Non renormalizability non predictiveness

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Discussion Overview

The discussion revolves around the concept of non-renormalizability in the context of pure gravity, particularly focusing on the implications of divergent contributions at higher loop levels and the challenges in making predictions within such a framework. Participants explore theoretical methods, such as the BPHZ method, and the nature of counter terms required for addressing divergences.

Discussion Character

  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant expresses confusion about non-renormalizability, noting that the BPHZ method seems to allow for the subtraction of divergences at each loop level, suggesting that only a finite number of counter terms should be needed.
  • Another participant argues that requiring an infinite number of independent terms indicates an infinite number of degrees of freedom, which complicates making useful predictions.
  • A different viewpoint suggests that one could use an 'ansatz' for the infinite quantities or approximate divergent integrals with divergent series, potentially leading to finite values through resummation methods.
  • Concerns are raised about the need to fix arbitrary finite pieces when introducing new interactions to address divergences, implying that this requirement diminishes predictive power and necessitates additional measurements.
  • Some participants propose treating the theory as an effective field theory, noting that the dimensionful nature of Newton's constant suggests limitations in the applicability of gravity at high energy scales.

Areas of Agreement / Disagreement

Participants express differing views on the implications of non-renormalizability and the utility of effective field theories. There is no consensus on the best approach to handle divergences or the predictive power of the theory.

Contextual Notes

Participants highlight the complexity of addressing divergences and the dependence on energy scales, with some suggesting that only a finite number of divergences may be relevant at low energies, while others emphasize the challenges posed by infinite terms at higher energies.

Jim Kata
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Ok, I know Goroff and Sagnotti proved that the two loop contribution to pure gravity is divergent, but I guess I don't understand non - renormalizability. Looking, at the BPHZ method it seems that you could subtract out the divergences at each level. Although, you need an infinite number of counter terms for the entire path integral. It seems to me that at each loop level it would only take a finite number of counter terms to cancel the divergences. What am I missing here?
 
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Nothing. But if you need to subtract out an infinite number of independent terms than your theory has an infinite number of degrees of freedom, i.e. you would need to make an infinite number of observations to extract any useful prediction. This makes the theory less useful...
 
but although you need an infinite number of parameters could not try an 'ansatz' for the infinite quantities ? or if the problem of divergences is that integrals are divergent could not approximate these divergent integrals by divergent series and then use resummation methods to obtain finite values for these series ?? or at least assume that all the particles have a 'radius' proportional to their Compton wavelent [tex]\lambda = \hbar (mc)^{-1}[/tex] .

Also i am not sure but i think that at least for low energies we have only a finite number of divergences, in case you increase the energy scale then the infinite terms become relevant and spoil your prediction.
 
Jim Kata said:
Ok, I know Goroff and Sagnotti proved that the two loop contribution to pure gravity is divergent, but I guess I don't understand non - renormalizability. Looking, at the BPHZ method it seems that you could subtract out the divergences at each level. Although, you need an infinite number of counter terms for the entire path integral. It seems to me that at each loop level it would only take a finite number of counter terms to cancel the divergences. What am I missing here?

But the problem is that every time you bring in a new interaction that is required to patch up the divergences, you need to fix the finite piece as well! Sure, you can fix the divergent part of the bare coeffcient to insure that you result is finite but you are left with an arbitrary finite piece. How to fix this finite piece? By performing a new experiment. So you end up losing predictive power. If you can't make any prediction without requiring extra measurements to fix the constants of your theory, the theory is useless.

The way out is of course to treat the theory as an effective field theory.
 
The way out is of course to treat the theory as an effective field theory.

This much should be apparent just from the fact that Newton's constant is dimensionful. Just as we don't expect Fermi Theory to hold in the infinite energy (zero length) limit, we shouldn't expect gravity to hold in that regime either.
 

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