Generating Functional in Momentum Space -- QFT

[binbagsss]

Homework Statement

Hi,

Question attached:

inserting ##\phi (x)= \int \frac{d^4k}{(2\pi)^2}\phi(x)e^{-i k_u x^u}##
and similar for ##J(x) ## / ##J(k)## into the action and then integrating over ##k## gives:

Solution attached:

I AM STUCK on this part, completing the square ; so I see we get back the term ##\frac{-h}{i}J(k)\phi(-k)## however don't we also get an additional identiical term ##J(-k)\phi (k)##?What has happened to this?!

Many thanks in advance.

Homework Equations

see above

The Attempt at a Solution

see above

 

[mighty2000]

Hello,

Thank you for posting your question. It seems like you are on the right track with your solution. When completing the square, you should end up with two terms that look like ##\frac{-h}{i}J(k)\phi(-k)## and ##\frac{h}{i}J(-k)\phi(k)##. These two terms are actually the same, just with the variables switched. So, you can combine them into one term and simplify the expression. This is a common technique in completing the square, where the two terms that arise are actually the same and can be combined. I hope this helps! Let me know if you have any further questions.