QFT on the lattice and Green's functions

[Ramil]

Homework Statement

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I'm trying to write a program for caclulating Green's function using Monte Carlo method (Metropolis algorithm) in scalar field theory with a potential λφ⁴ in 4D. I'm writing it in python.

N_t, N_x, N_y, N_z - total number of lattice sites in each directions.
Field fi(i,j,k,l). Here (i, j, k, l) corresponds to (t, x, y, z).

I want to calculate two-point Green's function G(0) in the coincident limit. I want to get the dependency of G(0) on x coordinate.

Homework Equations

G(r,r')=<φ(r)φ(r')>, r→r'. (Here r is a space-time point r=(t,x,y,z))
G(0)=<φ²>

The Attempt at a Solution

Using Metropolis algorithm i do the following:
1. Generate an arbitrary configuration of the field fi, let it, for instance, be all zeros:
fi=np.zeros(N_t, N_x, N_y, N_z)
2. Then, using standard procedure I thermalise my configurations of the field, to get configurations fi(i, j, k, l) that satisfy a minima of an action.
3. Using obtained configurations fi(i, j, k, l) I want to calculate two-point Green's function G(0) in the coincident limit.

An excerpt of my code:
...
def Grin_0(fi):
N_x=10
N_y=10
N_z=10
N_t=60
for i in range(0,N_t):
for j in range(0,N_x):
for k in range(0,N_y):
for l in range(0,N_z):
G_0[i,j,k,l]=fi[i,j,k,l]*fi[i,j,k,l]
return G_0
...
Then i summirize G_0 along i,k,l indices and get the dependency of G_0 on index j.

Is it correct? How to find G_0 and its dependency on x correctly?

 

[mark726]

Your approach seems reasonable, but there are a few things you may want to consider or clarify:

1. In step 1, you mention generating an arbitrary configuration of the field, but then you specify that it will be all zeros. If you want to test your code, this may be a good starting point, but in general, you may want to generate a random configuration of the field to get a more realistic result.

2. In step 2, you mention thermalizing the configurations of the field, but you don't specify how you are doing this. It may be helpful to explain your thermalization procedure in more detail.

3. In step 3, you mention wanting to calculate the two-point Green's function in the coincident limit, but your code seems to calculate it for all points in the lattice. If you only want the coincident limit, you may want to specify how you are selecting the points for the calculation.

4. In your code, you define G_0 as a function, but then you don't use it as a function in your code. Instead, you seem to be calculating the two-point Green's function for all points in the lattice and then summarizing it later. If you want to use G_0 as a function, you may want to pass the field configuration as an argument and calculate the Green's function for each point in the lattice within the function.

5. Finally, you mention wanting to find the dependency of G_0 on the x coordinate, but your code only seems to be calculating the Green's function for each point in the lattice. If you want to find the dependency on the x coordinate, you may need to plot or analyze the results of your code in a way that shows this dependency.

Overall, your approach seems reasonable, but it may be helpful to provide more details and clarify some of the steps in your code. Additionally, you may want to consider how you will test or validate your results to ensure they are accurate.