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advanced_tools:renormalization [2017/12/17 16:56]
jakobadmin [Why is it interesting?]
advanced_tools:renormalization [2019/07/05 14:10] (current)
jakobadmin [Concrete]
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 ====== Renormalization ====== ====== Renormalization ======
  
-<tabbox Why is it interesting?> ​ 
-<WRAP group> 
  
-<WRAP half column> 
-<​blockquote>​ 
-The philosophy of renormalization is one of the most difficult, and controversial,​ in all of particle physics. Yet it underpins modern theory. 
  
-<cite>page 42 in The Infinity Puzzle ​by Frank Close</​cite>​ +<tabbox Intuitive>​  
-</​blockquote>​+<​blockquote>​The ​fact that unknown physics may be lurking at very small distances is related to the necessity for regularization and renormalization of quantum field theory. A local Lagrangian in a quantum field theory is not a priori well-defined precisely because interactions that involve products of fields at a single space-time point specify the physics down to arbitrarily small distances. In mathematical language, the fields are distributions rather than functions, and multiplying them together at the same space-time point is a dangerous act. In regularization and renormalization,​ the physics at small distances is modified in some way to make the theory well-defined. Then the dependence on the short distance physics is incorporated into a set of parameters that can be related to physical quantities at measurable distances. A renormalizable theory is one in which only a finite number of parameters are required to absorb all the dependence on short distance physics. [...] Eventually, we will make the idea of hiding unknown physics at small distances into one of our principle tools. <​cite>​[[http://​www.people.fas.harvard.edu/​~hgeorgi/​weak.pdf|Weak Interactions]] ​by Georgi</​cite></​blockquote>​
  
-<​blockquote>​ 
-For physicists, infinity is a code word for disaster, the proof that you 
-are trying to apply a theory beyond its realm of applicability. In the case 
-of QED, if you can’t calculate something as basic as a photon being absorbed by an electron, you haven’t got a theory—it’s as fundamental as 
-that. 
-<​cite>​page 4 in The Infinity Puzzle by Frank Close</​cite>​ 
-</​blockquote>​ 
-</​WRAP>​ 
-<WRAP half column> 
-<​blockquote>​ 
-It is possible to say that the UV divergences mean that the theory makes no physical sense, and that the subject of interacting quantum field theories is full of nonsense. ([[http://​aip.scitation.org/​doi/​abs/​10.1063/​1.33110|Dirac (1981)]]) 
-<​cite>​Renormalization by Collins</​cite>​ 
-</​blockquote>​ 
  
-<​blockquote>​ +<​blockquote>​Thuswhen ignoring gravity, which we can do in considering the physics ​of individual atoms or elementary particles because the gravitational force is so weak on the scale of atoms and elementary particleswe can simply ignore the infinite ground state energies of systems ​and calculate energy differences between the ground state and excited states. 
-Renormalization originated not from abstract theory but rather from the struggle to overcome a nasty technical problem. If one supposes that spacetime is continuumthen in any finite volume ​of space there is an infinite number ​of degrees of freedom, ​and in summing their contributions to physical processes one often finds divergent, and hence meaningless resultsRenormalization originated as a technical trick to absorb these divergences into redefinitions ​of the couplings: it relates so called “bare” couplingswhich appear ​in the fundamental Lagrangian and have no direct physical significance, ​to “renormalized” couplings, which correspond ​to what one actually measures in the laboratory.+This idea is at the heart of the mathematical methods that form the basis of renormalizationdescribed by Glashow when he discusses the numerous divergences that arise in quantum field theory. There is another physical basis for ignoring these infinities. The source of these infinities comes from extrapolating ​the mathematical algorithms that allow one to perform calculations with the theory down to arbitrarily small scales. But there is no reason ​to assume that no new physics will be encountered on ever-smaller scales—physics that would require one to change the nature of the calculations at small scales. Theories that make sense are therefore theories that are insensitive to changes associated with possible new physics on arbitrarily small scales—so-called renormalizable theories. In such theories ​one can discard ​the arbitrary infinities that arise from unknown effects at arbitrarily small scales with impunity. 
 +<​cite>​https://​inference-review.com/​letter/​anything-but-standard</​cite></​blockquote>​
  
-<​cite>​[[https://​arxiv.org/​pdf/​0910.5167v1.pdf|Gravity from a Particle Physicists’ perspective]] by Roberto Percacci</​cite>​ 
-</​blockquote>​ 
-</​WRAP>​ 
-</​WRAP>​ 
- 
-**See also:** 
- 
-  * [[advanced_tools:​renormalization:​bphz]] 
- 
- 
-<tabbox Layman> ​ 
 <​blockquote>​ <​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. 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.
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 would be infinitely dense. would be infinitely dense.
  
