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Questions tagged [renormalization]

Renormalization is an ensemble of techniques which serves to treat the infinities which appear in quantum field theory or statistical mechanics.

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What's wrong with lattice quantum gravity?

Assume one can write the metric field on a lattice, so on each lattice point one has a value of $g^{\mu\nu}$. Similar to the way lattice QCD is formulated. Then later taking the distance between ...
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Are problems with self-energy of point charge in classical electrodynamics solved by field quantization?

Classical electrodynamics gives strange results when considering a moving charge in its self generated field (Abraham-Lorentz equation). Some 50 years ago there were many efforts and publications ...
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Massless $\phi^4$ theory

Most of the standard textbooks on QFT discuss in some detail the massive $\phi^4$ theory in 4d space-time. I would be interested to see a discussion of massless $\phi^4$ theory (in fact other non-...
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Renormalization of sine gordon theory

So assume that we have a usual sine gordon theory in the the theory we have a term in the hamiltonian $$\frac{yu}{2\pi\alpha^2}\int dx \cos(\sqrt{8}\phi_\sigma(x))$$ where $\alpha$ is cut off ...
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Superficial degree of divergence in $\lambda\phi^4$

Ryder at the beginning of the chapter about renormalization defines the "superficial degree of divergence" of diagrams in $\lambda \phi^4$ theory. I'll recap the derivation. A diagram in $\lambda\phi^...
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Subtraction scheme invariance in QFT

I'm currently reading Schwartz's QFT text and I'm confused on how observables are supposed to be independent of subtraction schemes. In the text it seems that the renormalized loop diagrams are ...
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Why can we add counterterms?

I'm having a hard time understanding why renormalized perturbation theory works. Why is it permissible to add counterterms to the Lagrangian? Terms which are often divergent themselves and carry ...
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Running coupling constants within a highly compressed object

I wonder is it possible. in highly compressed objects, such as neutron stars and black holes, (I'm not sure that this applies to singularities), that the physical conditions within these objects ...
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Why does the Lagrangian Density have to be a polynomial of the field?

In a lecture, a professor appeared to have said that the Lagrangian can only contain terms that have powers of $\phi$ and a term with $\partial_\mu \partial^\mu \phi$ . I imagine this would make any ...
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Finding ground state energy using numerical real space renormalization group

I want to find ground state energy (as well as wavefunction) for spinless $tV$ model using Real-Space Renormalization Group (RSRG) approximation. The $tV$ model is defined as $$H=H_t+H_{int}=-t\sum_{i=...
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Is there any relation between density matrix renormalization group (DMRG) and renormalization group (RG)?

Probably I am going to receive many down-votes for this post but I really need to ask this question here. I am new to statistical mechanics. I wanted to learn Density Matrix Renormalization Group (...
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${\cal N} = 1$ SUSY Non-renormalization theorem

In Ref. 1, on Page 53, the ${\cal N} = 1$ SUSY non-renormalization theorem is derived. One first specifies the symmetries of the general ${\cal N} = 1$ SUSY action in the superspace formalism, and ...
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What would a non-perturbative renormalization group treatment for polymers look like?

I know that one can do perturbative renormalization for the polymer excluded volume problem or the self-avoiding walk problem corresponding to n=0 component field theory. Here in Hamiltonian, we have, ...
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What is the publication through which Zinn-Justin published what has come to be known as the “Zinn-Justin equation”?

does anybody know which publication contains the introduction of what has come to known as the Zinn-Justin equation?
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For dimensional regularization, why the arbitrary mass scale $\mu$ has the meaning of UV cutoff?

For sharp cut off regularization, we introduce the UV cutoff $\Lambda$. When we need to do momentum integral, we integrate the momentum ball with radius of $\Lambda$. This $\Lambda$ has the explicit ...
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Why is the standard model renormalizable if we believe it is an effective theory? [duplicate]

We believe that the standard model is only an effective field theory of its true UV completion. However, effective theories have dimensionful couplings and are not renormalizable. The standard model ...
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Closed set of operators under renormalization

While reading the article http://inspirehep.net/record/61135, I came across the concept of "closed set under renormalization". The definition they give is the following. In any renormalizable field ...
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Does the vanishing of the one loop beta function imply no running to all orders?

