Help with Correlation/Green's Function of Rotated Variables

In summary, the author is trying to find a correlation function between two fields, and has run into a roadblock. They provide an equation to calculate the correlation function, but are unsure of how to calculate the brackets. The brackets are the same as a Green's function. The author also found a typo in equation 8 in the paper.
  • #1
Hello (I'm reposting this from stack exchange, and thought this site may be more appropriate, so if you see it that's why),

I'm working through this paper, and have encountered "a little algebra shows that...", yet I'm not familiar enough with the topic at hand to figure this out. Here is the paper:https://arxiv.org/abs/cond-mat/0109316My issues start in sections (5) and (5) (strange numbering system), "Correlation functions" and "Keldysh Rotation" (pages 3 and 4), primarily focusing on equation 11. I'm pretty new to the topic of Green's functions, so perhaps I have missed something. I've come to this paper to try to understand something similar, and encountered this roadblock, so any help understanding how to obtain equation (11) would be greatly appreciated.
 
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  • #2
It looks like you should take a choice for ##\alpha##,##\beta##, then expand the quantum and classical field variables, ##\phi_{cl}## and ##\phi_q##, in the old field variables, ##\phi_f## and ##\phi_b##. This will give you a sum of correlation functions which are cited in equation 8. Now combine as necessary using equation 12.
 
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  • #3
Haborix said:
It looks like you should take a choice for ##\alpha##,##\beta##, then expand the quantum and classical field variables, ##\phi_{cl}## and ##\phi_q##, in the old field variables, ##\phi_f## and ##\phi_b##. This will give you a sum of correlation functions which are cited in equation 8. Now combine as necessary using equation 12.

How should the calculations be made in the correlation brackets? Or might you have a reference I could look at as a guide? Also, is the correlation bracket the same as a Green's function?
 
  • #4
Also, maybe this isn't part of confusion, do the different matrix elements correspond to different combos of cl and q?
 
  • #5
Sorry, for such a late reply. In this case the correlation function is the same as the Green's function. Mathematically, a Green's function is associated with some linear differential operator. In a field theory setting this linear differential operator will be the one appearing in the Euler-Lagrange equations derived from the action functional. In the case where your action functional appearing in the partition function is quadratic in the fields you have the 2-point correlation function and the Green's function (which is associated with the linear differential operator) being identical. The paper you are interested in fits this case. I would caution that the use of "Green's function" is sometimes used more liberally as a synonym for the correlation functions and vice versa. You will have to discover from the context what is intended.

As for calclating inside correlation brackets, you just need to realize that
##\langle(\sum\limits_i \phi_i)(\sum\limits_j \phi_j)\rangle=\sum\limits_i\sum\limits_j\langle\phi_i\phi_j\rangle##

Say you wanted to calculate ##-i\langle\phi_{cl}\bar{\phi}_{cl}\rangle## (which you have said you do), then you would expand these fields in the old fields and get

##
\begin{align}-i\langle\phi_{cl}\bar{\phi}_{cl}\rangle&=-{i\over 4}\langle\phi_{f}\bar{\phi}_{f}+\phi_{f}\bar{\phi}_{b}+\phi_{b}\bar{\phi}_{f}+\phi_{b}\bar{\phi}_{b}\rangle\\
&=-{i\over 4}\left(\langle\phi_{f}\bar{\phi}_{f}\rangle+\langle\phi_{f}\bar{\phi}_{b}\rangle+\langle\phi_{b}\bar{\phi}_{f}\rangle+\langle\phi_{b}\bar{\phi}_{b}\rangle\right)\end{align}
##
Now you are free to make substitution for these known correlation functions in the old field variables. I actually did this piece of the calculation and you do get the expected answer:
##
-i\langle\phi_{cl}\bar{\phi}_{cl}\rangle={1\over 2}\mathcal{D}^K(t,t')
##

As for your last question, yes.

EDIT: I forgot to say that I am not an expert in out-of-equilibrium field theories, so I do not know a reference which would be instructive within that topic. As for equilibrium statistical field theories, there is the second book in a two volume set by Kardar. Perhaps others will chime in with better suggestions.

EDIT#2: In the course of the doing that calculation I found a typo in equation 8 in the paper. The last correlation is for fields both having index ##b##
 
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  • #6
Haborix said:
In a field theory setting this linear differential operator will be the one appearing in the Euler-Lagrange equations derived from the action functional. In the case where your action functional appearing in the partition function is quadratic in the fields you have the 2-point correlation function and the Green's function (which is associated with the linear differential operator) being identical. The paper you are interested in fits this case.

The action functional contains the integrand ##\bar{\phi}(t)\mathcal{D}^{-1}\phi(t)##. By quadratic in the fields, do you mean ##\bar{\phi},\phi## or do you mean ##\bar{\phi},\mathcal{D}^{-1}\phi##?

For the second half of your response, thank you very much! I didn't know if ##\langle ab + cd \rangle = \langle ab \rangle + \langle cd\rangle##, so that is very helpful to know for this paper! I'll check out the Kardar texts. I've been stuck on this issue for awhile, and thankful for figuring it out. It is relieving to see it wasn't so terrible.
 
  • #7
I mean the former. I have some general advice for working with field theory and that is to always return to the fundamentals/basics when you get confused. Sometimes the calculations get long and tedious and you might forget what you were trying to do in the first the place. This is probably obvious, and a good practice outside of field theory, but it is worth actively recalling as you do work in field theory.
 

1. What is the purpose of using correlation and Green's function for rotated variables?

The purpose of using correlation and Green's function for rotated variables is to understand the relationship between two variables that have been rotated or transformed in some way. This can help identify patterns and connections that may not be apparent when looking at the original variables.

2. How do you calculate correlation for rotated variables?

To calculate correlation for rotated variables, you first need to transform the variables back to their original orientation. Then, you can use the standard correlation formula, which involves calculating the covariance and standard deviations of the two variables.

3. What is Green's function and how is it related to correlation for rotated variables?

Green's function is a mathematical tool used to solve differential equations, and it can also be applied to the analysis of correlation for rotated variables. Green's function allows us to understand how a change in one variable affects the other variable, which is similar to how correlation measures the relationship between two variables.

4. Can correlation and Green's function be used for non-linearly related variables?

Yes, correlation and Green's function can be used for non-linearly related variables. However, it is important to note that the results may not be as accurate as when used for linearly related variables. In these cases, it may be more useful to use non-parametric correlation measures or alternative methods for analyzing the relationship between variables.

5. How can correlation and Green's function be used in practical applications?

Correlation and Green's function can be used in practical applications such as data analysis, signal processing, and machine learning. They can help identify patterns and relationships between variables, and can also be used to make predictions or decisions based on these relationships.

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