Linked Questions
18 questions linked to/from Intuition behind Linked Cluster Theorem: connected vs. non-connected diagrams
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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,...
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3
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In what sense is the proper/effective action $\Gamma[\phi_c]$ a quantum-corrected classical action $S[\phi]$?
There is a difference between the classical field $\phi(x)$ (which appears in the classical action $S[\phi]$) and the quantity $\phi_c$ defined as $$\phi_c(x)\equiv\langle 0|\hat{\phi}(x)|0\rangle_J$$ ...
- 27.2k
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Use my example to explain why loop diagram will not occur in classical equation of motion?
We always say that tree levels are classical but loop diagrams are quantum.
Let's talk about a concrete example：
$$\mathcal{L}=\partial_a \phi\partial^a \phi-\frac{g}{4}\phi^4+\phi J$$
where $J$ is ...
- 6,081
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Proof that the effective/proper action is the generating functional of one-particle-irreducible (1PI) correlation functions
In all text book and lecture notes that I have found, they write down the general statement
\begin{equation}
\frac{\delta^n\Gamma[\phi_{\rm cl}]}{\delta\phi_{\rm cl}(x_1)\ldots\delta\phi_{\rm cl}(x_n)}...
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4
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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 ...
- 2,104
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How to prove useful property of logarithm of generating functional in QFT?
How to prove that $\ln(Z(J))$ generates only connected Feynman diagrams? I can't find the proof of this statement, and have only met its demonstrations for case of 2- and 4-point.
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Connected and strongly connected Feynman diagrams
Recently I read, that only connected Feynman diagrams give contribution of nonzero values into the scattering amplitude. Why it is so and what is the physical sense of connected diagrams (due to their ...
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3
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Correlation function of single annihilation/creation operator vanishes
I could not find anything on that on google, or here on physics stack exchange, which surprises me. My problem is, that I do not see, why exactly
$$\left<a\right> = \left<a^\dagger\right> =...
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Summing over disconnected diagrams - Peskin and Schroeder
In Peskin & Schroeder, page 97, the following expression is given as part of the demonstration of how the $n$-point correlation function is calculated using connected diagrams:
$$\sum_{\text{...
user195631
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Computation of functional determinant using Feynman diagram
The above equation is from chapter 9.5 "Functional Quantization of the Spinor Field" of Peskin's and Schroeder's book $($page $305)$. I understand that the initial determinant equal to the ...
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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 ...
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5
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1
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Combinatorics geometric series for connected two-point function
In this answer Proof of geometric series two-point function it is said:
Now what about the coefficients in front of each Feynman diagram? Due to the combinatorics/factorization involved it ...
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Is gauge invariance necessary to have Ward identity hold for off-shell amplitudes?
In this other SE post: Is it really proper to say Ward identity is a consequence of gauge invariance? it is shown that the on-shell Ward identity is a consequence of global $U(1)$ symmetry for QED. ...
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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)
$$Z=\sum_{...
- 884
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2
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Why doesn't this diagram appear in the partition function in zero-dimensional QFT?
For the zero-dimensional QFT with action
$$S(\phi)=\frac{\alpha}{2}\phi^2+\frac{\lambda}{4!}\phi^4-J\phi,\tag{1}$$
we can perturbatively expand the partition function as
$$Z_\lambda(J)=\int_{-\infty}^{...
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