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18

You need nothing more than your understanding of $$ \int_{-\infty}^\infty f(x)\delta(x-a)dx=f(a) $$ Just treat one of the delta functions as $f(x)\equiv\delta(x-\lambda)$ in your problem. So it would be something like this: $$ \int\delta(x-\lambda)\delta(x-\lambda)dx=\int f(x)\delta(x-\lambda)dx=f(\lambda)=\delta(\lambda-\lambda) $$ So there you go.


17

Well, the Dirac delta function $\delta(x)$ is a distribution, also known as a generalized function. One can e.g. represent $\delta(x)$ as a limit of a rectangular peak with unit area, width $\epsilon$, and height $1/\epsilon$; i.e. $$\tag{1} \delta(x) ~=~ \lim_{\epsilon\to 0^+}\delta_{\epsilon}(x), $$ $$\tag{2} ...


16

These are all good questions. Perhaps I can answer a few of them at once. The equation describing the violation of current conservation is $$\partial^\mu j_\mu=f(g)\epsilon^{\mu\nu\rho\sigma}F_{\mu\nu}F_{\rho\sigma}$$ where f(g) is some function of the coupling constant. It is not possible to write any other candidate answer by dimensional analysis and ...


16

You're totally right. The Wikipedia definition of the renormalization is obsolete i.e. it refers to the interpretation of these techniques that was believed prior to the discovery of the Renormalization Group. While the computational essence (and results) of the techniques hasn't changed much in some cases, their modern interpretation is very different ...


16

This is a very interesting question which is usually overlooked. First of all, saying that "large scale physics is decoupled from the small-scale" is somewhat misleading, as indeed the renormalization group (RG) [in the Wilsonian sense, the only one I will use] tells us how to relate the small scale to the large scale ! But usually what people mean by that ...


15

Nowadays there exists a more fundamental geometrical interpretation of anomalies which I think can resolve some of your questions. The basic source of anomalies is that classically and quantum-mechanically we are working with realizations and representations of the symmetry group, i.e., given a group of symmetries through a standard realization on some space ...


15

I have written a pedagogical article about renormalization and renormalization group and I would be happy to have your opinion about it. It is published in American Journal of Physics. You'll find it also on ArXiv: A hint of renormalization. B. Delamotte


14

Lack of convergence does not mean there is nothing mathematically rigorous one can extract from perturbation theory. One can use Borel summation. In fact, Borel summability of perturbation theory has been proved for some QFTs: by Eckmann-Magnen-Seneor for $P(\phi)$ theories in 2d, see this article. by Magnen-Seneor for $\phi^4$ in 3d, see this article. by ...


12

Renormalization is absolutely not just a technical trick, it's a key part of understanding effective field theory and why we can compute anything without knowing the final microscopic theory of all physics. One good online source that explains a nice physical example is Joe Polchinski's "Effective field theory and the Fermi surface" (and you can also look up ...


10

By definition, a renormalizable quantum field theory (RQFT) has the following two properties (only the first one matters in regard to this question): i) Existence of a formal continuum limit: The ultraviolet cut-off may be taken to infinite, the physical quantities are independent of the regularization procedure (and of the renormalization subtraction ...


10

I think you raise a very important question, but I think you make it sound more trivial than it is. The point is: a lot of physicists would like to have alternative expansions, but it is very difficult to come up with one. If you've got some suggestions, don't hesitate to put it forward. The standard expansion starts from the time evolution operator ...


8

Are you essentially asking about non-perturbative approaches to QFT? Lattice QCD (based on Monte-Carlo sampling) and various strong-couplig/weak-coupling dualities (like AdS/CFT) come to mind as most prominent examples. This is more of a hint than a real answer, of course.


8

Check out the following 3 articles and 2 books: Regularization Renormalization and Dimensional Analysis: Dimensional Regularization meets Freshman E&M Published in the american journal of physics (can be found also on hep-ph, but slightly different with less references) Regularization, from Murayama's course of QFT at Berkeley A Hint of ...


8

I'm not sure about it, but my understanding of this is that the $\int_\Lambda^\infty$ term is essentially constant between different processes, because whatever physics happens at high energies should not be affected by the low-energy processes we are able to control. That way, we can meaningfully calculate differences between two integrals, and the ...


8

Could this imply there is a formulation where that value comes naturally... This sentence implicitly assumes that analytic continuation is "unnatural". But the truth is the other way around: analytic continuation is one of the most natural mathematical procedures in physics. On the contrary, it's functions – especially functions of momenta or energy – ...


