However, in my opinion IMHO Loop Quantum Gravity does a better job at solving the infinity problem than does String theory, because the infinity you are concerned with comes from General Relativity, amd LQG is more like GR than is SST.
Actually, there are infinities sprinkled through out all of particle physics. Quantum electrodynamics QED was the first to get rid of its infinities. The infinity occurred because particles in those theories are treated mathematically as points. So as you approach a point of finite mass, from Newton's law of gravitation, which has a 1/(x-x') in it, you get infinity automatically as x approaches x'.
Feymann, in deriving QED, just replaced the infinity by the value of the mass in a process called renormalization, which I have never done, so don't ask me any more about it. The same renormalization process was repeated in all the more complicated theories such as QCD and GUT. What made that possible is that all these theories use fields in the background of a flat spacetime.
GR does not do that. In GR the distortion of spacetime from flatness replaces the need for a gravitational field. And besides that the graviton, which is the particle that you get from collapse of gravitational field eigenfunctions, turns out to be spin two, rather than a linear spin one like the electron, or spin less than one like quarks. As a result the renormalization process, which really was just a trick anyway, does not work. The infinity at the point of point particles of spin greater than 1 is not removed.
The way to prevent such an infinity is then to stop assuming particles are points. There is no good reason why particles should be points. Points are vanishingly small and as such the properties of a point should also vanish. Points should not contain mass. The mass of a point should vanish in the limit of no space in a point. Right. Do that not make sense, even to us commoners.
So the obvious next step is to assume that particles have extent, so that any point on the extensive particle has zero mass, and so the infinity is eliminated; but if you integrate over the whole volume of the particle you get its mass, or any other property.
To me what is very interesting is that the assumption that particles are strings, which have extension in only one dimension, can remove the infinity. But if you think about it, stretching the mass of a particle in just one dimension removes the infinity at any point on the particle.
That is all that string theory does. It is the most simple next step. I do not like it because I think that particles should have extension in 3-d, that is a particle should have volume. The string still has no volume, even though the infinities are eliminated in 3-d space. So I think that string theory is doomed. But who am I to know. It's just my intuition. Just as physicists got hookey results from assuming pareticles were infinitisimal points. I think they will get hookey results when they assume that partilces are infinitisimal in two of the three dimensions of 3-d space.
LQG is different in that particles in those theories have volume. LQG is really just the string theory of space itself. It is assumed that the fundamental element of everything including space and time itself are loops- which for us commoners are just like strings. So you cannot define a point in space or space time. The smallest unit is a loop. So whatever properties space has, not mass, but it does have energy, is spread out over a loop.
LQG theorists then say that these loops weave together, just like cloth except it is a 3-d weave rather than 2-d, to create the area and volume of space. So far as I know they have not yet created any particles this way. But they have derived a nonsingular big bang; and they have derived Schroedingers equation, which is like deriving quantum mechanics; and they get a cosmological constant (or dark energy) with the correct sign, and a few other things that SST has not even come close to deriving. And they have done all this with just a fraction of the number of scientists doing SST.
Hope that all helps.