Hot answers tagged renormalization
17
To take a meaningful continuum limit, essentially, you need to be in regime where your field is smooth enough that a gradient expansion is possible. This is usually acheived by associating a very high energy cost to field configurations that take different values on nearest neigbours in the lattice.
The continuum limit of $O(n)$ models is worked out in ...
15
The most relevant tool: the Renormalization Group. You see the lattice model at larger and larger scales, and find out which terms get more relevant, and which get more irrelevant, as you zoom out. Once you reach a fixed point, the surviving terms make up your continuous system.
12
The possible applications I can think of are in determining the phases of various QFTs. There are tons of applications like that, here are some ideas:
-- If the solutions to 't Hooft's conditions are too complicated (entail too many fermions such that their contribution to the IR values of $a$ is greater that $a$ in the UV) there must be symmetry breaking, ...
11
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
11
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
If you are looking for a mathematical treatment for your question you need to look at
the book Fernandez-Frohlich and Sokal "Random walks, critical phenomena, and triviality in quantum field theory" Springer-Verlag, 1992. It might be out of print so if you can't get it
you can also try these freely accessible articles:
A. Sokal "An alternate constructive ...
10
the Standard Model just happens to be perturbatively renormalizable which is an advantage, as I will discuss later; non-perturbatively, one would find out that the Higgs self-interaction and/or the hypercharge $U(1)$ interaction would be getting stronger at higher energies and they would run into inconsistencies such as the Landau poles at extremely high, ...
10
The best way to explain renormalization is to consider what at first looks like a complete detour: Mandelbrot's fractal geometry. Developed in the 1960s and 1970s, Mandelbrot's geometry is the key idea behind major advances in statistical physics in the early 1970s, pioneered by Leo Kadanoff (mostly independently), but also associated with Alexander ...
10
The most straightforward use of the $a$-theorem is to determine what kinds of spontaneous symmetry breaking are possible. For example, in the usual QCD with three light flavors, at high energy one has a theory of fermions and gauge fields and at low energy one has a theory of pions. If you tried the same thing with a different, large enough of number of ...
9
You are conflating three conceptually different categories of "regularizations" of seemingly divergent series (and integrals).
The type of resummations that Hardy would talk about are similar to the zeta-function regularization - the example that is most familiar to the physicists. For example,
$$S=\sum_{n=1}^\infty n= -\frac{1}{12}$$
is the most famous ...
9
One approach is that of Seiberg
http://arXiv.org/abs/hep-ph/9309335
which is also expanded upon a little bit (and explained in a slightly different way) by Weinberg
http://arXiv.org/abs/hep-th/9803099
The old point of view is based on explicit supergraph computations
http://inspirehep.net/record/141168?ln=en
The disadvantage of the supergraphs approach ...
8
I believe one has to distinguish two kinds of dualities. AdS/CFT, even in the context where it describes an RG flow (so not the pure AdS_5xS^5 case), is an exact duality to a four-dimensional theory, which interpolates between one well-defined conformal field theory in the UV and another conformal field theory in the IR. So holographic renormalization is in ...
8
The running coupling $\lambda(\mu)$, as a function of renormalization scale $\mu$, does run negative for large $\mu$ in the SM if the Higgs is not too heavy. But "renormalization scales" are not particularly physical things to talk about. A more physical quantity is the renormalization-group improved effective Higgs potential, $V(H)$. For large values of ...
8
Let me take a stab at answering this (somewhat vague) question. You said you are interested in the analytic structure of QFT. But you also mentioned the RG, which is somewhat different. I will try to address the analytic structure of QFT and then emphasize that the renormalization group can be thought of as merely a trick to improve perturbation theory.
...
8
Whether you do your calculations using a cutoff regularization or dimensional regularization or another regularization is just a technical detail that has nothing to do with the existence of the hierarchy problem. Order by order, you will get the same results whatever your chosen regularization or scheme is.
The schemes and algorithms may differ by the ...
8
The counterterms at one loop would be $R^2$ operators, because loops are counted by powers of $G_N = 1/M_P^2$. The tree-level Lagrangian is the Einstein Hilbert action $M_P^2 R$, so the one-loop counterterms for logarithmic divergences should be terms that carry no powers of $M_P$ in front. Simply from dimensional analysis, then, these are $R^2$ terms, of ...
8
If you go back to the origins, the difficulty in merging gravity with the other forces mostly stems from general relativity being a purely geometric theory -- again, that's in its original form -- and all the other forces being quantum, by which I mostly mean they are conveyed by well-defined force particles. The photon as the particle that conveys the ...
7
I'm not an expert in this topic too, but I'm trying wrap my head around it.
Right now I'm trying to make an adequate hierarchy of concepts related to renormalization. Let me list them and tell how they are related:
Fields, Lagrangian (Hamiltonian) and coupling constants.
Perturbative calculations.
Different scales.
Self-similarity.
Quantum fields.
...
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 ...
6
I think you need to look for the following book, Finite Quantum Electrodynamics: this is not something "fringe" nor some "crackpot" off-shoot.
The name of the game is Causal Perturbation Theory, and was pioneered by Epstein, Glaser: "The role of locality in perturbation theory".
As far as i understand your question (in the context of your comments, etc), ...
6
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 ...
6
Your definition is quite good and works almost always. I'm quite sure it is rigorously true in 2D. You'll actually find it in some lecture notes. Remember that a theory is conformal if the trace of the stress tensor vanishes: $T \equiv T_\mu^{\mu} = 0.$ Indeed there is a folk theorem that states that
$T = \sum \beta_I \mathcal{O}^I$
where the sum runs over ...
5
Because we happen to be working at the right energy scale. In general, if there are renormalizable interactions around, they dominate over nonrenormalizable ones, by simple scaling arguments and dimensional analysis. Before the electroweak theory was developed, the Fermi theory of the weak interactions was nonrenormalizable because the leading interactions ...
5
Good question. The short answer is no, cutoff scales have no relevance to string theory.
Cutoff scales are given by maximum or minimum energies or distances where the given theory may be applied. This concept is only useful because in quantum field theory, such cutoffs are natural regulators to get rid of short-distance divergences. These divergences ...
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