DeWitt notation

Physics often deals with classical models where the dynamical variables are a collection of functions
_''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_ varphi\ \stackrel\ \fracA 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,

:$A^i B_i \ \stackrel\ \int_M \sum_\alpha A^\alpha(x) B_\alpha(x) d^dx$

References

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Quantum field theory
Mathematical notation

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