# Dimension of n-point Green function

• Einj
In summary, the dimension of an n-point Green function in QFT theory can be calculated by counting factors and using the dimensions of the fields and integration variables. For a scalar theory in D=2d dimensions, the dimension of the Green function is -n(d+1), where d is the dimension of the fields and -2d is the dimension of the integration variables. This can be derived from the expression for the n-point connected 1PI Green function.
Einj
Hi everyone. I have a very quick question. Can someone tell me how to compute the energy dimensions of an n-point Green function. Consider for example a $\lambda\phi^4$ scalar theory. I know that the dimensions of an n-pt Green function are $4-n$ (or something like that). How do I prove it?

Thanks

The dimension of anything in QFT theory can be calculated by counting factors. Each field derivative or integral contributes to the overall dimension.

Yes, I know that. For example in a scalar theory the dimension of the fields is 1 (in energy). My question is: how do I go from knowing the dimension of the field to knowing the dimension of the Green function?

Einj said:
Yes, I know that. For example in a scalar theory the dimension of the fields is 1 (in energy). My question is: how do I go from knowing the dimension of the field to knowing the dimension of the Green function?
If you are working in D=2d dimensions,then n-point connected 1PI Green function reads
Gn(p1,p2,...,pn)=∫∏i=1to nd2dxiei(p1x1+...pnxn)<0|$T\phi(x_1)...\phi(x_n)$|0>.
dim. of $\phi$ is d-1 here as you can check,and dim. of d2dx is -2d because length dimension is inverse of energy(mass) dimension.Hence Gn has dimension n(d-1)-2nd=-n(d+1)

## 1. What is the dimension of n-point Green function?

The dimension of n-point Green function is equal to the number of external points or particles in the specific physical system being studied. For example, in quantum field theory, the n-point Green function represents the correlation between n particles or fields.

## 2. How is the dimension of n-point Green function related to the complexity of a system?

The dimension of n-point Green function is directly related to the complexity of a system. As the number of external points or particles increases, the complexity of the system also increases. This is because more interactions and correlations need to be taken into account when calculating the n-point Green function.

## 3. Can the dimension of n-point Green function be calculated analytically?

In most cases, the dimension of n-point Green function cannot be calculated analytically. This is because the Green function depends on the specific physical system being studied and its interactions, which can be very complex and difficult to solve analytically. In some simplified systems, analytical solutions may be possible.

## 4. How is the dimension of n-point Green function used in theoretical calculations?

The dimension of n-point Green function is an important component in theoretical calculations, particularly in quantum field theory and statistical mechanics. It is used to calculate correlation functions, scattering processes, and other physical observables. The n-point Green function is also used to study the behavior of a system at different energy scales.

## 5. How does the dimension of n-point Green function relate to renormalization?

The dimension of n-point Green function plays a crucial role in renormalization, a technique used to remove divergences and make physical predictions in quantum field theory. The dimension of the Green function helps determine the divergences in a theory and allows for the calculation of counterterms to cancel them out. Renormalization is essential in making precise and reliable predictions in quantum field theory.

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