# Singularities and quantized space time

Discrete space time quanta would solve the problems of infinite densities for singularities in General Relativity and Quantum Gravity by imposing a non zero limit on the minimum radius of black holes. This would also prevent the "Violet Catastrophe" where the size of "Quantum Foam" energy fluctuations approach infinity as the scale approaches zero. Despite this, there appears to be a great deal of reluctance to even mention this as a possible solution to these problems in popular science magazines and documentaries, despite the idea having been around for half a century or more.

Is this because it is inconsistent with some popular theories, or are there other implications which make it unappealing?

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Ive learned that a fixed smallest possible value of a "spacetime volume" would lead to an issue with Lorentz invariance for example... – Dilaton Sep 11 '11 at 7:39
@Dilaton: not at all --- the existence of a smallest angular momentum in quantum mechanics doesn't cause problems with rotational symmetry. – genneth Sep 11 '11 at 9:09

Everybody knows that space-time has to go, but the reason discrete space-time short-distance grains are not discussed is because the naive idea are all wrong. If you give a heuristic space-time foam, you can get hand-waving justifications for finite theory, but there is no real argument there, because the path-integral for quantum gravity is a mess.

You can try to define the path integral as a sum over different manifolds, but how do you do a sum over manifolds? There are problems of principle, like the fact that reducing the sum over topologies counting each topology only once is known to be computationally undecidable. So you need to cut off the integral somehow at short distances so you can't have arbitrary handles pop up. Then perhaps you can make sense of the integral, but how can you cut off small distances when the distances are dynamical?

The path-integral is a dead end without new ideas. One such idea is loops, and this gives a direct picture of space-time graininess by spin-foams and things like that. But this idea is very hard to complete.

The best way out of the impasse is to change perspective completely. You look at the experimental questions you can ask. If you want to know what is happening in a microscopic region of space, all you can do is shoot particles at it, and get some particles out, in a quantum mechanical way. So you ask what S-matrix is happening there? That's the complete description of space-time.

This question sidesteps the issues completely, because the S-matrix is defined asymptotically on the boundary of the original space, using plane waves at infinity, so it is well defined using only the asymptotic symmetries of the big space. It doesn't make any difference if there is microscopic foam, or a big mess, or whatever, the S-matrix is well defined and stable to quantum gravity. This is why it is a good foundation.

So you look for an S-matrix theory, and this is string theory. Within string theory, you can find formulations in which you can see the space between black holes built up from the oscillations of those black holes. This is very close to a full atomic description of space. The quantities you have are only quantum mechanics in 0 space and 1 time dimension, and they build up a full 11 dimensional reality in the limit of many atoms.

The atomic description in string theory has no simple space-time description, because it is asymptotic, it respects the holographic principle. You can't say that space has been chopped up into volumes, because it is really only the boundary of space that needs chopping up. The interior is just a reconstruction. These ideas supplant more primitive notions of space-time graininess, and really, those old ideas are obsolete.

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Wow, that's quite an interesting and "radical" view of things; I did not know that path integrals are "out" for example ... – Dilaton Sep 12 '11 at 11:35
@Dilaton: quantum gravity path integrals have been "out" for twenty years. Only Hawking still uses them. Unlike field theory path integrals, which make sense after wick rotation and regularization, it is hard to do either in quantum gravity. Maybe they'll come back, but I doubt it. – Ron Maimon Sep 12 '11 at 13:14
Ok @Ron, I kind of like Your answer so Ill give it a point :-). I perfectly understand Your arguments about why the S-matrix remains well behaved even in the context of quantum gravity. But when You say that such an S-matrix theory IS string theory I can not quite follow this (apart from the holography argument)... But maybe its just because Ive not (yet?) read any mildly technical texts (like my Demystified Book) about ST ;-P. – Dilaton Sep 12 '11 at 16:09
One more great answer. Congratulations! – Diego Mazón Nov 29 '12 at 0:19

Apparently a discrete space-time is not a good idea because it does not follow from experiments but serves to shut up crying problems in an artificial way.

Many minimal "distances" may appear quite physically and it does not mean that they are truly minimal. Look, for example, at atom: there is a natural size - Bohr radius $a_0$, but shorter distances are reachable too. Make a zoom in atom: you will see a positive charge cloud of the size $\left(\frac{m_e}{M_A}\right)a_0$ that may be larger than the nucleus proper size (in light atoms and in excited atomic states with $a_n>>a_0$). There is no Coulomb singularity here due to quantum mechanical smearing, but shorter distances are again reachable. You see, the physical characteristic distances do not prevent us from probing shorter ones or forbid them as non reachable.

Similarly with quantum gravity. In a correct theory there will not be singular solutions. But long as there is no experimental evidences and as long as we build our theories "by analogy with self-acting electrodynamics", all QG theories will be highly speculative. Our speculations, due to rush for the ultimate theory of everything, has already gone too far. They dominate our physical picture despite huge conceptual, physical, and mathematical problems.

Introducing fundamental lengths (or cut-offs) is not a new idea and it is insufficient. In renormalizable theories people perform renormalizations and get rid of the cut-off length completely. Factually, cut-off-dependent terms are discarded and the reminder is, as they say, independent on physics of short distances. They call it "universality". It means whatever physics is at short distances, after discarding its contributions the reminder is the same. It's a big science of cheating and self-fooling. In non-renormalizable theories this discarding does not help to carry out calculations but they say anyway: "everything is OK, our theory is very "effective"".

No, we miss something more basic and we need more profound reformulation of our theories than just doctoring wrong answers.

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This looks like a rant. Can you reference a paper to back up this scathing critique? – quant_dev Sep 12 '11 at 10:07
I used to give references to QFT giants but then I would get in response that the giants, the fathers of QFT, did not understand the meaning of this or that. So I appeal to common sense mostly. – Vladimir Kalitvianski Sep 12 '11 at 10:45
I can see that you have formulated your own approach. Did you attempt to publish it in a journal? – quant_dev Sep 12 '11 at 10:53
Yes, I did. Look, for example, at this one: docs.google.com/… – Vladimir Kalitvianski Sep 12 '11 at 10:59