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--- advanced_tools:renormalization_group [2017/10/30 17:50]
jakobadmin
+++ advanced_tools:renormalization_group [2018/05/05 12:46] (current)
jakobadmin
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 ====== Renormalization Group ======
+<tabbox Intuitive>
+<blockquote>
+You are arranging the theory in such a way that only the right degrees of freedom, the ones that are really relevant to you, are appearing in your equations. I think that this is in the end what the renormalization group is all about. It's a way of satisfying the Third Law of Progress in Theoretical Physics, which is that //you may use any degrees of freedom you like to describe a physical system, but if you use the wrong ones, you'll be sorry//.
+<cite>[[https://www.physics.ohio-state.edu/~ntg/8805/refs/weinberg_rg_good_thing.pdf|Why the Renormalization Group Is a Good Thing]] by Steven Weinberg</cite>
+</blockquote>
+----
+  * A layman's introduction is [[https://websites.pmc.ucsc.edu/~wrs/Project/2014-summer%20seminar/Renorm/Wilson-many%20scales-Sci%20Am-79.pdf|Problems in Physics with Many Scales of Length]] by Kenneth Wilson and also
+  * http://frankwilczek.com/Wilczek_Easy_Pieces/172_Unification_of_Couplings.pdf
+<tabbox Concrete>
+**Recommended Textbooks:**
+  * P. Pfeuty and G. Toulouse: Introduction to the Renormalization Group
+  * “Renormalization Methods : a Guide for Beginners” by David Mc Comb
+  * [[http://amzn.to/2hMQTCP|Renormalization: An Introduction to Renormalization, the Renormalization Group and the Operator-Product Expansion]] by J. Collins
+----
+--> Navier-Stokes Equation#
+<blockquote>
+The idea of nineteenth century hydrodynamic physics was that, upon systematically integrating out the high frequency, short wavelength modes associated with
+atoms and molecules, one ought to be able to arrive at a universal long wavelength
+theory of fluids, say, the Navier-Stokes equations, regardless of whether the fluid
+was composed of argon, water, toluene, benzene. [...] The existence of atoms
+and molecules is irrelevant to the profound and, some might say, even fundamental
+problem of understanding the Navier-Stokes equations at high Reynold's numbers.
+We would face almost identical problems in constructing a theory of turbulence if
+quantum mechanics did not exist, or if matter first became discrete at length scales
+of Fermis instead of Angstroms.
+<cite>Michael Fisher in Conceptual Foundations of Quantum Field Theory, Edited by Cao</cite>
+</blockquote>
+<--
+<tabbox Abstract>
+<blockquote>
+It is worthwhile to stress, at the outset, what a "renormalization group" is not! Although in many applications the particular renormalization group employed may be invertible, and so constitute a continuous
+or discrete, group of transformations, it is, in general, only a semigroup. In other words a renormalization
+group is not necessarily invertible and, hence, cannot be 'run backwards' without ambiguity: in short it is
+not a "group". The misuse of mathematical terminology may be tolerated since these aspects play, at best,
+a small role in RG theory.
