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1
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1answer
56 views

Interacting Lagrangian - Coupling constant and cutoff factor

I have a general question concerning a given interacting Lagrangian: $$\mathfrak{L}_I = \frac{g}{\Lambda^2} \bar{\chi} \ \gamma^\mu \gamma_5 \ \chi \ \partial^\nu F_{\mu\nu}$$ where $F_{\mu\nu}$ is ...
8
votes
1answer
107 views

Difference between regularization and renormalization?

In my studies on quantum field theory we have come up with the concepts of regularization and renormalization. I'm a little confused about these two. In my understanding regularization is a way to ...
1
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0answers
37 views

Apparent elimination of overlapping divergences

The integral, $$ \iint_{\mathbb{R}^{2+}}\frac{xy}{1+x+y} \mathrm{d}y \, \mathrm{d}x$$ possesses an overlapping divergence when $ x \to \infty $ and $ y \to \infty $. However, under a change of ...
1
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0answers
15 views

What properties does the conductor making the plates for Casimir effect have?

Reading this explanation, I've understood that the divergence in computation of Casimir force on two parallel conducting plates is because of an unphysical model of ideal conductor, which makes EM ...
5
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0answers
52 views

What is the probability of a Brownian path?

Suppose I have a Brownian particle with a diffusion constant $D$ starting out from a given position at time $0$ and follow it until time $\tau$. What is the probability (distribution) that it takes ...
2
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0answers
49 views

renormalization by differentiation how does it work?

i mean let be the integral $$ \int_{0}^{\infty} \frac{p^{3}}{(p^{2}+m^{2})^{2}} $$ logartihmic divergence but if a apply differentiation with respect to $ m^{2} $ i get $$ \int_{0}^{\infty} ...
4
votes
1answer
269 views

What areas of physics depend on the sum $1 + 2 + 3 + 4 + 5 + 6+ 7+\ldots= -1/12$? [duplicate]

This youtube video from Numberphile, http://youtu.be/w-I6XTVZXww shows how the value is derived. In the video, one interviewee claims that "this result is used in many areas of physics". In the ...
1
vote
1answer
76 views

Why can we simply absorb the positive coefficient of $i\epsilon$ in a propagator?

As far as I know, absorbing of the positive coefficient of $i\epsilon$ in a propagator seems to be a trivial operation without even the need of justification. In Peskin page 286, he did this: ...
9
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2answers
307 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 ...
8
votes
1answer
115 views

Evaluating the Einstein-Hilbert action

The Einstein-Hilbert action is given by, $$I = \frac{1}{16\pi G} \int_{M} \mathrm{d}^d x \, \sqrt{-g} \, R \, \, + \, \, \frac{1}{8\pi G}\int_{\partial M} \mathrm{d}^{d-1}x \, \sqrt{-h} \, K$$ ...
2
votes
1answer
104 views

Branch cuts in two-point function

The propagator of a QFT is known to have a branch cut as a function of the (complex) external momentum. The branch point (as done by, say, Peskin & Schroeder in eqn.7.19 section 7.1) is ...
1
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0answers
34 views

Regulating a particular function

I am interested in computing the integral of this function: \begin{align} \int_0^\infty\frac{2du(u^2+1)}{(1-e^{2\pi u})}, \end{align} which of course at first sight, does not converge. But in QFT ...
6
votes
0answers
104 views

Which values of the Riemann zeta funtion 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. ...
1
vote
0answers
95 views

How should I regularize this integral?

I need to calculate the following integral (which is divergent): \begin{equation} I(m,C)=\int_{-\infty}^\infty {\rm d}\omega\int_{\rm space}{\rm d^3 ...
4
votes
1answer
84 views

3 to 3 scattering in massless $\phi^4$ theory

During my QFT study I faced a problem of calculating amplitude of 3 to 3 scattering in massless $\phi^4$ theory in zero momenta limit at tree level. One of topologically distinct diagrams ...
6
votes
2answers
139 views

A question to clarify the use of divergent series in calculating the casimir effect

Some time ago I posted a question here on this forum. I would like to ask some questions regarding the way the energy per unit area between metallic plates is calculated. The full calculation is on ...
7
votes
1answer
176 views

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 ...
4
votes
0answers
133 views

How does the renormalization scale $\mu$ cancel in all finite observables?

In dimensional regularization, we must shift the dimensionless coupling $g$ by the renormalization scale $\mu$ (which has unit mass dimension): \begin{equation} g \rightarrow \mu^{4-d} g \tag{1} ...
2
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0answers
31 views

Observable which dependes on the cutoff

In arXiv:0710.4330v1 Balitsky calculate the eikonal scattering of dipole composed of quark anti-quark, $Tr(U_{x}U^{\dagger}_{y})$, to NLO accuracy. The result he found is: Where $\mu$ is the ...
10
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2answers
261 views

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 ...
6
votes
0answers
115 views

Srednicki's book chapter 8

Reading first page in chapter 8 of Srednicki's it reads: To employ the $\epsilon$ trick, we multiply $H_0$ with $1-i\epsilon$. The results are equivalent to replacing $m^2$ with $m^2-i\epsilon$. ...
4
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0answers
97 views

can we PHYSCALLY (not by mathematics) justify that $ \zeta (-s)= 1+2^{s}+3^{s}+4^{s}+… $

the values $ \zeta (-1)= -1/12 $ and $ \zeta (-3)= 1/120 $ give accurate results for casimir and to evaluate the dimension in bosonic string theory so is there a physcial JUSTIFICATION to justify ...
0
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0answers
32 views

