Questions tagged [regularization]

In QFT, regularization is a method of addressing divergent expressions by introducing an arbitrary regulator, such as a minimal distance *ϵ* in space, or maximal energy *Λ*. While the physical divergent result is obtained in the limit in which the regulator goes away, *ϵ* → 0 or *Λ* → ∞, the regularized result is finite, allowing comparison and combination of results as functions of *ϵ, Λ*. Use for dimensional regularization as well.

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112
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1answer
7k views

Superfields and the Inconsistency of regularization by dimensional reduction

Question: How can you show the inconsistency of regularization by dimensional reduction in the $\mathcal{N}=1$ superfield approach (without reducing to components)? Background and some references: ...
67
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1answer
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Classical and quantum anomalies

I have read about anomalies in different contexts and ways. I would like to read an explanation that unified all these statements or points of view: Anomalies are due to the fact that quantum field ...
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5answers
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What exactly is regularization in QFT?

The question. Does there exist a mathematicaly precise, commonly accepted definition of the term "regularization procedure" in perturbative quantum field theory? If so, what is it? Motivation and ...
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2answers
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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? I....
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7answers
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Rigor in quantum field theory

Quantum field theory is a broad subject and has the reputation of using methods which are mathematically desiring. For example working with and subtracting infinities or the use of path integrals, ...
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14answers
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Suggested reading for renormalization (not only in QFT)

What papers/books/reviews can you suggest to learn what Renormalization "really" is? Standard QFT textbooks are usually computation-heavy and provide little physical insight in this regard - after my ...
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6answers
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Fourier transform of the Coulomb potential

When trying to find the Fourier transform of the Coulomb potential $$V(\mathbf{r})=-\frac{e^2}{r}$$ one is faced with the problem that the resulting integral is divergent. Usually, it is then argued ...
29
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5answers
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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 ...
27
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1answer
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Instantons, anomalies, and 1-loop effects

A symmetry is anomalous when the path-integral measure does not respect it. One way this manifests itself is in the inability to regularize certain diagrams containing fermion loops in a way ...
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Why Does Renormalized Perturbation Theory Work?

I've read about renormalization of $\phi^4$ theory, ie. $\mathcal{L}=\frac{1}{2}\partial_\mu\phi\partial^\mu\phi-m^2\phi^2-\frac{\lambda}{4!}\phi^4\,,$ particularly from Ryder's book. But I am ...
25
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1answer
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Regulator-scheme-independence in QFT

Are there general conditions (preservation of symmetries for example) under which after regularization and renormalization in a given renormalizable QFT, results obtained for physical quantities are ...
23
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Don't understand the integral over the square of the Dirac delta function

In Griffiths' Intro to QM [1] he gives the eigenfunctions of the Hermitian operator $\hat{x}=x$ as being $$g_{\lambda}\left(x\right)~=~B_{\lambda}\delta\left(x-\lambda\right)$$ (cf. last formula on ...
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1answer
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Difference between regularization and renormalization?

In quantum field theory we have the concepts of regularization and renormalization. I'm a little confused about these two. In my understanding regularization is a way to make divergent integrals ...
19
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1answer
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What's the relation between Wilson Renormalization Group (RG) in Statistical Mechanics and QFT RG?

What's the relation between Wilson Renormalization Group(RG) in Statistical Mechanics and QFT RG? For easier to compare, I choose scalar $\phi^4$ in both cases. Wilson RG: Given $\phi^4$ model, $$Z=...
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3answers
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Does renormalization make quantum fields into (slightly) nonlinear functionals of test functions?

Quantum fields are presented as operator-valued distributions, so that the operators in the theory are linear functionals of some test function space. This works well for free fields, giving us a ...
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Can dimensional regularization solve the fine-tuning problem?

I have recently read that the dimensional regularization scheme is "special" because power law divergences are absent. It was argued that power law divergences were unphysical and that there was no ...
17
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800 views

Other processes than formal power series expansions in quantum field theory calculations

I am not sure if this question is too naive for this site, but here it goes. In QFT calculations, it seems that everything is rooted in formal power series expansions, i.e. , what dynamical systems ...
16
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497 views

How do we know that analytic continuation agrees with UV regulators?

