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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. $$\tag{1} \delta(x) ~=~ \lim_{\epsilon\to 0^+}\delta_{\epsilon}(x),$$ $$\tag{2} ... 23 You need nothing more than your understanding of$$ \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) $$So there you go. 20 This is a very interesting question which is usually overlooked. First of all, saying that "large scale physics is decoupled from the small-scale" is somewhat misleading, as indeed the renormalization group (RG) [in the Wilsonian sense, the only one I will use] tells us how to relate the small scale to the large scale ! But usually what people mean by that ... 19 You're totally right. The Wikipedia definition of the renormalization is obsolete i.e. it refers to the interpretation of these techniques that was believed prior to the discovery of the Renormalization Group. While the computational essence (and results) of the techniques hasn't changed much in some cases, their modern interpretation is very different ... 19 Nowadays there exists a more fundamental geometrical interpretation of anomalies which I think can resolve some of your questions. The basic source of anomalies is that classically and quantum-mechanically we are working with realizations and representations of the symmetry group, i.e., given a group of symmetries through a standard realization on some space ... 17 These are all good questions. Perhaps I can answer a few of them at once. The equation describing the violation of current conservation is$$\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 g. It is not possible to write any other candidate answer by dimensional analysis ... 16 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 ... 14 First: There is no rigorous construction of the standard model, rigorous in the sense of mathematics (and no, there is not much ambivalence about the meaning of rigor in mathematics). That's a lot of references that Daniel cited, I'll try to classify them a little bit :-) Axiomatic (synonymous: local or algebraic) QFT tries to formulate axioms for the ... 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 ... 11 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'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 ... 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 ... 9 Here is my answer from a condensed matter physics point of view: Quantum field theory is a theory that describes the critical point and the neighbor of the critical point of a lattice model. (Lattice models do have a rigorous definition). So to rigorously define/classify quantum field theories is to classify all the possible critical points of lattice ... 9 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 ... 9 Dimensional regularization (i.e., dim-reg) is a method to regulate divergent integrals. Instead of working in 4 dimensions where loop integrals are divergent you can work in 4-\epsilon dimensions. This trick enables you to pick out the divergent part of the integral, as using a cutoff does. However, it treats all divergences equally so you can't ... 9 You seem to be confusing regularization with renormalization. Regularization is the process of removing (or, more properly, parameterizing) infinities in loop integrals. Often in elementary texts a "cutoff" representing an energy scale above which the theory is assumed to be invalid is discussed, and counterterms are added to the Lagrangian in order to make ... 8 The definite answer to your question is: There is no mathematicaly precise, commonly accepted definition of the term "regularization procedure" in perturbative quantum field theory. Instead, there are various regularization schemes with their advantages and disatvantages. Maybe you'll find Chapter B5: Divergences and renormalization of my theoretical ... 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 – ... 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 The answer to both questions is that string theory is completely free of any ultraviolet divergences. It follows that its effective low-energy descriptions such as the Standard Model automatically come with a regulator. An important "technicality" to notice is that the formulae for amplitudes in string theory are not given by the same integrals over loop ... 8 Weinberg's presentation is not pedagogically ideal, because the steps are too formal, and he puts emphasis on ones that can be misleading to a student. The presentation might lead you think that the determinant of the U factors are somehow not equal to 1 naively because of some phase business, and this is categorically not true. This is what is confusing ... 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 ... 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 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 ... 7 There are several books that approach QFT (and/or Gauge Theory) from different levels of 'mathematical rigor' (for some definition of "mathematical rigor" — that Moshe would approve ;-). So, let me give you some sort of 'preliminary list'… it's by no means complete, and it's in no particular order either, but i think it may pave the way to further work. ... 7 I) Problem. We put (four pi times) the permittivity 4\pi\varepsilon=1 equal to one from now on (in the SI unit system) for simplicity. Let us first rephrase the question a bit. Instead of starting from the potential$$\tag{1} \Phi~=~\frac{1}{r} \qquad \mathrm{and} \qquad \Phi~=~\frac{1}{2r^2}, \qquad r\neq 0,  respectively, let us assume that the ...