About the n-point Green functions in position-space

[kmyzzmy]

When we want to calculate the n-point function in the position space, it's always very difficult. For example, when we're calculate the 3-point function of $$\phi^3$$ theory in position space, we would get an integral
$$\int d^4 z \frac{1}{|z-x_1|^2|z-x_2|^2|z-x_3|^2}$$
It seems hard to integrate it.
I'm wondering if anyone has already done this before. Or is there any theory to calculate these kind of integrals?

 

[stone]

Usually done numerically

 

[A. Neumaier]

Usually done numerically

But then you don't get it as a function of x_1, x_2, and x_3!

 

[stone]

Of course, but you can do it various values of x_i's
Although I am pretty sure there must be some analytical forms for phi^3 theory. But in general for arbitrary perturbations, it is always done numerically.

 

[A. Neumaier]

Of course, but you can do it various values of x_i's
Although I am pretty sure there must be some analytical forms for phi^3 theory. But in general for arbitrary perturbations, it is always done numerically.

Could you please give some online references for the numerical approach to n-particle Green's functions (not necessarily of Phi^3)?

 

[stone]

Most of the applications are in condensed matter systems
users.physik.fu-berlin.de/~pelster/Papers/green.pdf
www.fz-juelich.de/nic-series/volume32/assaad.pdf

There are hundreds of papers around, but I think the best would be look at a book.
The book by Atland and Simmons is a comprehensive one
https://www.amazon.com/dp/0521769752/?tag=pfamazon01-20

 

[A. Neumaier]

Most of the applications are in condensed matter systems
users.physik.fu-berlin.de/~pelster/Papers/green.pdf
www.fz-juelich.de/nic-series/volume32/assaad.pdf

Thanks. But one needs to reduce the dimension to make the numerics tractable.

By Euclidean invariance, one can reduce the integral of the OP to the case where x_1=0, x_2=(u,0,0,0), x_3=(v,w,0,0). Thus one is left with a 3-parameter integral. The integration domain can also be reduced to 3D by using polar coordinates in the last two components, and integrating over the angle analytically.

Then one is left with a 3-parameter family of 3D integrals. Still, to map out the resulting function reliably requires a lot of numerical integrations.

 

[RedX]

Thanks. But one needs to reduce the dimension to make the numerics tractable.

By Euclidean invariance, one can reduce the integral of the OP to the case where x_1=0, x_2=(u,0,0,0), x_3=(v,w,0,0). Thus one is left with a 3-parameter integral. The integration domain can also be reduced to 3D by using polar coordinates in the last two components, and integrating over the angle analytically.

Then one is left with a 3-parameter family of 3D integrals. Still, to map out the resulting function reliably requires a lot of numerical integrations.

So normally when you go Euclidean space, you calculate $$<0|\phi(-iu_0,u)\phi(-ix_0,x)\phi(-iy_0,y)|0>$$, and then the integral over the internal vertex d⁴z :


$$\int d^4 z \frac{1}{|z-u|^2|z-x|^2|z-y|^2}$$

can be reduced from 4 dimensions to 3 via contour integration and selecting the poles which are now off axis? Once you get your answer this way, wherever you see u₀, you replace it with iu₀, and similarly with x₀ and y₀?

But now since you are working with numerical values, then you don't know how to back substitute since all the variables became numbers, so you can't do things this way?

So you have to work directly with this integral:

$$\int d^4 z \frac{1}{|z-u|^2|z-x|^2|z-y|^2}$$

where u,x, and y are all real time instead of imaginary time. So how do you make the squared terms Euclidean?

 

[A. Neumaier]

But now since you are working with numerical values, then you don't know how to back substitute since all the variables became numbers, so you can't do things this way?

I had assumed that the integral was already Euclidean, since one wouldn't use absolute values in Minkowski space.

The numerical approach would evaluate the integral at a number of points, fit some rational function with the correct symmetry to the values of the integrals, and then analytically continue the resulting rational function back to Minkowski space.