Why point particle interaction has UV divergence,but string interaction does not?

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Discussion Overview

The discussion revolves around the differences in ultraviolet (UV) divergences between point particle interactions in quantum field theory (QFT) and string interactions. Participants explore the implications of these differences for calculating probability amplitudes and the underlying mathematical structures involved.

Discussion Character

  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant questions why point particle interactions in QFT exhibit UV divergences while string interactions do not, suggesting that momentum integration leads to these divergences.
  • Another participant explains that in string theory, momentum integration is replaced by integration over the moduli space of Riemann surfaces, which introduces a different structure that avoids UV divergences.
  • A participant raises a concern about the scattering of D0-branes, which are pointlike objects, questioning whether this leads to the reintroduction of short-distance divergences due to the lack of smearing of interaction vertices.
  • A later reply acknowledges that open strings stretched between D0-branes can lead to massless states when the branes coincide, suggesting that this scenario may introduce on-shell divergences.

Areas of Agreement / Disagreement

Participants express differing views on the implications of D0-brane interactions and whether they reintroduce divergences, indicating that the discussion remains unresolved regarding the treatment of these specific cases in string theory.

Contextual Notes

The discussion highlights the complexity of integrating over different mathematical structures in string theory compared to QFT, as well as the potential for divergences in specific scenarios involving pointlike objects.

ndung200790
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Please teach me this:
Why point particle interaction(QTF theory) has UV divergences,but string interaction does not have UV divergences.Because I think that calculating the probability amplitude of a process,we must sum(or integrate) over all possible value of momentum.So how we avoid the UV divergence?
Thank you very much in advance.
 
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ndung200790 said:
Because I think that calculating the probability amplitude of a process,we must sum(or integrate) over all possible value of momentum

That is true in particle QFT, but not in string theory. What replaces momentum integration is an integration over the moduli space ("shapes") of Riemann surfaces, and this is substantially different to particle theory. One may loosely say there appears a UV cutoff for the momentum integration. Actually things are more complicated and the stringy way of integration is more than a cutoff, and thus cannot be expressed in terms of particle QFT supplemented by a naive momentum cutoff. Pertaining to the geometry of Riemann surfaces, there are notions like modular invariance which have no analog in particle physics, so string theory is "more" than just a collection of infinitely many particle theories. That's why it can do things that cannot be achieved in particle QFT.
 
One thing I have always wondered in this context is what happens when one talks about the scattering of D0-branes, which are pointlike objects. In that case, no smearing out of the interaction vertices occurs, as it does for extended objects like strings. So doesn't this bring short-distance divergences back into the game?
 
Orbb said:
One thing I have always wondered in this context is what happens when one talks about the scattering of D0-branes, which are pointlike objects. In that case, no smearing out of the interaction vertices occurs, as it does for extended objects like strings. So doesn't this bring short-distance divergences back into the game?

Yes, from open strings stretched between the branes, which become massless when the branes coincide. So in a sense these are on-shell divergences.
 

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