Does Renormalization group tell you if a theory is Renormalizable or not ?

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Discussion Overview

The discussion revolves around the relationship between the Renormalization Group (RG) and the renormalizability of various theories, including Quantum Electrodynamics (QED) and gravity. Participants explore whether the RG can indicate if a theory is renormalizable based on its behavior at different energy scales.

Discussion Character

  • Debate/contested

Main Points Raised

  • One participant questions if the RG equation can determine the renormalizability of a theory at different energy scales.
  • Another participant asserts that if a theory is non-renormalizable, it cannot have an RG equation by definition.
  • Some participants suggest that it is not possible to know a theory's renormalizability 'a priori' and propose that an RG equation might still be obtainable, citing gravity as an example.
  • It is mentioned that the RG can indicate the existence of a continuum limit, which is relevant for gravity, and that a UV fixed point is necessary for this limit.
  • There is a note that while some theories are renormalizable, they may not possess continuum limits, with QED cited as an example where the fixed point is infrared rather than UV.
  • A participant expresses confusion regarding the connection between renormalizability and the presence of an infrared fixed point.

Areas of Agreement / Disagreement

Participants express differing views on whether the RG can provide insights into the renormalizability of theories, indicating that multiple competing perspectives remain unresolved.

Contextual Notes

There are limitations in understanding the definitions and implications of renormalizability, RG equations, and fixed points, which may affect the clarity of the discussion.

zetafunction
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Does Renormalization group tell you if a theory is Renormalizable or not ??

the idea is this, using the Renormalization group equation for our theory (QED, Gravity, Gauge theories..) can tell this RG equation if our theory is renormalizable or not for big or small energies ??
 
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No, if the theory is non-renormalizable then by definition there is no RG equation for that theory.
 


but you would not know if a given theory is renormalizable or not 'a priori' so perhas you could obtain a renormalization group equation or similar, for example i think there is a RG equation fro gravity..
 


zetafunction said:
but you would not know if a given theory is renormalizable or not 'a priori' so perhas you could obtain a renormalization group equation or similar, for example i think there is a RG equation fro gravity..

The RG can tell you if the theory has a continuum limit, and that's what's being asked for gravity. For a continuum limit to exist, the RG equation must have a UV fixed point. If a continuum limit exists, then the theory is renormalizable.

However, there are renormalizable theories that don't have continuum limits, such as QED. The QED fixed point is infrared, not UV. I don't understand the relationship (if any?) between renormalizability and an infrared fixed point.
 

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