Questions tagged [1pi-effective-action]

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Conditions for renormalization of effective potential

Currently I am reading Das' Finite Temperature Field Theory. After computing the first order correction to the potential with a cut-off (eq. 6.77) Das states that, "we add counterterms and ...
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65 views

Effective action quantum field

I am reading Peskin and Schroeder Section 11.4. They derive a formula for the effective action p.372 Equation 11.63 using a scalar field interaction, $$ \Gamma \left ( \phi _{cl} \right )=\int d^{4}...
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Why does the background field effective action generate only vacuum graphs?

I refer to LF Abbott's "Introduction to the background field method". The background field generating functional is $$ \tilde{Z}[J,\phi] = \int \mathcal{D}Q \exp i[S[Q+\phi] + J.Q], \text{ ...
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How to compute the quantum effective action from 1PI Feynman diagrams?

On page 33 of these notes by David Skinner, it is claimed that [starting from a connected graph and removing the bridges] tells us how to compute $\Gamma(\Phi)$ perturbatively from the original ...
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Confusions on effective action and 1PI 3-point Green's function [duplicate]

In Peskin's QFT book chapter 11.5, the author gives a graph(see below) and claims that the third functional derivative for effective action $\Gamma$ gives the 1PI (one-particle-irreducible) 3-point ...
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Why don't counterterms appear in one-loop correction for 1PI effective action?

In Zee's "QFT in a Nutshell: Second Edition", section IV.3, the author calculates the 1PI effective potential for a single real scalar field. The full Lagrangian is given by equation (1): $$\mathcal{...
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29 views

Scattering matrix element working with an effective action

I've obtained the Euler-Heisenberg effective action and I'm trying to obtain it's photon-photon scattering cross section in order to compare it with the complete QED cross section. I was able to ...
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1answer
120 views

Srednicki's explanation of the 1PI quantum action

In chapter 21 (p.127-129) of Srednicki's book the quantum action $\Gamma(\phi)$ is defined in formula (21.1) I won't repeat here (it's quite long). Then he considers the following path integral: $$...
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QED electron self energy in effective action

The electron self energy at one loop is given by the one particle irreducible graph I know how to calculate it using the Feynman rules but I was wondering how this diagram appears in the QED ...
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55 views

Feynman rules for space-dependent coupling

Let's say I have an effective action which looks like (I got this action from large $N$ method for $\varphi^4$ theory): $$\int \frac{d^4x}{2g}\phi^2(x)+\int d^4x \ \log(-\nabla^2+\mu^2+i\phi(x)). $$ ...
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Must the mean field, in the context of the background field method, satisfy the classical equations of motion?

When deriving the effective action $\Gamma$ in the background field method, one splits the field $\phi = \phi_b + \phi_f$ into a background (or mean field) $\phi_b$ and fluctuations $\phi_f$, then ...
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1answer
165 views

1-loop Correction

Given a Lagrangian: $$L = \frac{1}{2}(\partial_u\phi)^2 + g (\partial_u\phi)^4.$$ Does anyone have idea how to write down the 1-loop correction? The derivative coupling is the part that confuses me.
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Why do the diagrams in $\Gamma[\Phi]$ differ from those in $\Phi\Gamma^{\rm int}_{\Phi}[\Phi]$ only by numerical prefactors?

Suppose $W$ is the generator of connected Feynman diagrams in $\Phi^4$ theory. We define $$\Gamma[\Phi]=W[j]-W_jJ,\tag{13.37}$$ where $$W_jJ=\int{dxW_j(x)j(x)}\tag{13.38}$$ and $$ \Phi\equiv\frac{\...
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29 views

Substitution of propagator to a product in Zinn-Justin

I'm reading Quantum Field Theory and Critical Phenomena, 4th ed., by Zinn-Justin and on page 127 he defines an action $$S_{\epsilon}[\phi]=\int{dxdy} \phi(x)\phi(y)[K(x,y) + \epsilon ] + V(\phi).$$ ...
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Any connected diagram is a tree of full propagators

In P. Etingof, Geometry & QFT, MIT 2002 online lecture notes; Lemma 3.11 (https://physics.stackexchange.com/users/7266/abdelmalek-abdesselam).) He says that any connected diagram is a tree of ...
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313 views

Quantum Scalar Field Theory with cubic and quartic interaction

If I have a scalar Lagrangian with and interaction term given by cubic and quartic terms (so a scalar theory + $φ^3+φ^4$ interaction), what are the possible divergent 1PI diagrams at one and two loops?...
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43 views

Effective potential and radiation corrections

I'm a bit confused on the idea of adding corrections to the classical potential of $\phi^4$ theory in QFT. From what I understand is that one should add corrections to the potential in order to ...
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64 views

How does the generalized effective action in Wetterich's exact RG scheme relate to observables at different scales?

