>''α''</sup>}<sub>''α''</sub> over a d-dimensional space/spacetime [[manifold]] ''M'' where ''α'' is the "[[flavor (particle physics)|flavor]]" index. This involves [[functional (mathematics)|functional]]s over the ''φ'''s, [[functional derivative]]s, [[functional integral]]s, 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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