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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How does the sum of the series “1 + 2 + 3 + 4 + 5 + 6…” to infinity = “-1/12”? [duplicate]

How does the sum of the series “1 + 2 + 3 + 4 + 5 + 6…” to infinity = “-1/12”, in the context of physics? I heard Lawrence Krauss say this once during a debate with Hamza Tzortzis ...
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Assymptotic freedom significance

So I have read a bit on this, and get the idea and mathematical machinery leading up to this. I get that it sheds light on the relationship between coupling strengths and length scales. Can someone ...
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Quantum corrections to massless fermionic field

in QED the corrections to electron propagator change the bare electron mass from $m_0$ to $m=m_0+δm=m_0+∑(\not{p}=m)$ (Peskin, formula 7.27). This is the consequence of the fact, that the quantum ...
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QED coupling constant at one loop

On page 257 in Peskin's QFT book a qualitative sketch of the QED coupling is given (see the picture below). Why should I expect such a behavior from QED? The QED beta function is ...
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Renormalization beta-function

I want to know the other forms of beta-function that make manifest certain properties of renormalization group, for instance dependence on poles/residue and more. If possible can you state a ...
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independence of the bare parameters on μ for beta function

So I know re-normalization has bean "beaten to death". I want to understand something a bit specific which might seem trivial. Independence of the bare parameters on $\mu$ and relevance to the beta ...
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Divergent Series

Why is it that divergent series make sense? Specifically, by basic calculus a sum such as $1 - 1 + 1 ...$ describes a divergent series (where divergent := non-convergent sequence of partial sums) ...
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Regularization of the Casimir effect

For starters, let me say that although the Casimir effect is standard textbook stuff, the only QFT textbook I have in reach is Weinberg and he doesn't discuss it. So the only source I currently have ...
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How to show that tensor gravity is nonrenormalizable?

Let's have the tensor gravity theory, which represented the massless spin-2 field: $$ L = -\frac{1}{32 \pi G}\left( \frac{1}{2}(\partial_{\alpha}h_{\nu \beta}) \partial^{\alpha}\bar {h}^{\nu \beta} - ...
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Derivative with respect to ${\not}{p}$

When studying renormalization of QED in standard textbooks, we typically encounter derivatives with respect to ${\not}{p}=p^\mu \gamma_\mu$, i.e., $\partial/\partial{\not}p$. As far as I understand, ...
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Basic question about the S-Matrix, Unitarity and Effective Field Theory

Consider scattering some particles in a state collectively denoted by $i$ to a final state denote by $f$. The scattering amplitude, S-matrix is then defined by: $S_{fi}\equiv \langle ...
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Peskin's book page 334 proof of $Z_1=Z_2$ to all orders in QED perturbation theory

Peskin in his QFT page 334 argued that $Z_1=Z_2$ to all orders in QED perturbation theory, but I couldn't understand his argument: ... With a generalization of the argument given there (section ...
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Explicit calculation of bosonic string Weyl invariance at one loop

I have been trying to do all the calculations in the Green, Schwarz and Witten Superstring Theory textbook. At the end of chapter 3, the author did one-loop calculation for Weyl invariance for the ...
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Semantic problem about renormalizability

This post relates to this previous one. My question is, what is the actual meaning of a theory being renormalizable? There might be at-least two possibilities (correct me if I am wrong) ...
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One-loop beta functions of the Standard Model

For my master's research on energy scale independent combinations of renormalization group equations in supersymmetric theories, I need an overview of all the one-loop beta functions of the Standard ...
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Renormalization of Composite Chiral Superfields

On page 24 of these lecture notes http://arxiv.org/abs/hep-th/0309149 it is stated that products of chiral superfields do not suffer from short distance singularities. In other words, if I want to ...
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Anomalous dimension for bare actions with a standard kinetic term

In this paper on p42, it is explained that when starting with a bare action that contains a standard kinetic term, this kinetic term attains a correction in the course of the RG flow which can be ...
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Running chargino/neutralino masses in MSSM

