Building upon the last post, I finally realize the difference between connected moments and disconnected moments: It has to do with conservation of momentum in the Feynman diagrams we have been using to represent the Gram-Charlier expansion. (I've recently realized the story becomes more complicated in Statmech, and connected moments are quickly defined to be the cumulants, but more on that next time.)
The correlation between two localized excitations in the field are given by derivatives the generating function:
\begin{eqnarray}
\langle J_{i_1}\dots J_{i_l} \rangle &=& \frac{\partial}{\partial J_{i_1}}\cdots \frac{\partial}{\partial J_{i_l}} \log\left(Z(\mathbf{J})\right)
\end{eqnarray}
But we found, from before, that the generating function was actually built out of a sum of Green's functions, which are basically correlators in momentum space:
\begin{eqnarray}
Z(\mathbf{J}) &=& \sum_s=0^l J_{i_1}\cdots J_{i_l} G_{i_1 \cdots i_l} \\
G_{ij} &=& \langle q_i q_j \rangle = \int d^Nq \left(q_i q_j\right)e^{\mathbf{J}\cdot \mathbf{q} - \mathbf{q}\cdot \mathbf{A} \cdot \mathbf{q}-\frac{\lambda}{4!} \mathbf{q}^4}
\end{eqnarray}
The above is written for a random anharmonic field -- thus the lambda J $q^4$ term -- and we see that this is just our two-point green's function from before. If we expand this integral out in powers of lambda, we will get our one loop and two loop terms:
\begin{eqnarray}
G_{ij} &=& \langle q_i q_j \rangle = \int d^Nq \left(q_i q_j\right) \left(1-\frac{\lambda}{4!}\sum_n (q_n)^4+\frac{\lambda^2}{4!4!}\sum_m \sum_n (q_n)^4(q_m)^4 + \dots \right)e^{\mathbf{J}\cdot \mathbf{q} - \mathbf{q}\cdot \mathbf{A} \cdot \mathbf{q}}
\end{eqnarray}
We see by Wick contraction this leads to terms like:
\begin{eqnarray}
G_{ij} &=& \langle q_i q_j \rangle + \sum_n \frac{\lambda}{4!} \langle q_i q_j q_n q_n q_n q_n\rangle + \sum_m \sum_n \frac{\lambda^2}{4!4!} \langle q_i q_j q_n q_n q_n q_n q_m q_m q_m q_m\rangle + \dots \\
&=& \mathbf{A}^{-1}_{ij} + \frac{\lambda (4\cdot 3)}{4!}\sum_n(\mathbf{A}^{-1}_{in}\mathbf{A}^{-1}_{nj}\mathbf{A}^{-1}_{nn}+\mathbf{A}^{-1}_{ij}\mathbf{A}^{-1}_{nn}\mathbf{A}^{-1}_{nn} ) + \frac{\lambda (8 \cdot 7)}{4! 4!}\sum_m \sum_n \left( \dots \right)+\dots
\end{eqnarray}
The disconnected terms are any that contain
\begin{eqnarray}
\mathbf{A}^{-1}_{ij}
\end{eqnarray}
in the n and m summations. The word "connected" in momentum space means that the sum of our q's must be equal to zero. Or, that the ingoing and outgoing momenta must sum to zero. (Disconnected diagrams correspond to "vacuum fluctuations" where a random variable or particle appears out of nowhere, and then disappears at some later point in time.) This restriction on our available terms in the two-point Green's function can be written as:
\begin{eqnarray}
G_{ij\ \ \rm{connected}} &=& \langle q_i q_j \rangle \delta(q_i+q_j) + \sum_n\frac{\lambda}{4!} \langle q_i q_j q_n q_n q_n q_n\rangle \delta(q_i+q_j+q_n) \\ &&+ \sum_m \sum_n \frac{\lambda^2}{4!4!} \langle q_i q_j q_n q_n q_n q_n q_m q_m q_m q_m\rangle \delta(q_i+q_j+q_n+q_m) + \dots
\end{eqnarray}
And this discrete sum looks a lot like the one loop "integrations" I have been working with for the power spectrum and bispectrum! Trouble is, in Standard perturbation theory, we are using an explicit recursion relation to expand the random variable -- over density -- in powers of the scale factor, not through some coupling constant lambda.
The anharmonic term may be written slightly incorrectly above, but it is difficult to represent the potential of a self-interacting field without a textbook. I'll have to look up the approximation.
-------------------------------------------------------------------------------------------------------------------------
Translational invariance of correlation functions and "connectedness"
-------------------------------------------------------------------------------------------------------------------------
It is also interesting to note that, translationally invariant correlation functions, such as the Power Spectrum and Bispectrum, carry a natural dirac delta function of the form
\begin{eqnarray}
G_{ij} &=& \langle q_i q_j \rangle \delta(q_i +q_j) \\
G_{ijk} &=& \langle q_i q_j q_k \rangle \delta(q_i +q_j+q_k) ,
\end{eqnarray}
written above. This means that translationally invariant correlations automatically require connected diagrams? After reading about this in some Statmech textbooks, it seems a translationally invariant system already defines connected moments as the cumulants due to a graphical expansion of the partition function. Will write more about this later....