This question sounds ridiculous, but bear with me. I am having a hard time reconciling the following two facts: Classical global symmetries can become anomalous upon quantization, and the anomalous ...
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How to fuse quantum mechanics and general relativity?

I am very new to this topic but I have started reading Kevin Wray's lecture notes about string theory (PDF) and in the introduction he says: "Sometimes it is said that we don’t understand how to ...
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Dimension of gamma matrices in dimensional regularization

When performing loop integrals in theories containing Dirac fermions, one almost always confronts terms of the form $$\text{Tr}\left[\gamma^{\mu_1}\cdots\gamma^{\mu_n}\right].$$ For instance, in $d$ ...
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Viewing anomalous dimensions in RG as a quantum anomaly

Other than sharing the word “anomalous”, both the anomalous dimension in RG and the more well-known quantum anomalies (such as chiral anomaly) share a common feature. These are violations of classical ...
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Quantum field theory, super-renormalizable interactions

Is it really necessary to renormalize a super-renormalizable theory?
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Confusion on real space renormalization group for Ising model on lattice

For the Ising model with only nearst neighbor interaction on square lattice, if we do the RG by integrating out half degree of freedom, then we would get a new Ising model with many kind of ...
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Computing the Wilsonian Action

Equation 12.5 of Peskin&Schroeder reads $$Z = \int\left[\mathcal{D}{\phi}\right] e ^{-\int d^dx \, \frac{1}{2} (\partial \phi)^2 + \frac{m^2}{2}\phi^2 + \frac{\lambda}{4!}\phi^4} \cdot \...
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Why are RG flow fixed points associated with different phases?

Why are RG flow fixed points associated with different phases? I thought the RG makes only statements about behavior near to critical points... a definite phase is far away from the critical point, ...
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Why does the fine-structure constant $α$ have the value it does?

This is a follow-up to this great answer. All of the other related questions have answers explaining how units come into play when measuring "universal" constants, like the value of the speed of ...
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How many counterterms does QED have?

I have read the statement that QED has four counterterms to cancel divergences. However, I have learnt that there are only three counterterms (vertex, electron propagator, photon propagator), which is ...
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Understanding how is a field theory with a negative mass dimensional coupling becomes nonrenormalizable?

A quantum field theory with a negative mass dimensional coupling (or equivalently, an operator having mass dimension $>4$) is nonrenormalizable. Therefore, it must be the case that (i) the ...
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Interaction term in free energy for Gaussian Fixed Point

In general in statistical field theory, the free energy $F_0$ as a function of our order parameter $\phi$ can be written as $$F_0[\phi]=F_0[\phi^-]+F_0[\phi^+]+F_I[\phi^-,\phi^+]$$ where the last ...
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How does renormalization relate to emergence?

In statistical mechanics renormalization is often related to coarse-graining which in turn allows to calculate some macroscopic states. The resulting macroscopic description is sometimes called ...
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2answers
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Regularization: Evaluating the one-loop $\phi^4$ integral to order $\lambda^2$

I'm currently on the chapter of regularization on Zee's QFT book. For the $\phi^4$ theory, an amplitude for a single loop correction to order $\lambda^2$ is given by a diagram Following the Feynman ...
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Why is vanishing beta function associated with scale-invariance?

Why is vanishing beta function associated with scale-invariance? Coupling constants have change rate of zero at some scale, but how is that related to scale-invariance? Association of vanishing beta ...
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“Running with momentum $p$” v.s. “running with renormalization scale $\mu$”

The renormalized charge/coupling in QFT is usually phrased as renormalization scale $\mu$ dependent $e(\mu)$ in the renormalization group setting. But can we take the more elucidating angle of "...
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Normal ordering in 2D thermal CFT

I am trying to understand the notion of normal ordering in thermal CFT in 2D CFT, for instance I consider a two-point function of scalar primary operator with $\Delta$ dimension at finite temperature $...
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When was the phrase “beta function” of renormalization first used?