Consider the plot below, showing the running of different masses due to renormalization for a certain point of the (c)MSSM. I am able to exactly reproduce the plot, including the running of M1, M2, ...
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Amplitudes in renormalized perturbation theory

This question arose while reading Peskin and Schroeder, specifically, it arose in regards to the sum of diagrams above their Eq. (10.20) on pg. 326. The context is $\phi ^4$ theory and they are using ...
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Link between anomalous dimensions and fractal dimensions

I just realized that anomalous dimensions in quantum/statistical field theory is not that different from fractal dimensions of objects. They both describe how quantitaive objects transform under a ...
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Why does the counterterm's propagator have inverse units of the propagator? $\phi^4$-theory

According to Peskin & Schroeder (page 325), the Feynman rule for the counterterm ------(x)----- for $$ \frac12 ...
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Why do three dimensional gauge theories flow to conformal theories in the infrared?

What is meant with the fact that Super Yang-Mills flows to a conformal field theory in the infrared? Also, is this a general fact or does this depend on the fact of considering a certain class of ...
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Misunderstanding Wick Ordering

In M. Salmhofer's "Renormalization, An Introduction" Wick ordering is defined as follows: Let $C = C_\Gamma$ be a nonnegative symmetric operator on $\mathbb{C}^\Gamma$. For $J: \Gamma \to ...
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Why isn't renormalization called dark physics?

In QED there is extra energy that has to be gotten rid of to match observations. Kind of the opposite to GR/ND where you have to add extra energy/matter to match observations. Why isn't ...
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Evaluating propagator without the epsilon trick

Consider the Klein–Gordon equation and its propagator: $$G(x,y) = \frac{1}{(2\pi)^4}\int d^4 p \frac{e^{-i p.(x-y)}}{p^2 - m^2} \; .$$ I'd like to see a method of evaluating explicit form of $G$ ...
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IR non-renormalizable theory

can be a theory with an infinite number of divergent integrals of the form $$ \int \frac{d^{p}k}{k^{m}} $$ for m=1 , 2 , 3 , 4 ,...... so the theory would be IR non renormalizable and you would need ...
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Beta function from exact renormalization group equation

I'm trying to calculate beta functions from exact (or functional) renormalization group equation (I mean Wilson-Polchinski RGE). I've got the equations but I don't know exactly how to use the ...
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Renormalizability of the Polyakov Action

I was told today that the Polyakov action for a $p$-brane is (superficially) re-normalizable iff $p\leq 1$. Of course, when I went to check for myself, I screwed up my power-counting, and I'm having ...
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Why are only logarithmic divergence relevant for the Callan-Symanzik equation? Intuitive understanding?

I may be wrong, but it seems that only logarithmic divergences need to be retained when using the Callan-Symanzik equation, finding running couplings, etc. Why is this the case? Is there some simple ...
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If Renormalization Scale is Arbitrary, Why Do We Care about Running Couplings?

For the bounty please verify the following reasoning [copied from comment below] Ah right, so the idea is that overall observable quantities must be independent of the renormalization scale. But at ...
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What happens when you apply the path integral to the Einstein-Hilbert action?

The Einstein Field Equations emerge when applying the principle of least action to the Einstein-Hilbert action, and from what I understand the path integral formulation generalizes the principle of ...
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Why do we expect our theories to be independent of cutoffs?

Final edit: I think I pretty much understand now (touch wood)! But there's one thing I don't get. What's the physical reason for expecting the correlation functions to be independent of the cutoff? ...
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Is the effective Lagrangian the bare Lagrangian?

In standard (non-Wilsonian) renormalization we split the bare Lagrangian $\mathcal{L}_0$ into a physical Lagrangian $\mathcal{L}_p$ with measurable couplings and masses counterterms ...
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Sharp cut-off, quadratic corrections and naturalness

When introducing the fine-tuning problem, a sharp cut-off as a regulator in the calculation of the Higgs mass corrections is used. Since this regulator breaks translational and gauge invariance, up to ...
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What is the definition of a “UV-complete” theory?