- 
-An experiment to determine what a hydrogen atom consists of, or 
-even as straightforward as measuring the mass or electric charge of a 
-free electron, is analogous to viewing something on a computer screen, 
-where the image is made of pixels. At low resolution you see some level 
-of detail, but if you increase the density of pixels, the picture becomes 
-sharper. For example, if an image with a few pixels shows a river, repre- 
-sentations built from larger numbers of pixels will reveal finer details of 
-its swirling waters. Likewise, at low resolution, an atom of hydrogen can 
-just be made out to contain a powerful electric field, which grips a single 
-electron in the atom’s outer regions. The electron itself can be discerned 
-as a fuzzy lump of electricity,​ filling a single pixel and seemingly just be- 
-yond the limit of resolution. 
-In order to see what that haze of charge is really like, we must take a 
-closer look. Suppose we increased the pixel density by a hundred, so that 
-for every individual large pixel that was present in the previous case, we 
-now have a hundred fine-grain ones. Where before the electric charge 
-was located within a single large pixel, we now find that it is situated inside 
-the single minipixel at the center, the ninety-nine surrounding minipixels 
-containing negatives and positives: the whirlpools of virtual electrons and 
-positrons in the vacuum. 
-Now increase the density by another factor of a hundred. The charge 
-of the electron, which previously filled a single minipixel, is now found to 
-be concentrated into a central micropixel, which is surrounded by further 
-vacuum whirlpools. And each of the whirlpools that had previously 
-shown up in the image made of minipixels is revealed to contain yet finer 
-eddies. 
-And so it continues. The image of an electron is like a fractal, repeating 
-over and over at finer detail, forever. The electric charge of the electron is 
-focused into a single pixel of an ever-decreasing size, surrounded by a 
-vacuum of unimaginable electrical detail. 
-I cannot imagine it, nor, I suspect, can any other physicist. We are con- 
-tent to follow what the mathematics and experimental data reveal. Any- 
-way, it is not the surrounding vacuum that concerns us here; our quarry 
-is the electric charge of the original electron. 
-As we zoom in toward the kernel, and the smeared lump of that elec- 
-tron’s charge is concentrated into a smaller and smaller pixel, the density 
 Calculations in QED, which gave infinity as the answer, were in some cases doing so because they had summed up the contributions of all pixel sizes—from large scale all the way down to the infinitesimal extreme. However, this is not what a real experiment measures. What you see depends on the resolving power of your microscope. In our voyage into inner space, current technology enables us to resolve matter at a scale as fine as one-billionth the size of a single atom of hydrogen. This is very small, to be sure, but it is not truly infinitesimal. In practice, what we want to be able to calculate are the values of measurements made at this particular scale of resolution. Adding up the contributions of all pixel sizes is like a travel agent charging you for trips to outer space, and including in the bill exotic destinations that are as yet impossible to reach. For example, you may have traveled on the International Space Station, the limit of what is currently on offer, and expressed an interest in Lunar exploration and even a trip to Mars when these become practical. The travel agent includes these in the bill, along with twenty- first century trips to deep space that are presently science fiction. With the potential trips extrapolated into the indefinite future, the bill comes to infinity. By any rationale, this is absurd. What you really want to know is the cost of a trip to the moon or to Mars. To do so, we may agree with the accountant that the price for a trip to the space station (plus