Linked Questions

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Connected Diagrams [duplicate]

The generating functional for the connected part of the Green functions is defined as $$iW[j] = \log Z[j].$$ From this the four-point connected Green's function is then given by $G_c(x_1,x_2,x_3,...
amilton moreira's user avatar
8 votes
4 answers
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Proof of Connected Diagrams

If $Z[J]$ is the generating functional for the path-integral, could any prove (or more reasonably, refer me to a proof) that $$W[J]\equiv\frac{\hbar}{i}\log\left(Z[J]\right)$$ "generates" only ...
PPR's user avatar
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13 votes
2 answers
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Intuition behind Linked Cluster Theorem: connected vs. non-connected diagrams

Within statistical physics and quantum field theory, the linked cluster theorem is widely used to simplify things in the calculation of the partition function among other things. My question has the ...
KF Gauss's user avatar
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4 votes
2 answers
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Interpreting generating functional as sum of all diagrams

The generating functional is defined as: $$Z[J] = \int \mathcal{D}[\phi] \exp\Big[\frac{i}{\hbar}\int d^4x [\mathcal{L} + J(x)\phi(x)]\Big].$$ I know this object is used as a tool to generate ...
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4 votes
1 answer
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Zee's Nutshell: Feynman diagrams "baby problem": Connected vs. Disconnected

On page 47 of A. Zee's QFT in a Nutshell, he explains how disconnected Feynman diagrams can be built from lower-order connected diagrams: I don't know how to understand formula $(6)$. I understand ...
Bass's user avatar
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6 votes
1 answer
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Cluster Expansion

In the cluster expansion (section 5.2 in M. Kardar "Statistical Physics of Particles") we write the grand canonical partition function. During the expansion, we do the following switch between a sum ...
golanor's user avatar
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9 votes
1 answer
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Summing over disconnected diagrams - Peskin and Schroeder

In Peskin & Schroeder, page 97, the following expression is given as part of the demonstration of how the $n$-point correlation function is calculated using connected diagrams: $$\sum_{\text{...
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6 votes
1 answer
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What is the physical meaning of $W[J]=\frac{\hbar}{i}\ln Z[J]$?

The quantity $Z[J]$ (which is the generating functional for all Green functions) physically represents the probability amplitude for a system to remain in the vacuum state. Can we find a similar ...
Solidification's user avatar
4 votes
0 answers
410 views

S-Matrix Generating Functional (Problem 4.1 in Weinberg)

I'm currently working through Weinberg's QFT book, but I'm somewhat stuck at problem 4.1, which states: Define generating functionals for the S-matrix and its connected part: \begin{equation} F[\...
Lurianus's user avatar
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0 answers
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Free Energy vs. Partition Function in QFT

The partition function of QFT is defined as $$Z=\int\mathcal{D}\varphi e^{iS[\varphi]}.$$ Now, it is a general fact that this formal path integral can be computed perturbatively as (sketchy) $$Z=\sum_{...
B.Hueber's user avatar
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Summing graphs in the partition function (statistical physics)

I am looking at Tong's lecture notes on statistical physics, and I wanted to understand a step in his cluster expansion better. The goal here is to calculate the partition function in the canonical ...
Bedge's user avatar
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1 vote
0 answers
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Connected diagrams and Keldysh

I have seen that: In the ground state ($T = 0$) formulation of the Green’s function written in terms of operators in the interaction picture, the Green’s function reads: $$G(r,t;r',t') = -i\frac{\...
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