+<cite>Michael Fisher in Conceptual Foundations of Quantum Field Theory, Edited by Cao</cite>
+</blockquote>
+<blockquote>
+in Wilson's conception RG theory crudely says: "Well, (a) there is a flow in some space, $\mathcal{H}$, of Hamiltonians (or "coupling constants"); (b) the critical point of a system is associated with a fixed point (or stationary point) of that flow; ( c ) the flow operator - technically the RG transformation $\mathbb{R}$ - can be linearized about that fixed point; and (d) typically, such a linear operator (as in quantum mechanics) has a spectrum of discrete but nontrivial eigenvalues, say $\lambda_k$; then (e) each (asymptotically independent) exponential term in the flow varies as $e^{\lambda_k l}$ where $l$ is the flow (or renormalization) parameter and corresponds to a physical power law, say $|t|^{\phi_k}$, with critical exponent $\phi_k$ proportional to the eigenvalue $\lambda_k$." How one may find suitable transformations $\mathbb R$, and why the flows matter, are the subjects for the following chapters of our story. Just as quantum mechanics does much more than explain sharp spectral lines, so RG theory should also explain, at least in principle, (ii) the values of the leading thermodynamic and correlation exponents, $\alpha, \beta, \gamma, \delta, \nu, \eta$ (to cite those we have already mentioned above) and (iii) clarify why and how the classical values are in error, including the existence of borderline dimensionalities, like $d_x = 4$, above which classical theories become valid. Beyond the leading exponents, one wants (iv) the correction-to-scaling exponent $0$ (and, ideally, the higher-order correction exponents) and, especially, (v) one needs a method to compute crossover exponents, $\phi$, to check for the relevance or irrelevance of a multitude of possible perturbations. Two central issues, of course, are (vi) the understanding of universality with nontrivial exponents and (vii) a derivation of scaling: see (16) and (19). And, more subtly, one wants (viii) to understand the breakdown of universality and scaling in certain circumstances - one might recall continuous spectra in quantum mechanics - and (ix) to handle effectively logarithmic and more exotic dependences on temperature, etc.
+<cite>Michael Fisher in Conceptual Foundations of Quantum Field Theory, Edited by Cao</cite>
+</blockquote>
+[{{ :rgepicture.png?nolink |Source: Conceptual Foundations of Quantum Field Theory, Edited by Cao}}]
+**Recommended Papers:**
+  * [[https://arxiv.org/pdf/cond-mat/0702365.pdf|An introduction to the nonperturbative renormalization group]] by Bertrand Delamotte
+  * [[http://cps-www.bu.edu/hes/articles/s99a.pdf| Scaling, universality, and renormalization: Three pillars of modern critical phenomena]] by H. Eugene Stanley
+  * [[http://www-fp.usc.es/~edels/SUSY/Hollowood.pdf|6 Lectures on QFT, RG and SUSY]] by Timothy J. Hollowood
+  * For a derivation of the renormalization group using //solely// dimensional analysis, see [[http://www.sciencedirect.com/science/article/pii/0003491681900725|Dimensional Analysis in field theory]] by P.M Stevenson
 <tabbox Why is it interesting?>
+<blockquote>What does a JPEG have to do with economics and quantum gravity? All of them are about what happens when you simplify world-descriptions. A JPEG compresses an image by throwing out fine structure in ways a casual glance won't detect. Economists produce theories of human behavior that gloss over the details of individual psychology. Meanwhile, even our most sophisticated physics experiments can't show us the most fundamental building-blocks of matter, and so our theories have to make do with descriptions that blur out the smallest scales. The study of how theories change as we move to more or less detailed descriptions is known as renormalization. <cite>https://www.complexityexplorer.org/tutorials/67-introduction-to-renormalization</cite></blockquote>
 <blockquote>
 Furthermore, [[http://press.princeton.edu/titles/5772.html|it]] forthrightly acknowledges the breadth of the RG approach citing as
@@ Line 69: / Line 144: @@
 <cite>Michael Fisher in Conceptual Foundations of Quantum Field Theory, Edited by Cao</cite>
 </blockquote>
-<tabbox Layman>
-  * A layman's introduction is [[https://websites.pmc.ucsc.edu/~wrs/Project/2014-summer%20seminar/Renorm/Wilson-many%20scales-Sci%20Am-79.pdf|Problems in Physics with Many Scales of Length]] by Kenneth Wilson and also
-  * http://frankwilczek.com/Wilczek_Easy_Pieces/172_Unification_of_Couplings.pdf
-<tabbox Student>
-  - First, have a look at http://philosophy.wisc.edu/forster/Percolation.pdf
-  - Then read http://jakobschwichtenberg.com/renormalization-group-flow/
-  - Then, the best complete introduction can be found in Critical point phenomena: universal physics at large length scales by Bruce, A.; Wallace, D. published in the book The new physics, edited by P. Davies
-  - Afterwards read https://www.andrew.cmu.edu/user/kk3n/found-phys-emerge.pdf which explains lucidly many of the most important "advanced" concepts.