an option on possible future trips to infinitely far other destinations) is some amount: X. Then we might expect to get a sensible answer, in terms of multiples of X, for the cost of a trip to the moon or Mars. In fact, it would be deter- mined by the finite difference in the distances between those destinations. ​ Analogously,​ infinity emerges from a typical sum in QED because the calculation has included the effects of scales of length—pixel sizes, if you prefer—that are infinitesimally small. With hindsight, this too is absurd. And as in the case with the travel bill, we need some way in QED to do the accounts for what we can measure and not include in the sums the wonders of an inner space that lie beyond vision. If we have some point of reference, where we know the correct value, the equivalent of the known cost of going to the space station, then we may be able to compute another quantity relative to the first one. That was at the core of what Feynman and Schwinger were developing. This technique of using known values, such as the charge of the electron (the cost of going to the space station in our analogy) to calculate other values, such as its magnetism (the cost to the moon or Mars), became known as renormalization. 2 Calculations in QED, which gave infinity as the answer, were in some cases doing so because they had summed up the contributions of all pixel sizes—from large scale all the way down to the infinitesimal extreme. However, this is not what a real experiment measures. What you see depends on the resolving power of your microscope. In our voyage into inner space, current technology enables us to resolve matter at a scale as fine as one-billionth the size of a single atom of hydrogen. This is very small, to be sure, but it is not truly infinitesimal. In practice, what we want to be able to calculate are the values of measurements made at this particular scale of resolution. Adding up the contributions of all pixel sizes is like a travel agent charging you for trips to outer space, and including in the bill exotic destinations that are as yet impossible to reach. For example, you may have traveled on the International Space Station, the limit of what is currently on offer, and expressed an interest in Lunar exploration and even a trip to Mars when these become practical. The travel agent includes these in the bill, along with twenty- first century trips to deep space that are presently science fiction. With the potential trips extrapolated into the indefinite future, the bill comes to infinity. By any rationale, this is absurd. What you really want to know is the cost of a trip to the moon or to Mars. To do so, we may agree with the accountant that the price for a trip to the space station (plus an option on possible future trips to infinitely far other destinations) is some amount: X. Then we might expect to get a sensible answer, in terms of multiples of X, for the cost of a trip to the moon or Mars. In fact, it would be deter- mined by the finite difference in the distances between those destinations. ​ Analogously,​ infinity emerges from a typical sum in QED because the calculation has included the effects of scales of length—pixel sizes, if you prefer—that are infinitesimally small. With hindsight, this too is absurd. And as in the case with the travel bill, we need some way in QED to do the accounts for what we can measure and not include in the sums the wonders of an inner space that lie beyond vision. If we have some point of reference, where we know the correct value, the equivalent of the known cost of going to the space station, then we may be able to compute another quantity relative to the first one. That was at the core of what Feynman and Schwinger were developing. This technique of using known values, such as the charge of the electron (the cost of going to the space station in our analogy) to calculate other values, such as its magnetism (the cost to the moon or Mars), became known as renormalization. 2
  
