Linked Questions
15 questions linked to/from Proof of geometric series for connected two-point function
25
votes
3
answers
8k
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Self-energy, 1PI, and tadpoles
I'm having a hard time reconciling the following discrepancy:
Recall that in passing to the effective action via a Legendre transformation, we interpret the effective action $\Gamma[\phi_c]$ to be the ...
- 6,511
12
votes
2
answers
11k
views
How to correctly understand these "1-particle-irreducible insertions"?
In QED, when dealing with the vacuum polarization and the photon propagator, some authors like Peskin & Schroeder introduce the so-called "1-particle irreducible" diagrams. These are defined as:
...
- 37.4k
19
votes
2
answers
3k
views
Defining quantum effective/proper action (Legendre transformation), existence of inverse (field-source)?
Given a Quantum field theory, for a scalar field $\phi$ with generic action $S[\phi]$, we have the generating functional $$Z[J] = e^{iW[J]} =
\frac{\int \mathcal{D}\phi e^{i(S[\phi]+\int d^4x J(x)\...
- 533
7
votes
3
answers
982
views
How can the mass of an unstable composite particle become complex?
To show where the resonances in cross sections come from, one usually considers the exact propagator in the interacting theory, which for a scalar is
$$iG(p^2)=\frac{i}{p^2-m_R^2+\Sigma(p^2)+i\epsilon}...
- 163
3
votes
3
answers
1k
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Why only need to change external legs in minimal-subtraction renormalization?
For the Ch. 27 in book QFT by Srednicki, in the modified minimal-subtraction renormalization scheme($\overline{MS}$), the residue for pole at $-m_{ph}^2$ is $R$, instead of one.
However, I can not ...
- 1,652
5
votes
2
answers
347
views
What is, diagrammatically, the 2-vertex $\Gamma^{(2)}$?
I know that the 2-vertex $\Gamma^{(2)}$ is the second derivative of the effective action, but I fail to see what it is diagrammatically: is it the truncated 1PI diagram? The non-truncated one?
If this ...
- 6,049
2
votes
3
answers
529
views
Disconnected Feynman diagram for the 2-point correlation function
In Peskin&Schroeder they explain in a graphical way why the Schwinger functional generates only connected diagrams. However I don‘t understand why they get 2 diagrams since the first diagram is ...
- 405
5
votes
1
answer
349
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Fourier transformation of the inverse Klein-Gordon propagator
On Peskin & Schroeder's QFT, page 30, the scalar field propagator as the retarded Green function is defined as
$$(\partial^2+m^2)D_R(x-y)=-i\delta^4(x-y) \tag{2.56}$$
The Fourier transformation is ...
- 1,461
3
votes
1
answer
815
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The LSZ formula in Peskin and Schroeder
I'm working on the Eq.(7.57) in Peskin(page 236).
So I try to verify it with LSZ formula.
According to Eq (7.42)
So $\mathcal{M}(p \rightarrow p)=-Z M^{2}\left(p^{2}\right)$
In this I have two ...
- 179
5
votes
1
answer
374
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Resummation of single class of diagrams vs all 1PI diagrams
Maggiore considers on page 136 in Section 5.6 Renormalization in the book A Modern Introduction to Quantum Field Theory, the resummation of tadpole diagrams as its own individual geometric series to ...
- 2,152
2
votes
2
answers
532
views
Exact propagator - 1PI diagrams
Above diagram can be written in terms of series:
$$i\Delta = -\frac{i}{p^2 + m^2} + \Big(-\frac{i}{p^2 + m^2}\Big)(i\Pi)\Big(-\frac{i}{p^2 + m^2}\Big)+ \Big(-\frac{i}{p^2 + m^2}\Big)(i\Pi)\Big(-\frac{...
- 3,494
1
vote
1
answer
552
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Proof of 1-particle irreducible (1PI) diagrams
If we split the effective action into
$$Γ[Φ] =\frac{1}2ΦiG_0^{-1}Φ + Γ^{int} [Φ]\tag{1}$$
we can show that the full propagator is given by
$$G= i[iG − Σ]^{-1}\tag{2}$$
With
$$Σ=-Γ_{ΦΦ}^{int} [Φ]\tag{3}...
- 485
5
votes
1
answer
549
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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 ...
- 485
0
votes
0
answers
163
views
Geometric series of two point function and self energy
This question is related to this question Proof of geometric series two-point function.
Suppose we have a graph $A$ with a symmetry factor $s_1$. According to Srednicki (chapter 9, eq. (9.13)) for a ...
- 485
2
votes
1
answer
121
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Understanding $W^{(n)}$, $\Gamma^{(n)}$, and $\Sigma$ in Feynman diagrams
In quantum field theory (specifically $\phi^4$ theory), $W$ is the sum of all connected Feynman diagrams and the effective action $\Gamma$ is the sum of all 1PI Feynman diagrams. They are related by a ...
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