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advanced_tools:renormalization [2019/02/03 10:53]
jakobadmin
advanced_tools:renormalization [2019/07/05 12:10] (current)
jakobadmin [Concrete]
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 <tabbox Intuitive> ​ <tabbox Intuitive> ​
 +<​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>​
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 <​blockquote>​Thus,​ when 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 particles, we can simply ignore the infinite ground state energies of systems and calculate energy differences between the ground state and excited states. <​blockquote>​Thus,​ when 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 particles, we can simply ignore the infinite ground state energies of systems and calculate energy differences between the ground state and excited states.
 This idea is at the heart of the mathematical methods that form the basis of renormalization,​ described 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. This idea is at the heart of the mathematical methods that form the basis of renormalization,​ described 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.
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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.
advanced_tools/renormalization.1549191201.txt.gz · Last modified: 2019/02/03 10:53 by jakobadmin