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 <​cite>"​A different Universe"​ by Robert Laughlin</​cite>​ <​cite>"​A different Universe"​ by Robert Laughlin</​cite>​
 </​blockquote>​ </​blockquote>​
 +
 +----
  
 ** Recommended Books:** ** Recommended Books:**
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   * The Infinity Puzzle by F. Close   * The Infinity Puzzle by F. Close
  
-<​tabbox ​Student+<​tabbox ​Concrete
 <​blockquote>​ <​blockquote>​
 Another well-known problem is connected with the fact that field theory has infinitely many degrees of freedom, and all of them have some zero point oscillations. Even for a free (photon) field we have a strongly divergent vacuum energy ​ Another well-known problem is connected with the fact that field theory has infinitely many degrees of freedom, and all of them have some zero point oscillations. Even for a free (photon) field we have a strongly divergent vacuum energy ​
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 **Recommended Resources:​** **Recommended Resources:​**
  
 +  * The best explanation of how different renormalization schemes conceptually work can be found in [[https://​arxiv.org/​abs/​0812.3578|Regularization,​ Renormalization,​ and Dimensional Analysis]] by Fredrick Olness & Randall Scalise
   * [[ http://​math.ucr.edu/​home/​baez/​renormalization.html|Renormalization Made Easy]] by John Baez   * [[ http://​math.ucr.edu/​home/​baez/​renormalization.html|Renormalization Made Easy]] by John Baez
   * A good short summary of how renormalization works in practice can be found in Collin'​s "​Renormalization"​ page 11   * A good short summary of how renormalization works in practice can be found in Collin'​s "​Renormalization"​ page 11
 +  * Moreover, a concise over of different regularization methods can be found [[http://​hitoshi.berkeley.edu/​230A/​regularization.pdf|here]].
   * Another good book is "​Renormalization Methods : a Guide for Beginners"​ by David Mc Comb   * Another good book is "​Renormalization Methods : a Guide for Beginners"​ by David Mc Comb
-  * The best introduction is http://​www.physics.umd.edu/​courses/​Phys851/​Luty/​notes/​renorm.pdf+  * A great discussion of why infinities arise in the first place can be found in http://​www.physics.umd.edu/​courses/​Phys851/​Luty/​notes/​renorm.pdf ​and [[http://​philsci-archive.pitt.edu/​16072/​1/​Renormalization%20scrutinized%20-%20Revised%20manuscript.pdf|Renormalization Scrutinized]] by Sebastien Rivat
   * See also [[https://​arxiv.org/​abs/​hep-th/​0212049|A hint of renormalization by B. Delamotte]] and   * See also [[https://​arxiv.org/​abs/​hep-th/​0212049|A hint of renormalization by B. Delamotte]] and
-  * http://​www.mat.univie.ac.at/​~neum/​ms/​ren.pdf and+  * http://​www.mat.univie.ac.at/​~neum/​ms/​ren.pdf and http://​philsci-archive.pitt.edu/​16072/​1/​Renormalization%20scrutinized%20-%20Revised%20manuscript.pdf
   * http://​www.mat.univie.ac.at/​~neum/​physfaq/​topics/​ren1 .   * http://​www.mat.univie.ac.at/​~neum/​physfaq/​topics/​ren1 .
   * Also the video course: [[https://​www.complexityexplorer.org/​tutorials/​67-introduction-to-renormalization/​segments/​5681|Introduction to Renormalization]] by Simon DeDeo is highly recommended.   * Also the video course: [[https://​www.complexityexplorer.org/​tutorials/​67-introduction-to-renormalization/​segments/​5681|Introduction to Renormalization]] by Simon DeDeo is highly recommended.
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- +<​tabbox ​Abstract
- +
-<​tabbox ​Researcher +
- +
- +
  
  
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   * [[http://​www2.mathematik.hu-berlin.de/​~kreimer/​wp-content/​uploads/​BerghoffPhD.pdf|Wonderful Renormalization]] by Berghoff   * [[http://​www2.mathematik.hu-berlin.de/​~kreimer/​wp-content/​uploads/​BerghoffPhD.pdf|Wonderful Renormalization]] by Berghoff
   * P.A.M. Dirac, The inadequacies of quantum field theory   * P.A.M. Dirac, The inadequacies of quantum field theory
 +  * [[https://​arxiv.org/​abs/​1406.4532|Renormalization for Philosophers]] by J Butterfield
  
-<​tabbox ​Examples+<​tabbox ​Why is it interesting?​
  
---Renormalization ​in Classical Physics#+<​blockquote> 
 +The philosophy of renormalization is one of the most difficult, and controversial, ​in all of particle physics. Yet it underpins modern theory.
  
-See https://​en.wikipedia.org/​wiki/​Regularization_(physics)#​Classical_physics_example +<​cite>​page 42 in The Infinity Puzzle by Frank Close</cite> 
-  +</​blockquote>​
-<--+
  
---Example2:#+<​blockquote> 
 +For physicists, infinity is a code word for disaster, the proof that you 
 +are trying to apply a theory beyond its realm of applicability. In the case 
 +of QED, if you can’t calculate something as basic as a photon being absorbed by an electron, you haven’t got a theory—it’s as fundamental as 
 +that. 
 +<​cite>​page 4 in The Infinity Puzzle by Frank Close</​cite>​ 
 +</​blockquote>​ 
 + 
 +<​blockquote>​ 
 +It is possible to say that the UV divergences mean that the theory makes no physical sense, and that the subject of interacting quantum field theories is full of nonsense. ([[http://​aip.scitation.org/​doi/​abs/​10.1063/​1.33110|Dirac (1981)]]) 
 +<​cite>​Renormalization by Collins</​cite>​ 
 +</​blockquote>​
  
