- #1
jfy4
- 649
- 3
Hi,
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.
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.
Last edited: