Why are no DIVERGENT quantities (infinities) in String Theory ?

[zetafunction]

Why are no DIVERGENT quantities (infinities) in String Theory ??

why String theory is FREE of infinities ?? ... why there are no divergent integrals in string theory whereas in normal Quantum Field theory there are infinities ??

 

[bcrowell]

As far as I know, it is only a conjecture that string theory converges to all orders. Here is a review article with some discussion of the state of knowledge about this as of 2003: http://arxiv.org/abs/hep-th/0303185v2 See p. 34, where he states that the conjecture has not been proved. I don't think the conjecture had been proved as of 2006, which was when Smolin's The Trouble with Physics was published; on pp. 278-281 of that book he states that finiteness to all orders still had not been proved.

As far as the reasons why it might be expected to be finite, the arxiv paper says,
"There are intuitive arguments that suggest that ultraviolet divergences of the kind that plague conven-
tional quantum field theory cannot occur in string theory. The main reason is that the
interactions of strings involve the breaking and joining of strings and these do not take
place at points. However, a string theory can fail to be consistent for other reasons.
There may be infrared divergences, or ambiguities in the definition of the amplitudes,
there can be anomalies in the action of the lorentz boosts, or the theory may fail to be
unitary."

 

[Demystifier]

why String theory is FREE of infinities ?? ... why there are no divergent integrals in string theory whereas in normal Quantum Field theory there are infinities ??

Simple answer:
Because string (unlike a point particle) has a finite length, so high momenta corresponding to distances shorter than the string length are cut off. This removes the UV divergences present in QFT. IR divergences (which are benign) are not removed.

A more correct answer is related to the fact that string theory contains an additional symmetry (2D conformal symmetry) not present in QFT, and this additional symmetry removes more degrees of freedom as redundant. The removal of these additional degrees of freedom can be reduced to a removal of UV degrees of freedom present in QFT.

 

[haushofer]

Very nice question!

String theory also contains an "infinite tower of particles" with arbitrary high mass and spin; is this also relevant for looking at the finiteness of string theory?

 

[bcrowell]

Very nice question!

String theory also contains an "infinite tower of particles" with arbitrary high mass and spin; is this also relevant for looking at the finiteness of string theory?

These would be described as short-wavelength modes of vibration of a string, so I think your question boils down to whether string theory has ultraviolet divergences. The quote in #2 suggests that this is considered one of the less likely reasons to get infinities in string theory.

 

[MTd2]

less *likely* reasons to get infinities in string theory.

Isn't it certain?

 

[bcrowell]

Isn't it certain?

You mean certain that there are infinities, or certain that there aren't? Nothing is known for certain, if the two Lee Smolin references in #2 are accurate and not too out of date. In the second reference, Smolin describes being surprised to find that there was no proof of a fact that a lot of string theorists were assuming had been proved, and then surprised at the hostile reaction when he tried to accurately summarize the state of knowledge in a review paper he'd been asked to write (which I believe is what the first reference is).

 

[petergreat]

Lubos Motl has a long post about finiteness of string theory:
http://motls.blogspot.com/2006/12/patience-medal-for-jacques-distler.html"

 

[bcrowell]

Motl's post is mainly personal insults, but it does link to a discussion here, http://asymptotia.com/2006/11/10/more-scenes-from-the-storm-in-a-teacup-vi/ , which has quite a bit of reasoned debate on this issue. It sounds like this is an issue on which experts disagree. Some people think finitness has been proved, and others don't.

 

[petergreat]

I've been told that it's exactly this infinite tower of particles that make the total scattering amplitude fall off exponentially above Planck scale. So it seems that this tower is considered to be something nice rather than a potential trouble-maker.