Questions tagged [1pi-effective-action]

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12
votes
3answers
3k views

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)}...
13
votes
3answers
2k views

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 $$\phi_c(x)\equiv\langle 0|\hat{\phi}(x)|0\rangle_J$$ ...
14
votes
2answers
2k 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)\...
28
votes
1answer
4k views

Difference between 1PI effective action and Wilsonian effective action?

What is the simplest way to describe the difference between these two concepts, that often go by the same name?
2
votes
1answer
360 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)}...
5
votes
2answers
875 views

What is classical Lagrangian? The Bare one or renormalized one? Are counterterms quantum corrections to renormalized Larangian?

EDIT When one talks about the “classical Lagrangian” of a field, does one mean the renormalized Lagrangian with physical/renormalized masses and physical/renormalized couplings and without ...
2
votes
1answer
149 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 ...
1
vote
0answers
286 views

The locality of Wess-Zumino terms

Suppose the simple theory with chiral fermions possessing non-trivial gauge anomalies cancellation: $$ S = \int d^4 x \big(\bar{\psi}i\gamma_{\mu}D^{\mu}_{\psi}\psi + \bar{\kappa}i\gamma_{\mu}D^{\mu}_{...
24
votes
2answers
6k views

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 ...
6
votes
2answers
5k 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: ...
13
votes
1answer
2k views

Coleman-Weinberg potential: resum at 2 loops?

Say we want to compute the Coleman-Weinberg potential at 2 loops. The general strategy as we know is to expand the field $\phi$ around some background classical field $\phi \rightarrow \phi_b + \phi$...
10
votes
1answer
4k views

Understand “Quantum effective action” in Weinberg's book “The quantum theory of fields”

In Weinberg's book "The Quantum theory of fields", Chapter 16 section 1: The Quantum Effective action. There is an equation (16.1.17), and several lines of explanation, please see the Images....
8
votes
2answers
727 views

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 ...
8
votes
1answer
562 views

Confusion about the calculation of 1PI effective action using path-integrals

The Lagrangian of the $\phi^4$-theory can be written in terms of bare parameters as $$\mathcal{L}=\frac{1}{2}(\partial_\mu\phi_0)^2-\frac{1}{2}m_0^2\phi_0^2+\frac{\lambda_0}{4!}\phi_0^4\tag{1}.$$ The ...
2
votes
1answer
272 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 ...
2
votes
1answer
196 views

How does one ensure that effective action includes all possible quantum corrections to the clasical action?

Consider a classical scalar field theory for a real scalar field $\phi$ given by $$\mathcal{L}=\frac{1}{2}(\partial_\mu\phi)^2-V(\phi)$$ where $V(\phi)$ is the classical potential. In quantum field ...
4
votes
1answer
977 views

Resources for 2 Particle Irreducible (2PI) or Cornwall-Jackiw-Tomboulis (CJT) formalism

I'am currently learning the 2 particle irreducible (2PI) or Cornwall-Jackiw-Tomboulis (CJT) formalism. Does anybody know a textbook or a review that treats this subject? As far I only found the ...
3
votes
1answer
135 views

Rotating away a constant gauge field

In a few papers (see, for example, here, the bottom of the left column on the page 6, or here, the upper part of the page 5) I've met the strange calculations using the constant gauge field $$ A_{\mu}(...
2
votes
1answer
810 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 ...
1
vote
2answers
748 views

Does the connected Green's function's decomposition into 1PIs have connected contributions, or can it be written exclusively using 1PIs?

While reading this article by Abbot on the background field method, in Fig 5. on page 38 (page 6 in the pdf file), we can see the relation between connected contributions to the two point function and ...