-  - See also "[[https://www.physics.ohio-state.edu/~ntg/8805/refs/weinberg_rg_good_thing.pdf|Why the Renormalization Group Is a Good Thing]]" by S. Weinberg
-  - [[https://inspirehep.net/record/109631|Critical Phenomena for Field Theorists]] by Steven Weinberg
-  - Also the video course: [[https://www.complexityexplorer.org/tutorials/67-introduction-to-renormalization/segments/5681|Introduction to Renormalization]] by Simon DeDeo is highly recommended to understand how the renormalization group can be used in other fields than physics.
-  - http://www.damtp.cam.ac.uk/user/tong/sft.html
-**Recommended Textbooks:**
-  * P. Pfeuty and G. Toulouse: Introduction to the Renormalization Group
+<tabbox FAQ>
-  * “Renormalization Methods : a Guide for Beginners” by David Mc Comb
-**FAQ:**
 --> What are bare charges?#
@@ Line 125: / Line 183: @@
 <--
-<tabbox Researcher>
-<blockquote>
-It is worthwhile to stress, at the outset, what a "renormalization group" is not! Although in many applications the particular renormalization group employed may be invertible, and so constitute a continuous
-or discrete, group of transformations, it is, in general, only a semigroup. In other words a renormalization
-group is not necessarily invertible and, hence, cannot be 'run backwards' without ambiguity: in short it is
-not a "group". The misuse of mathematical terminology may be tolerated since these aspects play, at best,
-a small role in RG theory.
-<cite>Michael Fisher in Conceptual Foundations of Quantum Field Theory, Edited by Cao</cite>
-</blockquote>
-<blockquote>
-in Wilson's conception RG theory crudely says: "Well, (a) there is a flow in some space, $\mathcal{H}$, of Hamiltonians (or "coupling constants"); (b) the critical point of a system is associated with a fixed point (or stationary point) of that flow; ( c ) the flow operator - technically the RG transformation $\mathbb{R}$ - can be linearized about that fixed point; and (d) typically, such a linear operator (as in quantum mechanics) has a spectrum of discrete but nontrivial eigenvalues, say $\lambda_k$; then (e) each (asymptotically independent) exponential term in the flow varies as $e^{\lambda_k l}$ where $l$ is the flow (or renormalization) parameter and corresponds to a physical power law, say $|t|^{\phi_k}$, with critical exponent $\phi_k$ proportional to the eigenvalue $\lambda_k$." How one may find suitable transformations $\mathbb R$, and why the flows matter, are the subjects for the following chapters of our story. Just as quantum mechanics does much more than explain sharp spectral lines, so RG theory should also explain, at least in principle, (ii) the values of the leading thermodynamic and correlation exponents, $\alpha, \beta, \gamma, \delta, \nu, \eta$ (to cite those we have already mentioned above) and (iii) clarify why and how the classical values are in error, including the existence of borderline dimensionalities, like $d_x = 4$, above which classical theories become valid. Beyond the leading exponents, one wants (iv) the correction-to-scaling exponent $0$ (and, ideally, the higher-order correction exponents) and, especially, (v) one needs a method to compute crossover exponents, $\phi$, to check for the relevance or irrelevance of a multitude of possible perturbations. Two central issues, of course, are (vi) the understanding of universality with nontrivial exponents and (vii) a derivation of scaling: see (16) and (19). And, more subtly, one wants (viii) to understand the breakdown of universality and scaling in certain circumstances - one might recall continuous spectra in quantum mechanics - and (ix) to handle effectively logarithmic and more exotic dependences on temperature, etc.
