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

In summary, the absence of divergent quantities in String Theory may be due to the presence of additional symmetries in the theory, and the finite length of strings may be a contributing factor.
  • #1
zetafunction
391
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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 ??
 
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  • #2


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."
 
  • #3


zetafunction said:
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.
 
  • #4


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?
 
  • #5


haushofer said:
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.
 
  • #6


bcrowell said:
less *likely* reasons to get infinities in string theory.

Isn't it certain? :confused:
 
  • #7


MTd2 said:
Isn't it certain? :confused:

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).
 
  • #8


Lubos Motl has a long post about finiteness of string theory:
http://motls.blogspot.com/2006/12/patience-medal-for-jacques-distler.html"
 
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  • #10


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.
 

1. Why are there no infinities in String Theory?

One of the main reasons for the absence of infinities in String Theory is that it is a finite theory. Unlike other theories such as Quantum Field Theory, which involves point particles, String Theory involves extended objects which have a finite size. This helps to avoid the occurrence of infinite values in calculations.

2. How does String Theory avoid infinities?

String Theory incorporates the concept of supersymmetry, which states that every bosonic (force-carrying) particle has a corresponding fermionic (matter) particle, and vice versa. This symmetry helps to cancel out the infinities that would otherwise arise in the calculations.

3. Can infinities still arise in String Theory?

While String Theory is able to avoid infinities in most cases, there are some situations where they can still arise. For example, when there are strong gravitational fields or when the strings are interacting at very high energies. However, these infinities are not as severe as those in other theories and can often be resolved through further mathematical techniques.

4. How does String Theory address the issue of singularities?

Singularities, such as the ones found in black holes, are points in space-time where the laws of physics break down and infinities occur. In String Theory, the strings are able to probe these singularities and provide a more complete understanding of them. This has led to the idea that singularities may not actually exist in String Theory, but rather are a result of our limited understanding of space-time.

5. Can the absence of infinities in String Theory be tested?

Currently, there is no experimental evidence for String Theory and therefore no way to test the absence of infinities. However, there are ongoing efforts to develop more advanced mathematical techniques and experimental methods that may help to test the predictions of String Theory in the future.

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