My question is a historical one: when was the phrase "beta function", as it pertains to the renormalization-group equations, used in physics? I am talking about this beta function: $$\beta_g\equiv \...
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Renormalized mass

I am reading Schwarz QFT and I reached the mass renormalization part. So he introduces, after renormalization, a physical mass, defined as the pole of the renormalized propagator, and a renormalized ...
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Can a unified physics theory have a smaller number of couplings than its effective field theory?

Suppose that we have a QFT that has $n$ number of physical coupling constants, or there are $n$ coupling constants required to perturbatively renormalize the given QFT. Suppose this QFT to be an ...
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Perturbative vs Wilsonian renormalization in 2D

In 2D a scalar field is dimensionless so terms in the Lagrangian $\phi^n$ of arbitrary power are renormalizable (indeed we can even have two derivatives). This has two consequences that seem to be in ...
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Mistake in Peskin & Schroeder, Renormalization of Linear Sigma Model?

In section 11.4 of Peskin & Schroeder's "Introduction to Quantum Field Theory", the authors calculate the effective potential of the linear sigma model to one-loop order: $$\begin{align*} V_{\...
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Why do we have freedom to choose a renormalization scale in massless QFT theories?

I thought that the freedom to choose renormalization conditions arises from the freedom to choose the arbitrary renormalization parameters. Let me exemplify this in a Massive $\phi^4$ scalar theory ...
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How do the renormalization factors disappear from the computation recipe of the S-matrix in Peskin & Schroeder (p. 229 (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) $\phi_r= (\sqrt{Z})^{-1} \phi$ in eq. (...
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The significance of the infinite electric charge relating to renormalisation in QED

This question is related Significance of electrical fields of infinite objects but is not a duplicate, imo. My question is based on reading Frank Close's book "The Infinity Puzzle", which on page 42 ...
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In charge renormalization equation, $e=e_{0}^{2}\left[1-e_{0}^{2}A\right]$, how can an infinite $e_{0}$ and $A$ give finite $e$ in any limit?

In Griffiths elementary particle book (chapter 7, 'Quantum electrodynamics', equation 7.188), one gets the following equation for the vacuum polarization calculated to one loop correction. $$\frac{e_{...
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How to extract a finite answer after applying dimensional regularization in QED?

When one applies dimensional regularization in QED, in the end, one often gets an expression like $$\Gamma(n/2)\left(\frac{s}{\mu^{2}}\right)^{-n/2}$$, where $n$ is a small number, $\Gamma()$ is the ...
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QED vertex correction, proper vertex function and meaning

I might be making great confusion in trying to interpret proper vertex function. I'm studying QED vertex correction. I'm just going to write down the pieces of the puzzle. So I know that the ...
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Diagrams Contributing to Electric Charge Correction (M. Schwartz)

In M. Schwartz's Quantum Field Theory and the Standard Model, he talks about the correction to the $g$-factor in Chapter 17. In Section 17.2 (Pg. 318), he is evaluating the diagram that gives the ...
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Sign of counterterm vertex factor (Srednicki)?

My question is about a mere minus sign which, although irrelevant in my specific problem (as will be shown), I fear may bite me later on. In Srednicki chapter 14, the author is computing the 1-loop ...
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Anomalous dimensions and RG flow - Polchinski 1984

Usually while defining a renormalizable field theory in the UV, we can assume it to have a simple form, i.e only containing a few operators(the relevant and marginal ones). Now, as we start flowing ...
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Running of gravitational constant versus. other coupling constants, GUT and TOE

Is there any theory that proves convincingly that the gravitational constant runs with energy AND it is like the other coupling constants and converge at a single point at a given energy? Or is that a ...
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Momentum Shell RG for Ising Model

I am studying the book "Introduction to Functional RG" by Kopietz. Having introduce all needed concepts, authors obtained the following RG-flow equations: $$\partial_l\bar{r}_l=2\bar{r}_l+\frac{1}{2}\...