-<cite>Michael Fisher in Conceptual Foundations of Quantum Field Theory, Edited by Cao</cite>
-</blockquote>
-[{{ :rgepicture.png?nolink |Source: Conceptual Foundations of Quantum Field Theory, Edited by Cao}}]
 --> How is the renormalization group used in QFT?#
@@ Line 292: / Line 327: @@
 <--
+<tabbox History>
+The renormalization group equations were discovered by Gell-Mann and Low in their paper [[http://journals.aps.org/pr/abstract/10.1103/PhysRev.95.1300|Quantum Electrodynamics at Small Distances]]. They studied the Coulomb potential $V$ and discovered that in the high momentum limit, where the mass of the electron shouln't matter, the potential does not scale as one would expect naively from dimensional analysis. They explained this by noting that this is a result of the renormalization procedure. If we, instead define the fine structure constant $ \alpha$ simply by
-**Recommended Papers:**
+$$ \alpha \equiv r V(r) ,$$
-  * [[https://arxiv.org/pdf/cond-mat/0702365.pdf|An introduction to the nonperturbative renormalization group]] by Bertrand Delamotte
+which means that $\alpha$ is simply the coefficient of $1/r$ in the Coulomb potential, there is no problem at all. The interpretation is then that $\alpha$ is scale dependend. The scale dependence is described by the famous $\beta$ functions.
-  * [[http://cps-www.bu.edu/hes/articles/s99a.pdf| Scaling, universality, and renormalization: Three pillars of modern critical phenomena]] by H. Eugene Stanley
-  * [[http://www-fp.usc.es/~edels/SUSY/Hollowood.pdf|6 Lectures on QFT, RG and SUSY]] by Timothy J. Hollowood
-  * For a derivation of the renormalization group using //solely// dimensional analysis, see [[http://www.sciencedirect.com/science/article/pii/0003491681900725|Dimensional Analysis in field theory]] by P.MStevenson
-<tabbox Examples>
---> Navier-Stokes Equation#
+In a similar analysis by Callan and Symanzik they explained the "anomaly in the trace of the energy-momentum tensor". This trace should be zero in the no-mass limit, but isn't in higher order perturbation theory. This problem came up through the studying of the scaling behaviour of higher order contributions to scattering ampliteds. Analogously, to the Coulomb potential problem described above, these also did not posses the "naive" scaling behaviour one would expect. The analogous to the Gell-Mann-Low $\beta$ functions for this problem are the famous Callan-Symanzik equations.
-<blockquote>
+The common theme of both analysis is that the naive dimensional analysis breaks down, because of renormalization.
-The idea of nineteenth century hydrodynamic physics was that, upon systematically integrating out the high frequency, short wavelength modes associated with
-atoms and molecules, one ought to be able to arrive at a universal long wavelength
-theory of fluids, say, the Navier-Stokes equations, regardless of whether the fluid
-was composed of argon, water, toluene, benzene. [...] The existence of atoms
-and molecules is irrelevant to the profound and, some might say, even fundamental
-problem of understanding the Navier-Stokes equations at high Reynold's numbers.
-We would face almost identical problems in constructing a theory of turbulence if
-quantum mechanics did not exist, or if matter first became discrete at length scales
-of Fermis instead of Angstroms.
-<cite>Michael Fisher in Conceptual Foundations of Quantum Field Theory, Edited by Cao</cite>
-</blockquote>
-<--
---> Example2:#
+<blockquote>
-<--
+The beginning of modern field theory in Russia I would associate with the great work by Landau, Abrikosov and Khalatnikov [1]. They studied the structure of the logarithmic divergences in QED and introduced the notion of the scale dependent coupling. This scale dependence comes from the fact that the bare charge is screened by the cloud of the virtual particles, and the larger this cloud is the stronger screening we get.