Consider the divergent series $$S = 1 + 1 + 1 + \ldots$$ which may appear in some calculations involving the Casimir effect. There are two main ways to evaluate this series. One can perform analytic ...
15
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6answers
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Why are we scared of singularities? [closed]

I often hear people say that general relativity predicted its own demise because of the singularities it predicts. If I'm not mistaken, this is also a problem in QFT. I wonder why singularities garner ...
15
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2answers
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Massless integrals in dim-reg

Consider the massless divergent integral $$ \int dk^4 \frac{1}{k^2}, $$ which occurs in QFT. We can't regularize this integral with dim-reg; the continuation from the massive to the massless case is ...
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Why regularization?

In quantum field theory when dealing with divergent integrals, particularly in calculating corrections to scattering amplitudes, what is often done to render the integrals convergent is to add a ...
14
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4answers
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Gelfand-Yaglom theorem for functional determinants

What is the 'Gelfand-Yaglom' Theorem? I have heard that it is used to calculate Functional determinants by solving an initial value problem of the form $Hy(x)-zy(x)=0$ with $y(0)=0$ and $y'(0)=1$. ...
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2answers
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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) but,...
14
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1answer
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Chiral anomalies à la Fujikawa: Why don't we just take another measure?

When deriving the chiral anomaly in the non perturbative approach for a theory of massless Dirac fermions, you start by showing that the path-integral measure is not invariant unter the chiral ...
14
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1answer
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Renormalizing IR and UV divergences

In lectures on effective field theory the professor wanted to find the correction to the four point vertex in massless $\phi^4$ theory by calculating the diagram, $\hspace{6cm}$ We consider the zero ...
14
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1answer
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Dimensional regularization: removing more than just logarithmic divergencies?

I have followed two courses on QFT, which both involved renormalization by dimensional regularization. My confusion is that one of the professors claimed that dimensional regularization can only be ...
14
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1answer
964 views

Dimensional Regularization and Massless Integrals

Consider the following integral: \begin{equation} \int \frac{d^3 k}{(2\pi)^3} \frac{1}{k^3} \end{equation} In dim-reg, such integrals evaluate to $0$. However, if we instead consider \begin{...
14
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1answer
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Which values of the Riemann zeta function at negative arguments come up in physics?

For my bachelor's thesis, I am investigating Divergent Series. Apart from the mathematical theory behind them (which I find fascinating), I am also interested in their applications in physics. ...
13
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1answer
834 views

Renormalization is a Tool for Removing Infinities or a Tool for Obtaining Physical Results?

Quoting Wikipedia: renormalization is any of a collection of techniques used to treat infinities arising in calculated quantities. Is that true? to me, it seems better to define renormalization ...
13
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2answers
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Are there books on Regularization and Renormalization in QFT at an Introductory level?

Are there books on Regularization and Renormalization, in the context of quantum field theory at an Introductory level? Could you suggest one? Added: I posted at math.SE the question Reference ...
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Laplacian of $1/r^2$ (context: electromagnetism and Poisson equation)

We know that a point charge $q$ located at the origin $r=0$ produces a potential $\sim \frac{q}{r}$, and this is consistent with the fact that the Laplacian of $\frac{q}{r}$ is $$\nabla^2\frac{q}{r}~...
12
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3answers
700 views

“Dimensional analysis” arguments in quantum field theory

I'm uncomfortable with dimensional analysis arguments made in quantum field theory, particularly those related to renormalization. For example, in section III.2 of Zee's QFT book, it says: Consider ...
12
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2answers
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Critical Dimension of Bosonic Strings and Regularization of $\sum_{n=1}^\infty n$

If $D$ is critical dimension of Bosonic strings, a particular derivation goes like the following, where we arrive finally at $$ \frac{D-2}{2}\sum_{n=1}^\infty n + 1 = 0. $$ Now mathematically this is ...
12
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2answers
751 views