The correlation between two localized excitations in the field are given by derivatives the generating function:
\begin{eqnarray}
\langle J_{i_1}\dots J_{i_l} \rangle &=& \frac{\partial}{\partial J_{i_1}}\cdots \frac{\partial}{\partial J_{i_l}} \log\left(Z(\mathbf{J})\right)
\end{eqnarray}
But we found, from before, that the generating function was actually built out of a sum of Green's functions, which are basically correlators in momentum space:
\begin{eqnarray}
Z(\mathbf{J}) &=& \sum_s=0^l J_{i_1}\cdots J_{i_l} G_{i_1 \cdots i_l} \\
G_{ij} &=& \langle q_i q_j \rangle = \int d^Nq \left(q_i q_j\right)e^{\mathbf{J}\cdot \mathbf{q} - \mathbf{q}\cdot \mathbf{A} \cdot \mathbf{q}-\frac{\lambda}{4!} \mathbf{q}^4}
\end{eqnarray}
The above is written for a random anharmonic field -- thus the lambda J $q^4$ term -- and we see that this is just our two-point green's function from before. If we expand this integral out in powers of lambda, we will get our one loop and two loop terms:
\begin{eqnarray}
G_{ij} &=& \langle q_i q_j \rangle = \int d^Nq \left(q_i q_j\right) \left(1-\frac{\lambda}{4!}\sum_n (q_n)^4+\frac{\lambda^2}{4!4!}\sum_m \sum_n (q_n)^4(q_m)^4 + \dots \right)e^{\mathbf{J}\cdot \mathbf{q} - \mathbf{q}\cdot \mathbf{A} \cdot \mathbf{q}}
\end{eqnarray}
We see by Wick contraction this leads to terms like:
\begin{eqnarray}
G_{ij} &=& \langle q_i q_j \rangle + \sum_n \frac{\lambda}{4!} \langle q_i q_j q_n q_n q_n q_n\rangle + \sum_m \sum_n \frac{\lambda^2}{4!4!} \langle q_i q_j q_n q_n q_n q_n q_m q_m q_m q_m\rangle + \dots \\
&=& \mathbf{A}^{-1}_{ij} + \frac{\lambda (4\cdot 3)}{4!}\sum_n(\mathbf{A}^{-1}_{in}\mathbf{A}^{-1}_{nj}\mathbf{A}^{-1}_{nn}+\mathbf{A}^{-1}_{ij}\mathbf{A}^{-1}_{nn}\mathbf{A}^{-1}_{nn} ) + \frac{\lambda (8 \cdot 7)}{4! 4!}\sum_m \sum_n \left( \dots \right)+\dots
\end{eqnarray}
The disconnected terms are any that contain
\begin{eqnarray}
\mathbf{A}^{-1}_{ij}
\end{eqnarray}
in the n and m summations. The word "connected" in momentum space means that the sum of our q's must be equal to zero. Or, that the ingoing and outgoing momenta must sum to zero. (Disconnected diagrams correspond to "vacuum fluctuations" where a random variable or particle appears out of nowhere, and then disappears at some later point in time.) This restriction on our available terms in the two-point Green's function can be written as:
\begin{eqnarray}
G_{ij\ \ \rm{connected}} &=& \langle q_i q_j \rangle \delta(q_i+q_j) + \sum_n\frac{\lambda}{4!} \langle q_i q_j q_n q_n q_n q_n\rangle \delta(q_i+q_j+q_n) \\ &&+ \sum_m \sum_n \frac{\lambda^2}{4!4!} \langle q_i q_j q_n q_n q_n q_n q_m q_m q_m q_m\rangle \delta(q_i+q_j+q_n+q_m) + \dots
\end{eqnarray}
And this discrete sum looks a lot like the one loop "integrations" I have been working with for the power spectrum and bispectrum! Trouble is, in Standard perturbation theory, we are using an explicit recursion relation to expand the random variable -- over density -- in powers of the scale factor, not through some coupling constant lambda.
The anharmonic term may be written slightly incorrectly above, but it is difficult to represent the potential of a self-interacting field without a textbook. I'll have to look up the approximation.
-------------------------------------------------------------------------------------------------------------------------
Translational invariance of correlation functions and "connectedness"
-------------------------------------------------------------------------------------------------------------------------
It is also interesting to note that, translationally invariant correlation functions, such as the Power Spectrum and Bispectrum, carry a natural dirac delta function of the form
\begin{eqnarray}
G_{ij} &=& \langle q_i q_j \rangle \delta(q_i +q_j) \\
G_{ijk} &=& \langle q_i q_j q_k \rangle \delta(q_i +q_j+q_k) ,
\end{eqnarray}
written above. This means that translationally invariant correlations automatically require connected diagrams? After reading about this in some Statmech textbooks, it seems a translationally invariant system already defines connected moments as the cumulants due to a graphical expansion of the partition function. Will write more about this later....
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