7

I've been thinking about divergent series on and off, so maybe I could chip in. Consider a sequence of numbers (in an arbitrary field, e.g. real numbers) $\{a_n\}$. You may ask about the sum of terms of this sequence, i. e. $\sum a_n$. If the limit $\lim_{N\rightarrow\infty} \sum^N |a_n|$ exists then the series is absolutely convergent and you may talk ...


7

It took the insights of Wilson and Kadanoff to answer this question. Universality. It doesn't matter all that much what the precise details in the ultraviolet are. Under the renormalization group, only a small number of parameters are either relevant or marginal. All the rest are irrelevant. As long as you take care to match up the relevant and marginal ...


7

Suppose I want to show $$\int \delta(x-a)\delta(x-b) dx = \delta(a-b) $$ To do that , I need to show $$\int g(a)\int \delta(x-a)\delta(x-b) dx da = \int g(a)\delta(a-b) da$$ for any function $g(a)$. $$LHS = \int \int g(a) \delta(x-a)da \ \delta(x-b) dx $$ $$=\int g(x)\delta(x-b)dx $$ $$=g(b) $$ But RHS clearly = $g(b)$ too. The result follows putting ...


6

It looks like a delta-function. However, because $\epsilon / (\epsilon^2+t^2)$ - you should omit one $\epsilon$ in the numerator, to get the right integral equal to one, by the way - decreases too slowly as $|t|\to\infty$, as $1/t^2$, it will only work as the Dirac delta distribution for test functions that decrease at infinity or at least increase slower ...


6

Check this review on the cosmological constant problem for a nice discussion of what hierarchies look like in different regularizations. Here is the rough idea. In cutoff or Pauli-Villars regularization the counterterms are sensitive to the cutoff scale(s) $\Lambda$. But there is no such scale when using dim.reg. (only the renormalization point $\mu$, which ...


6

We put the permittivity $\varepsilon=1$ to one from now on. Let us first rephrase the question a bit. Instead of starting from the potential $$\Phi=\frac{1}{r} \qquad \mathrm{and} \qquad \Phi=\frac{1}{2r^2}, \qquad r\neq 0, $$ respectively, let us assume that the electric field has be given as $$\vec{E}=\frac{\vec{r}}{r^3} \qquad \mathrm{and} ...


6

In some sense, i understand this question of yours as regarding to more "mathematically precise" approaches to QFT: in the end of the day, your question implies "non-perturbative definitions of QFT" in a form or another — afterall, if you can use some other tool, why not turn the problem around and define your theory based on how you can use such tool? ...


6

Although, I do not know if a general proof exists, I think that the Casimir effect of a renormalizable quantum field theory should be completely understood by means of a theory of renormalization on manifolds with boundary. The key feature is that one cannot, in general, neglect the renormalization of the coupling constants in the boundary terms. Using this ...


6

The role of holomorphic functions (and their generalizations in the form of holomorphic sections of vector bundles) in physics is invaluable. Please see for example the following review by B.C. Hall, discussing holomorphic methods in mathematical physics, especially in quantum mechanics. It should be emphasized that these theories cover important parts of ...


6

Yes, the infrared divergences are guaranteed to be present in any meaningful computational scheme for a loop diagram, including dimensional regularization, because their existence is an objective question. In dim reg, UV divergent integrals have to be calculated in $4-\epsilon$ dimensions so we imagine that the dimension is lower than the dimension of ...


6

The quantity $k_{NL}$ in their paper isn't a "cutoff scale"; it is a scale at which the nonlinearities (therefore NL) in some quantities become substantial. There is no cutoff scale $\Lambda$ in dimensional regularization; it's one of the main features and virtues of the dimensional regularization. Instead, $\Lambda\to \infty$ is replaced by the $\epsilon = ...


5

Any analytic function is defined everywhere on its Riemannian surface just by its values in an arbitrarily small neighborhood of a point. The expression to be ''analytically continued'' therefore just specifies which function is meant, but it has ''direct and natural'' values everywhere on its Riemann surface. Except that not all of these values can be ...


5

If e.g. $b=1$, it is obvious that the delta-functional – you should really call it $\Delta$ and not $\delta$ – "cancels" against the integral, giving $$\int L(\phi,-a\phi) {\mathcal D}\phi $$ where I just substituted the right value for $\phi'$. Similarly for $a=1$. Now, for general $a,b$, the Jacobian is just the simple power $b^{-N}$ where $N$ is the ...



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