# What does “Integrating out field” mean?

In Schwartz’s QFT book, there is a couple of exercise problems of particle polarization in chapter 3. I have trouble with finding interaction terms from the given Lagrangians. Is it just okay to express the Lagrangians in momentum space?

• sorry for adding pictures three time :( – JunginYu Jun 14 at 15:00
• Your title refers to the phrase "integrating over field" but that phrase doesn't appear anywhere in any of the questions. Are you asking what "integrating out" a field means? Because that is defined in the question. – Charlie Jun 14 at 15:10
• Hi and welcome to physics.SE! Please do not post images of texts you want to quote, but type it out instead so it is readable for all users and so that it can be indexed by search engines. For formulae, use MathJax instead. – ACuriousMind Jun 14 at 15:11

By substituting $$A_{\mu}$$ for it's solution in terms of its Equation of motion, you are essentially removing $$A_{\mu}$$ as a dynamical degree of freedom, you only see it's imprint as it's effect on the current $$J_{\mu}$$.