Moments, Cumulants, and effective action?

[TriTertButoxy]

Hi,
I'm studying quantum mechanics and statistical mechanics, and they make heavy use of the 'correlation functions/green's functions' which are merely the moments of the distribution of some variable.

I have very intuitive understanding of moments and cumulants in terms of the distribution. The moment of a distribution is the sum (or integral) over the distribution weighted by the power of that variable, and the cumulants carry the 'essence' of the shape of the distribution (mean, variance, skewness, ...). In fact, it is very useful to define generating functions of moments Z(j) and cumulants W(j). But then, there's this new generating function (the effective action), defined as the Legendre transform of the generating function of cumulants, W(j):

$$\Gamma(x)=xj-W(j)=\sum_{n=0}^\infty\gamma_n \frac{1}{n!}x^n$$

The diagrammatic interpretation of this is clear -- they generate 1PI graphs. But what do they mean in statistics? What do they characterize about the distribution?
Or more specifically, what do the individual $\gamma_n$ characterize about the distribution?

Thanks! and even more thanks!

 

[mmwave]

Hello,

First of all, it's great to hear that you are studying quantum mechanics and statistical mechanics. The use of correlation functions/green's functions in these fields is indeed very important and can provide valuable insights into the behavior of systems.

To answer your question, the effective action (or generating functional) is a mathematical tool that is used to simplify the calculation of correlation functions in quantum field theory. It is essentially a Legendre transform of the generating function of cumulants, as you have mentioned.

In terms of statistics, the effective action can be seen as a way to characterize the distribution of a variable by its cumulants. The individual cumulants, represented by \gamma_n, can provide information about the shape of the distribution, such as its mean, variance, skewness, etc. Just like how the moments of a distribution can be used to describe its behavior, the cumulants can also give us insights into the distribution.

The diagrammatic interpretation of the effective action in terms of 1PI (one-particle irreducible) graphs is very useful in understanding the behavior of systems in quantum field theory. In statistics, the effective action can also be used to simplify the calculation of correlation functions and provide a deeper understanding of the distribution.

I hope this helps to clarify the significance of the effective action and its relationship to statistics. Keep up the good work in your studies and don't hesitate to ask any further questions. Best of luck!