I am not familiar with Wetterich's exact RG paradigm, and cannot understand the main idea behind it. I understand that if one could have solved the model and obtained the all the n-point functions ...
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1answer
229 views

2PI-effective action and functional derivatives

I'm trying to work out the 2PI-effective action for complex scalar fields. Introducing a multi field index $(a,b,c...)$ the complex conjugation and all other degrees of freedoms are suppressed, and ...
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1answer
275 views

Anomaly is due to the noninvariance of the path-integral under a symmetry. Is the noninvariance reflected on 1PI effective action?

When a symmetry is anomalous, the path integral $Z=\int\mathcal{D}\phi e^{iS[\phi]}$ is not invariant under that group of symmetry transformations $G$. This is because though the classical action $S[\...
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1answer
281 views

Proof of geometric series two-point function

In deriving the expression for the exact propagator $$G_c^{(2)}(x_1,x_2)=[p^2-m^2+\Pi(p)]^{-1}$$ for $\phi^4$ theory all books that i know use the following argument: $$G_c^{(2)}(x_1,x_2)=G_0^{(2)}...
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1answer
660 views

Definition of one-particle irreducible diagrams

Text books often defines one-Particle Irreducible diagram (1PI diagram) as a connected diagram which does not fall into two pieces if you cut one internal line. Is this internal line the full ...
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1answer
179 views

Proof of 1-particle irreducible (1PI) diagrams

If we split the effective action into $$Γ[Φ] =\frac{1}2ΦiG_0^{-1}Φ + Γ^{int} [Φ]$$ we can show that the full propagator is given by $$G= i[iG − Σ]^{-1}$$ With $$Σ=-Γ_{ΦΦ}^{int} [Φ]$$ Here $Γ_{...
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1answer
169 views

How do the renormalization factors disappear from the computation recipe of the S-matrix in Peskin & Schroeder (p. 229 eq. (7.45) & p.324)?

In the following I limit my considerations to 4-point diagrams. After the introduction of renormalized field operator (in renormalized perturbation theory) $$\phi_r= (\sqrt{Z})^{-1} \phi\tag{10.15}...
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The background field method of deriving a 2PI effective action. Calzetta and Hu book

I am going through "Nonequilibrium Quantum Field Theory" by Calzetta and Hu right now and it seems that I cannot fully understand the derivations in chapter 6.5. There, they consider the derivation of ...
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2answers
891 views

One-loop Correction to Effective Action

This might be a stupid question. In Bailin and Love's "Cosmology in gauge field theory and string theory", the authors are describing how to calculate the effective potential at a finite temperature ...
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1answer
188 views

Does $ℏ$ play a role in the 1PI effective action?

In most cases, people discuss the effective action or the effective potential in the convention $\hbar=1$. Occasionally, we see the expression at the 1-loop order as $$\Gamma[\phi]=S[\phi]+\frac{i\...
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1answer
498 views

Effective action of QED and the partition function

Given the partition function for QED $$ Z= \int \mathcal{D}A_{\mu}\mathcal{D}\Psi \mathcal{D}\bar{\Psi}\, \text{exp}\left(- \frac{i}{4}\int F_{\mu\nu} F^{\mu\nu} + i \int \bar{\Psi} (i {\not} D-m) \...
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1answer
154 views

What is the intuation of path integral in QFT? [closed]

It is known that the path integral in quantum mechanics means the summation of all probable classical trajectories between first and last measurement of the quantum state. In QFT this formalism leads ...
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1answer
347 views

Field renormalization of $\phi^4$ to second order

In Peskin & Schroeder Problem 10.3 pg. 345 they renormalize the field in $\phi^4$ theory using the following 2-loop sunset diagram. When looking at the correlation function $G^{(2)}_0$ this would ...
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3answers
567 views

Srednicki QFT Chapter 29: Feynman diagrams for calculating the effective action

I am trying to work my way through Srednicki Chapter 29 on Wilson's approach to renormalisation. However I am unsure why the Feynman diagrams Srednicki considers and calculates in this chapter are the ...
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1answer
231 views

Peskin and Schroeder Eq. 11.103: how does it relate to one-loop diagrams?

