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I am given an interaction lagrangian piece as

[tex]

\mathcal{L}_1 = \frac{1}{2} g \phi \partial^\mu \phi \partial_\mu \phi

[/tex]

Now normally when I have an interaction lagrangian piece I turn the field's into variations with respect to the source [itex]\delta_J[/itex], and take variations of the free partition function to get feynman diagrams, however in this case the partials confuse me. Am I allowed to use the EOM at this stage to simplify the interaction term? Or can I preform the following operations

[tex]

\frac{1}{2}g \phi \partial^\mu \phi \partial_\mu \phi \rightarrow \frac{1}{2}g \Box (\delta_J)^3

[/tex]

since typically the variation and the partial `commute'?

EDIT: I realized I can integrate by parts, that maybe helps a little, but I am still almost in the same boat as before. After integration I get

[tex]

\mathcal{L}_1 = -\frac{1}{4}g \phi^2 \Box \phi

[/tex]

since the total derivative vanishes.

Thanks, any help is appreciated.

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# Field Redefinition and EOM

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