-<tabbox History>
-https://arxiv.org/pdf/1310.5533.pdf
-See the great description in Wilson’s renormalization group : a paradigmatic shift by Edouard Brezin
+<cite>https://arxiv.org/pdf/hep-th/9211140.pdf</cite>
+</blockquote>
 <blockquote>
-The beginning of modern field theory in Russia I would associate with the great work by Landau, Abrikosov and Khalatnikov [1]. They studied the structure of the logarithmic divergencies in QED and introduced the notion of the scale dependent coupling. This scale dependence comes from the fact that the bare charge is screened by the cloud of the virtual particles, and the larger this cloud is the stronger screening we get.
+Unfortunately, when the book on quantum field theory by Bogoliubob and Shorkov was published in the late 1950s, which I believe contained the first mention in a book of these matters, Bogoliubov and Shorkov seized on the point about the invariance with respect to where you renormalize the charge, and **they introduced the term "renormalization group" to express this invariance. But what they were emphasizing, it seems to me, was the least important thing in the whole business.**
-<cite>https://arxiv.org/pdf/hep-th/9211140.pdf</cite>
+It's a truism, after all, that physics doesn't depend on how you define the parameters. I think readers of Bogoliubov and Shirkov may have come into the grip of a misunderstanding that if you somehow identify a group that then you're going to learn something physicsl from it. Of, course, this is not always so. For instance when you do bookkeeping you can count the credits in black and the debits in red, or you can perform a group transformation and interchange black and red, and the rules of bookkeeping will have an invariance under that interchange. But this does not help you to make any money.
+The important thing about the Gell-Mann-Low paper was the fact that they realized that quantum field theory has a scale invariance, that the scale invariance is broken by particle masses but these are negligible at very high energy or very short distances if you renormalize in an appropriate way, and that then the only thing that's breaking scale invariance is the renormalization procedure, and that one can take that into account by keeping track of the running coupling constant $\alpha_R$.
+<cite>[[https://www.physics.ohio-state.edu/~ntg/8805/refs/weinberg_rg_good_thing.pdf|Why the Renormalization Group Is a Good Thing]] by Steven Weinberg</cite>
 </blockquote>
-See also Critical Point: A Historical Introduction to the Modern Theory of Critical Phenomenaby CYRIL DOMB
+**Recommended Resources:**
+  * https://arxiv.org/pdf/1310.5533.pdf
+  * See the great description in Wilson’s renormalization group : a paradigmatic shift by Edouard Brezin
+  * See also Critical Point: A Historical Introduction to the Modern Theory of Critical Phenomena by CYRIL DOMB
+  * [[http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.129.3194&rep=rep1&type=pdf|Renormalization group theory: Its basis and formulation in statistical physics]] Michael E. Fisher
+<tabbox Roadmaps>
+Here's a plan of how to understand the renormalization group effectively and quickly:
+  - First, have a look at http://philosophy.wisc.edu/forster/Percolation.pdf
+  - Then read http://jakobschwichtenberg.com/renormalization-group-flow/
+  - Then, the best complete introduction can be found in Critical point phenomena: universal physics at large length scales by Bruce, A.; Wallace, D. published in the book The new physics, edited by P. Davies
+  - Afterwards read https://www.andrew.cmu.edu/user/kk3n/found-phys-emerge.pdf which explains lucidly many of the most important "advanced" concepts.
+  - See also "[[https://www.physics.ohio-state.edu/~ntg/8805/refs/weinberg_rg_good_thing.pdf|Why the Renormalization Group Is a Good Thing]]" by S. Weinberg
+  - [[https://inspirehep.net/record/109631|Critical Phenomena for Field Theorists]] by Steven Weinberg
+  - Also the video course: [[https://www.complexityexplorer.org/tutorials/67-introduction-to-renormalization/segments/5681|Introduction to Renormalization]] by Simon DeDeo is highly recommended to understand how the renormalization group can be used in other fields than physics.
+  - http://www.damtp.cam.ac.uk/user/tong/sft.html
+  - [[https://arxiv.org/abs/hep-th/0212049|A hint of renormalization]] by B. Delamotte
 </tabbox>

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