I am a bit confused by how Peskin & Schroeder describe the corrections to the two point function of the linear sigma model from the second functional derivative of the effective action. Without ...
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1answer
685 views

Difference Between Vertex Function and Self Energy

I am trying to understand the difference between the 2-point vertex function and the self energy. In many presentations, they are described in ways that seem nearly equivalent, yet as I work through ...
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1answer
228 views

About the calculation of one-particle-irreducible two-point diagram

This is derived from the answer and comments of this Phys.SE question concerning the calculation of two-point one-particle-irreducible diagram. On the one hand, according to the discussion on P.236 ...
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1answer
143 views

Is the sum of one-particle-irreducivle two-point diagrams always a real number?

On page 388 in section 11.6 of Peskin and Shroeder. There appears an equation of the inverse propagator(the second functional derivative of the effective action) for a theory that contains several ...
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2answers
251 views

Can one forget about the contribution of 1PR diagrams in computing a scattering amplitude?

From the LSZ reduction formula, it is clear that only the connected Feynman diagrams that contribute to a scattering amplitude. However, connected diagrams are of two types: 1PR and 1PI. 1PR diagrams ...
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Is Schrieffer-Wolff transformation equivalent to Feynman diagram and Path integral?

In high energy community, people usually use path integral (or Feynman diagram) to derive effective action (or effective Hamiltonian). However, in condensed matter or AMO community, people usually use ...
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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: ...
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1answer
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Concerning a functional of a functional of the former - classical fields in Quantum Action

Let $\varphi(x)$ and $j(x)$ be two field configurations. Let $\Gamma[\varphi]$ be a functional of the field $\varphi$ defined by: $$ \Gamma[\varphi] := \inf_j \ F[\varphi, j] = F[\varphi, j_\varphi] \...
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Probabilistic Intuition behind connected correlations and 1PI vertex function

In the context of statistical field or quantum field theory, one encounters so called generating function(al) for connected correlations, aka the following function(al): $$ W(J) = \ln (Z(J))$$ $$ Z(...
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1answer
160 views

A question concerning the effective quantum action for a scalar field

Define the quantum action $\Gamma[\varphi]$ by: $$ \Gamma[\varphi] := -\frac{1}2\int \frac{d^Dk}{(2\pi)^D} \varphi(-k)\Big(k^2 + m^2 - \Pi(k^2)\Big)\phi(k) \\+ \sum_{n=3}^\infty \frac{1}{n!}\int \...
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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 $$\phi_c(x)\equiv\langle 0|\hat{\phi}(x)|0\rangle_J$$ ...
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1answer
384 views

Defining a classical field corresponding to a quantum field

Why is the expectation value of the quantum field in the vacuum state $$\phi_c(x)=\langle0|\hat{\phi}(x)|0\rangle_J=\frac{\delta W}{\delta J}$$ referred to as the classical field? Why not the ...
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Why do we need the supremum when performing Legendre transformations?

Legendre transforms appear all over physics. For instance, in statistical mechanics, they allow us to move between descriptions in terms of different thermodynamic potentials. Similarly, in quantum ...
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1answer
62 views

Incorporation of adiabatic phase into quantum effective action

Suppose we have a system (or a subsystem) in the quantum state $|\text{in}\rangle$ and the same system in the state $|\text{out}\rangle$, which differs from $|\text{in}\rangle$ only by a phase: $$ \...
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Symmetry factor for 1PI Feynman diagrams in $\phi^4$ theory

I am trying to understand the various factors that the Feynman amplitude will carry corresponding to the Feynman diagrams of Fig. 1 of this reference. I understand that the $n^{th}$ diagram containing ...
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1answer
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Deriving the equality $\frac{\delta \Gamma[\phi_c]}{\delta\phi_c(x)}=0=\langle 0|\frac{\delta S[\phi,J]}{\delta\phi(x)}|0\rangle$?

I'm trying to convince myself that $$\Gamma[\phi_c]=W[J]-\int d^4x\hspace{0.2cm} j(x)\phi_c(x)$$ is the effective action i.e., it contains all quantum corrections to the classical action $S[\phi]$. ...
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1answer
1k views

A question about two-point 1-particle-irreducible diagram

I have a simple question about 1-particle-irreducible (1PI) diagram, I know I misunderstood something trivial but I just can not figure it out. Following Introduction to quantum field theory by ...
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160 views

Interpretation of the chiral anomaly a-la Alvarez-Gaume

In the paper "The topological meaning of non-abelian anomalies" written by Alvares-Gaume and Ginsparg they argue the appearing of the (gauge) anomaly in a theory with chiral fermions in the following ...
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180 views

Effective potential in Lagrangian

I have two question related to the steps in equations 3-7 in this paper: Question 1 They find the effective potential in eq. (5) as the negative of the effective Lagrangian (eq. (3)). I don't see how ...