Power divergences from loops

I do not know what I should think about power divergences from loops. Most QFT textbooks tell us how to deal with logarithmic divergences from loops $\sim\ln(\Lambda^2/\Delta)$: we can set a ...
12
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2answers
972 views

On the Axial Anomaly

I know that if we start with a massive theory, the chiral states $L$ and $R$ remain coupled to each other in the massless limit. Because a charged Dirac particle of a given helicity can make a ...
12
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0answers
374 views

Gauge invariant but not gauge covariant regularization

I'm not sure if someone's already asked this before, but I was wondering, in field theory, when we say that a certain field is gauge invariant but not gauge covariant, what does this mean? In ...
11
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4answers
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Is QFT mathematically self-consistent?

After recently going through a short program of self-study in quantum mechanics, I was surprised to find a quote attributed to Feynman essentially saying he was extremely bothered by the computational ...
11
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1answer
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Does string theory provide a physical regulator for Standard Model divergencies?

In other question, Ron Maimon says that he thinks string theory is the physical regulator. I did not know that string theory regularize divergencies. So, Q1: How does string theory regularize the ...
10
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2answers
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Is the step of analytic continuation unavoidable or can you model around it?

One sometimes considers the analytic continuation of certain quantities in physics and take them seriously. More so than the direct or actual values, actually. For example if you use the procedure ...
10
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2answers
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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 ...
10
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1answer
817 views

Iterated dimensional regularization

Given a 2-loop divergent integral $\int F(q,p)\,\mathrm{d}p\mathrm{d}q$, can it be solved iteratively? I mean I integrate over $p$ keeping $q$ constant Then I integrate over $q$ In both iterated ...
10
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1answer
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Symmetries in Wilsonian RG

I wanted to know if there is a theorem that in writing a Lagrangian if one missed out a term which preserves the (Lie?) symmetry of the other terms and is also marginal then that will necessarily be ...
10
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0answers
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Why does analytic continuation as a regularization work at all?

The question is about why analytical continuation as a regularization scheme works at all, and whether there are some physical justifications. However, as this is a relatively general question, I ...
9
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3answers
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Applications of analytic continuation to physics

I posted this on math.SE, but didn't get much response. It might fit better on this site. Holomorphic functions have the property that they can be uniquely analytically continued to (almost) the ...
9
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2answers
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What is IR CFT and UV CFT?

What is IR CFT and UV CFT? In many physics related materials, they often mention IR, and UV. I think it is related with regularization (I remember in QFT, there is UV cutoff in some regularization ...
9
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2answers
658 views

The integral is zero! $\int \frac{\mathrm{d}^d k}{(2\pi)^d} = 0$

In using dimensional regularization in QFT calculations, one comes across integrals over propagators, they might look like $(d = \text{dimension of spacetime}, n = \text{a number})$ $$\tag{1}I(d,n)=\...
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3answers
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Coulomb potential in 2D

I know that the Coulomb potential is logarithmic is two dimensions, and that (see for instance this paper: http://pil.phys.uniroma1.it/~satlongrange/abstracts/samaj.pdf) a length scale naturally ...
9
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1answer
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Why can consistent QFTs only arise from CFTs?

This is claimed by Jared Kaplan in his Lectures on AdS/CFT from the Bottom Up. He writes: It seems that all QFTs can be viewed as points along an Renormalization Flow (or RG flow, this is the ...
9
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2answers
779 views

Point splitting technique in Peskin and Schroeder

One of the cornerstones of point splitting technique of calculating chiral anomaly (Peskin and Schroeder 19.1, p.655) is a symmetric limit $\epsilon \rightarrow 0$. And this is the point that I don't ...
9
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1answer
428 views

Divergent integrals in QFT

I am starting to learn about QFT and something that I noticed is that integrals who would diverge otherwise are assigned a value if we do it by contour integration using the residues theorem and the ...