# QFT - Generating Functional

• binbagsss
In summary, the conversation is about finding the factor of n in a particular equation by taking derivatives. The person is confused about where the factor of n comes from and is trying to use the placement of the J term in the equation to figure it out.
binbagsss

## Homework Statement

Hi

I am looking at the attached question part c)

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..

#### Attachments

• generatingfunctional2.jpg
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many thanks

## 1. What is a generating functional in QFT?

A generating functional is a mathematical tool used in quantum field theory (QFT) to calculate correlation functions. It is a functional that maps a set of source terms to the expectation value of the corresponding quantum field operators.

## 2. How is a generating functional related to Feynman diagrams?

In QFT, the generating functional is used to calculate the probability amplitude of a scattering process, which is represented by a Feynman diagram. Each term in the generating functional corresponds to a specific Feynman diagram and the sum of all these terms gives the total probability amplitude.

## 3. What is the importance of the path integral in generating functionals?

The path integral is an integral over all possible paths of the system, and it is used to calculate the generating functional in QFT. It plays a crucial role in QFT as it allows for the calculation of correlation functions and scattering amplitudes, which are essential for understanding the behavior of quantum systems.

## 4. How is the generating functional used in renormalization?

In QFT, renormalization is a mathematical procedure used to remove infinities from calculations and obtain meaningful results. The generating functional is used in this process by introducing a cutoff parameter, which restricts the integration over certain momenta and removes the divergences.

## 5. Can the generating functional be extended to include interactions?

Yes, the generating functional can be extended to include interactions between quantum fields. This is achieved by introducing an interaction term in the functional, which accounts for the interactions between particles in the system. The resulting functional is then used to calculate the correlation functions and scattering amplitudes for the interacting particles.

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