I would like to know (1) what exactly is a UV-complete theory and (2) what is a confirmatory test of that? Is asymptotic freedom enough to conclude that a theory is UV-complete? Does it become ...
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Ambiguity in Beta Functions (2-loop)

Beyond one-loop, the beta function of a QFT is scheme dependent. I would like to understand better this ambiguity. The easiest thing to say is that you haven't calculated something physical, so of ...
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Does effective theory have the same meaning in particle and condensed matter physics

I have a naive question about the meaning of effective theory in particle physics and condensed matter physics. In particle physics, from what I know, the effective theory comes from the Wilsonian ...
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What does it mean for a QFT to not be well-defined?

It is usually said that QED, for instance, is not a well-defined QFT. It has to be embedded or completed in order to make it consistent. Most of these arguments amount to using the renormalization ...
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Relating the deformation of Calabi-Yau metrics and the conformal quantum field theories

(v2) As I read e.g. in this question, the nice holonomy group features of Calabi-Yau manifolds are valuable regarding supersymmetry (I suspect because it's a symmetry involving the target manifold, ...
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Why do people rule out zeta regularization for renormalization?

Using zeta regularization one can get a formula for regularizing the integral $ \int_{a}^{\infty}x^{m-s}\text dx $ for any $m$. However, I have not seen anywhere. For example, I do not know why in ...
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Gauge fields in Polyakov's treatment of renormalization for nonlinear sigma model

I am deriving the results of renormalization for nonlinear sigma model using Polyakov approach. I am closely following chapter 2 of Polyakov's book--- ``Gauge fields and strings''. Action for the ...
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What is energy in $z \neq 1 $ theories?

In a critical theory with dynamical critical exponent $z \neq 1 $, which amongst frequency, $\omega$, and dispersion, $E(\vec{k})$, may be referred to as ''energy''? I'm confused about this since in ...
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What are the details of the renormalization of Chern-Simons theory?

What is a good, simple argument as to why Chern-Simons theory' is renormalisable? Any good books/references dealing with this effectively? Why does the $\beta$-function vanish? Thanks!
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Two Loop ultraviolet divergence of $\phi^4$ theory

During the renormalization procedure of a massive $\phi^4$ theory at two loop level, one finds that the quantity $K_{\Lambda}(k^2,m^2)-K_{\Lambda}(0,m^2)$ that appears in the bare $\Gamma^{(2)}$ ...
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Why is renormalization necessary in finite theories?

Michael Brown made the following comment here: The modern understanding of renormalization (due to Kadanoff, Wilson and others) is hardly controversial and has nothing really to do with ...
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How to determine that the renormalization constant $Z_3$ must depend only on $g$ and $\Lambda/m$

In Le Bellac's book, Quantum and Statistical Field Theory, the renormalization constant $Z_3$ is introduced with the equation $$ \Gamma^{(2)}_R(k^2, m^2, g) = Z_3 \Gamma^{(2)}(k^2, m_0^2, g_0; ...
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Nonequilibrium themal QFT

Wick rotation to thermal of QFT in Minkowski space to thermal QFT, which is after this transformation analogue to statistical mechanics, does only describe equilibrium statistical mechanics. On page ...
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How does the Ward-Takahashi Identity imply that non-transverse photons are unphysical in QED?

Peskin and Schroeder say that the Ward Identity of QED proves that non-transverse photon polarizations can be consistently ignored, but I'm confused about the details. Setup One starts by ...
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Running of gauge couplings in the Standard Model [closed]

I'm sure many of us are familiar with the following plot showing the running of the inverse of the fine-structure constants of the SM. (I got the picture from google) At one-loop, the expressions ...
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Charge Renormalization and Photon Propagator

I'm trying to understand charge renormalization in QED. I know that one can write the full photon propagator as $$\frac{-i\eta_{\mu\nu}}{q^2(1-\Pi(q^2))}$$ where $\Pi$ is regular at $0$. Obviously ...