The sum of positive integers equals minus one twelfth [duplicate]

I was watching a lecture online from the american physicist Lawrence Krauss, when he made an off the cuff remark about the sum of all the positive integers being equal to one twelfth. My question is ...
7
votes
1answer
496 views

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) ...
3
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2answers
3k views

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 ...
20
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4answers
717 views

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 ...
7
votes
3answers
319 views

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 ...
3
votes
2answers
176 views

Dimensional regularization - integral

How can I derive the following formula? $$\int d^{d+1} k \frac{e^{i K X}}{K^2} = \frac{\Gamma (d-1)}{(4\pi)^{d/2} \Gamma (d/2) |X|^{d-1}}, \quad K^2 = k_0^2 + \vec k^2, KX = k_0 \tau + \vec k \vec ...
3
votes
1answer
125 views

Not satisfied with “trick” in zeta function regularization

I am not satisfied with the explanations of the $\sum_n \log \lambda_n = - \frac{d}{ds} \sum_n \lambda_n^{-s}|_{s=0}$ "trick" used in zeta function regularization, discussed here and here, or the ...
7
votes
2answers
196 views

A problematic integral in calculating the entanglement entropy in 1+1 D free massive bosonic field theory

I encountered a curious integration identity when I was reading the paper by Pasquale Calabrese and John Cardy on the entanglement entropy of 1+1D quantum field theory (arXiv). The identity is given ...
1
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0answers
61 views

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 ...
2
votes
1answer
174 views

Delta functional in path integral

I've recently encountered a path integral of the form $$\int \delta[a\phi+b\phi']\,L(\phi,\phi')\;\mathcal D\phi\mathcal D\phi'$$ (where $a$, $b$ are integers) and would like to eliminate one of the ...
3
votes
0answers
101 views

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 ...
1
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0answers
71 views

1D Foldy for the scalar wave equation

In 1D since the Green Function for the scalar wave equation is nonsingular, there is no need to exclude the self interaction term from Foldy's sum over scatterers. In fact you MUST include it since ...
23
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2answers
821 views

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? ...
2
votes
1answer
93 views

Caimir effect regularization for every divergent sum or series

can we use the tools of renormalization of casimir effect to get finite results for any divergent series in QFT ?? for example let be the divergent series $ \sum_{n=1}^{\infty}n^{l} $ for positive ...
5
votes
2answers
131 views

Zeta regularization and Renormalization group

Is there a physical method to prove for example when the zeta regularization of a series $$ 1+2^{k}+3^{k}+............= \zeta (-k) $$ gives the correct result: Casimir effect, vacuum energy and when ...
0
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0answers
45 views

how must i understand this 2-loop integral?

let be the 2-loop integral... $$ \int d^{d}l\int d^{d}k \frac{1}{k^{4}(k+p)^{2}(k+l)^{2}}=I(p)$$ dimensional regularization over the variable 'l0 to evaluate $$ \int ...
6
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1answer
179 views

Naturalness arguments and dimensional regularization?

How do issues of naturalness arise when regularizing QFT using dimensional regularization? I can only recall ever seeing naturalness arguments (hierarchy problem, cosmological constant problem, etc.) ...
4
votes
1answer
302 views

How to get the $i\epsilon$ prescription for a Faddeev-Popov ghost propagator?

In path integral formalism, for a physical field there will be an $i\epsilon$ term in the action, which comes from identifying the in and out vacuum, and in turn this $i\epsilon$ will naturally appear ...
1
vote
1answer
91 views

Dimensional regularization and the finite part

Let be a dimensional regularized integral $$ \int d^{4-\epsilon}kF(k,m,s)= \frac{2}{\epsilon}+\frac{m^{2}}{3}(\gamma +log(4\pi)-\frac{1}{\epsilon}))$$ then formally if we elmiinate the divergent ...
4
votes
0answers
106 views

Cutoff regularization: Why not cutoff exactly at the momentum reached in an experiment?

So far I have only actually calculated dimensional regularization and I just know about the idea of cutoff regularization. From what I understand, as the name suggests, you just ignore momentum as ...
9
votes
1answer
404 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 ...
1
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0answers
101 views

IR divergence and renormalization scale in dimensional regularization (part 2)

This is in continuation of my previous question, IR divergence and renormalization scale in dimensional regularization. Lubos gave a nice answer there but I want to get to a very specific example ...
8
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1answer
407 views

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 ...
4
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1answer
682 views

IR divergence and renormalization scale in dimensional regularization

Is it possible that if a certain (loop) integral is IR divergent then that will have effect on the dimensionally regularized answer for that? (..does the epsilon expansion see the IR divergence in ...
7
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2answers
230 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 ...
3
votes
1answer
250 views

What does it mean to renormalize an effective field theory?

This is in reference to slide 19 here http://cosmology.lbl.gov/talks/Pajer_13.pdf "As always in Effective Field Theory, the theory becomes predictive when there are more observables than parameters" ...
2
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1answer
88 views

Are these IR and UV divergences equal [closed]

let be 2 divergent integrals $$ \int_{0}^{\infty}\frac{p^{3}dp}{(p^{2}+m^{2})^{2}}= A $$ $$ \int_{0}^{\infty}\frac{dp}{p(p+q)^{2}}=B $$ B has a divergent as $ p \to 0 $ however i can use a change ...
0
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0answers
176 views

$\mathrm{i}\epsilon$ prescription makes a function analytical?

I've seen this everywhere where they say "Analytic continuation is obtained by the usual $\mathrm{i}\epsilon$ prescription..." but how is that? How do you analytically continue (say) $\ln x$ with ...