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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1answer
2k views

About calculation of anomalous dimension in Peskin and Schroeder's book.

This question is in reference to question 13.2 in the QFT book by Peskin and Schroeder. To put it in general - I would like to know how does one define "anomalous dimensions" if one is given the ...
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764 views

Two-loop regularization

Working out some quantum field theory computations, I have to find out the value of the two-loop Feynman integral $$ I(p)=\int\frac{d^4p_1}{(2\pi)^4}\frac{d^4p_2}{(2\pi)^4}\frac{1}{(p_1^2+m_1^2)(...
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132 views

mathematical explanation for UV divergences and $ \delta ^{(n)}(0) $

is there any mathematical explanation for the UV divergences ?? i have read that in the framework of Epstein-Glser theory :D these UV divergences appear from the product of distributions anyone does ...
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Is there a non-perturbative remormalization? If so, how does it work?

Is there a method to renormalize a theory without using perturbative expansions for the divergences? For example, is there a method to get masses and other renormalized quantities without using ...
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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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2answers
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Is QCD free from all divergences?

On page 8 in http://arxiv.org/pdf/hep-th/9704139v1.pdf David Gross makes the following comment: "This theory [QCD] has no ultraviolet divergences at all. The local (bare) coupling vanishes, and the ...
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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 ...
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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 ...
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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 ...
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A certain regularization and renormalization scheme

In a certain lecture of Witten's about some QFT in $1+1$ dimensions, I came across these two statements of regularization and renormalization, which I could not prove, (1) $\int ^\Lambda \frac{d^2 k}...
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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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Functional determinant approximation

Let the Hamiltonian in one dimension be $H+z$, then I would like to evaluate $\det(H+z)$. I have thought that if I know the function $Z(t) = \sum_{n>0}\exp(-tE_{n})$ I can use $$\sum_{n} (z+E_{n})...
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How are functional determinants of Laplace-type operators used in physics?

Many mathematical papers concerning the $\zeta$-regularized Determinant of Laplace-type operators refer for motivation to the broad use of such determinants in mathematical physics, especially in ...
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Is there a Non-perturbative renormalization algorithm? [duplicate]

Possible Duplicate: is there non-perturbative RENORMALIZATION ?? if so how it works? Is there a non-perturbative renormalization algorithm ???, for example to avoid the divergent integrals and so ...
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725 views

Regularisation of infinite-dimensional determinants

Can a regularisation of the determinant be used to find the eigenvalues of the Hamiltonian in the normal infinite dimensional setting of QM? Edit: I failed to make myself clear. In finite dimensions,...
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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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681 views

Dimensional Regularization Integral Formula

In the formula $$\int \frac {d^{4-2\epsilon} \ell} {(2\pi)^{4-2\epsilon}} \frac 1 {(\ell^2-\Delta)^2} = \frac i {(4\pi)^{2-\epsilon}} \Gamma(\epsilon) \left(\frac 1 \Delta\right)^\epsilon,$$ how ...
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Zeta-function regularization in QFT for heat kernels

When one is doing zeta-function regularization of the heat-kernel for QFT then one is doing these following steps, the integral over the imaginary time taking the trace of the heat-kernel or the ...
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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 ...
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1answer
297 views

Pauli-Villars (PV) regularisation breaks supersymmetry. How to see that?

Does the PV regulator breaks SUSY? Take for instance the 1-loop (top/stop loops) correction to the Higgs squared-mass parameter in the MSSM, and you'll get something like, $$\delta m^2_{h_u} = - ...
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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 ...
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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 ...
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$\frac{1}{(1-x)_+}$ type distributions and parton distribution functions

I am trying to get to grips with Altarelli-Parisi-type equations. In chapter 17 of Peskin/Schroeder, they first develop the equations for a similar problem in QED. Equation $(17.123)$ introduces the ...
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630 views

Pade Approximant

I have some questions about Pade approximants. Given a divergent power series $ \sum_{n >0} a(n)x^{n} $ can we use a Pade Approximant to it $ R(x)$ so we can obtain a SUM of the series for every $...
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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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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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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 ...
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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: ...
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Limit of Lorentzian is Dirac Delta

I have a quick question that just came up in my research and I could not find an answer anywhere so I thought I'd try here. So one of the definitions of the Dirac Delta is the limit of the Lorentzian ...
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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}~...
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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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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 ...
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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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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 ...