-  + 
-<--+ 
 +<​blockquote>​ 
 +Renormalization originated not from abstract theory but rather from the struggle to overcome a nasty technical problem. If one supposes that spacetime is continuum, then in any finite volume of space there is an infinite number of degrees of freedom, and in summing their contributions to physical processes one often finds divergent, and hence meaningless results. Renormalization originated as a technical trick to absorb these divergences into redefinitions of the couplings: it relates so called “bare” couplings, which appear in the fundamental Lagrangian and have no direct physical significance,​ to “renormalized” couplings, which correspond to what one actually measures in the laboratory. 
 + 
 +<​cite>​[[https://​arxiv.org/​pdf/​0910.5167v1.pdf|Gravity from a Particle Physicists’ perspective]] by Roberto Percacci</​cite>​ 
 +</​blockquote>​
  
 <tabbox FAQ> ​ <tabbox FAQ> ​
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   * https://​www.physicsforums.com/​threads/​why-do-we-need-to-renormalize-in-qft-really.917973/#​post-5793554   * https://​www.physicsforums.com/​threads/​why-do-we-need-to-renormalize-in-qft-really.917973/#​post-5793554
   * How I Learned to Stop Worrying and Love QFT by Mario Flory et. al.   * How I Learned to Stop Worrying and Love QFT by Mario Flory et. al.
 +
 +<​blockquote>​
 +The ultraviolet infinities appear as a consequence of space-time localization of
 +interactions,​ which occur at a point, rather than spread over a region. (Sometimes
 +it is claimed that field theoretic infinities arise from the unhappy union of quantum
 +theory with special relativity. But this does not describe all cases - later I shall discuss
 +a non-relativistic,​ ultraviolet-divergent and renormalizable field theory.) Therefore
 +choosing models with non-local interactions provides a way to avoid ultraviolet
 +infinities. The first to take this route was Heisenberg, but his model was not phenomenologically viable. These days in string theory non-locality is built-in at the start, so
 +that all quantum effects - including gravitational ones - are ultraviolet finite, but this
 +has been achieved at the expense of calculability:​ unlike ultraviolet-divergent local
 +quantum field theory, finite string theory has enormous difficulty in predicting definite
 +physical effects, even though it has succeeded in reproducing previously mysterious
 +results of quantum field theory - I have in mind the quantities associated with
 +black-hole radiance.
 +
 +<​cite>​[[https://​arxiv.org/​abs/​hep-th/​9602122|The Unreasonable Effectiveness of Quantum Field Theory]] by R. Jackiw</​cite>​
 +</​blockquote>​
  
 <​blockquote>"​ The solution is given by an iterated integral, known as Dyson’s series, U(s, t) = T(e −i R t s dτ HI (τ) ). Here time ordering is needed because the Hamiltonians evaluated at different times need not commute. **Since time ordering is defined with the generalized function θ, this is the point where (ultraviolet) divergences are inserted into the theory; in general one cannot multiply distributions by discontinuous functions, or equivalently,​ in the language of distributions,​ the product of two distributions is not well-defined.** There are two ways of dealing with this problem: Try to construct a well-defined version of T, or proceed with the calculation and try to get rid of the problems at the end. The first is the basic idea of the Epstein-Glaser approach, the second leads to renormalization,​ the art of removing these divergences in a physical meaningful manner. [...]  <​blockquote>"​ The solution is given by an iterated integral, known as Dyson’s series, U(s, t) = T(e −i R t s dτ HI (τ) ). Here time ordering is needed because the Hamiltonians evaluated at different times need not commute. **Since time ordering is defined with the generalized function θ, this is the point where (ultraviolet) divergences are inserted into the theory; in general one cannot multiply distributions by discontinuous functions, or equivalently,​ in the language of distributions,​ the product of two distributions is not well-defined.** There are two ways of dealing with this problem: Try to construct a well-defined version of T, or proceed with the calculation and try to get rid of the problems at the end. The first is the basic idea of the Epstein-Glaser approach, the second leads to renormalization,​ the art of removing these divergences in a physical meaningful manner. [...] 
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 <-- <--
   ​   ​
-<tabbox History> ​+<tabbox History> 
 + 
 +  * [[https://​arxiv.org/​abs/​hep-th/​9812203|The Glorious Days of Physics - Renormalization of Gauge theories]] by Gerard 't Hooft 
 + 
  
 </​tabbox>​ </​tabbox>​
  
  
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