DeWitt notation

In physics, we often deal with classical models where the dynamical variables are a collection of functions

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In the deWitt notation we write φα(x) as φi where i is now understood as an index covering both α and x.

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So, if we have a smooth functional A, A,i stands for the functional derivative

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:rac{delta}{delta phi^lpha(x)}A

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as a functional of φ. In other words, a "1-form" field over the infinite dimensional "functional manifold".

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The Einstein summation convention is used. In other words,

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:A^i B_iequiv int_M d^dx sum_lpha A^lpha(x) B_lpha(x)

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