QFT - Generating Functional

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Homework Statement



Hi

I am looking at the attached question part c)

generatingfunctional2.jpg


Homework Equations



below


The Attempt at a Solution



so if i take ##\frac{\partial^{(n-1)}}{\partial_{(n-1)}} ## of (2) it is clear I can get the ##\frac{i}{h} (\lambda_2 +\lambda_4 )## like-term, but I am unsure about the ##nG_{n-1}## .

There's obviously no other derivatives on the RHS so I will only yield a ##G_{n-1}## and that looks fine, I am a bit confused though, I can yield this from the ## Z[J] ## alone on the RHS, whereas the RHS is ##Z[J]## 'multiplied by' (it is already inside the integral) the extra term of ##S'[\Phi] + J## . So I suspect this extra term is the reason we get the ##n## factor but I am unsure how.

Looking at the LHS there is a single ##J## so it looks like this gives a factor of ##1## and then we take across ##(n-1)## from the RHS.

If I take a derivative wrt ##J##, on the LHS I can either act on the exponential or the single ##J## (but can only act on this ##J## once,) on the RHS it's the same story, with the difference that on the LHS the ##J## is outside the integral but on the RHS it is inside the integral, I'm trying to use this to deduce where the factor of ##n## comes from but I am struggling..
 

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  • #2
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bump
many thanks
 

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