I am trying to work my head around how renormalization works for quantum field theory. Most treatments cover perturbative renormalization theory and I am fine with this approach. But it is not the most general framework and is not intuitively related to the Wilsonian approach. I am also a bit lost with respect to the meaning of the associated notions in the QFT context like the anomalous dimension. So, I essence what I am asking what is the intuitive picture relating the coarse graining idea of statistical physics with the framework of quantum field theory. Also, how could one picture/understand the notion of proliferation, coupling constants, beta functions, fixed points, anomalous dimension and conformal invariance with respect to the way the ideas are interpreted in statistical physics. I find it easy to visualize the entire procedure in the latter context but not in the former. A complete analogy might not be possible but I am only asking to what extent it can be achieved.

  • $\begingroup$ You wrote : "Most treatments cover perturbative renormalization theory and I am fine with this approach." I wonder if you are comfortable with "bare particle physics"? $\endgroup$ – Vladimir Kalitvianski Jan 25 '13 at 13:56
  • $\begingroup$ Not sure if this is the sort of thing you are looking for but you might find this review. It uses the concept of the ERGE and starts of with familiar ideas of blocks of spins on a lattice from statistical physics. $\endgroup$ – Mistake Ink Jan 25 '13 at 17:15
  • $\begingroup$ Thank you for the review. Seems reluminating. "I am fine with this approach" = I see what it tries to accomplish and it makes sense even if maybe it's not entirely sound from some elementary mathematical point of view $\endgroup$ – Sebby Jan 25 '13 at 17:42
  • $\begingroup$ @Sebby: I did not ask about math, I asked about physics they propose you to accept (interaction of bare particles). Is it convincing to you? I just wonder how people perceive the notion of "bare particle". $\endgroup$ – Vladimir Kalitvianski Jan 25 '13 at 18:28
  • $\begingroup$ It is not discussed much. Naively it doesn't seem realistic, but luckily a slightly generalized form of the Lagrangian and some perturbation theory makes everything ok and we never really need to ask if our underlining formulation of QFT is wrong. This I sense/think is the general mood. $\endgroup$ – Sebby Jan 26 '13 at 23:14

As far as I understand it, a basic renormalization step consisting of

  1. Coarse graining (average or integrate out high energy degrees of freedom)
  2. recalculate the appropriate quantity defining the effective theory which describes the system at a certain scale
  3. rescale different quantities as needed

works in the same way for both, statistical mechanics and quantum field theoretic systems. The difference between them I have seen so far, is that for statistical mechanics one uses the partition function or the Hamiltonian to describe the effective theory, whereas in QFT the action or the Lagrangian is used. In both cases, considering an infinitesimal renormalization transformation, different renormalization group equations (or $\beta$ functions for specific coupling constants can be derived, and investigations of the renormalization group flow to find and characterize fixed points etc are done in a very similar spirit.

Concerning nonperturbative renormalization, a method which can do this (I dont know what other noperturbative renormalization methodes exist if any) is the Exact Renormalization Group (ERG), sometimes also called functional renormalization group, allows for this. A nice introduction to and overview of the ERG is given in this tutorial by Oliver J. Rosten, describes things (after reviewing block spin models in statistical mechanics) from a QFT point of view.

You ask about quite a number of different specific issues, as far as I have seen already, many things like beta functions, different fixed points, anomalous dimensions, etc are covered in Roston's text.

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