Is M-theory (a) self-consistent (b) externally consistent?

[nomadreid]

TL;DR

Although there is as of yet insufficient experimental backing for M-theory, aka string theory, is it at least as a mathematical system relatively consistent, and does it contradict any known experimental results?

I do not pretend to understand M-theory or even any of the string theories which make it up, so the answers to these questions do not need to get too much into detail. The two (related) questions:

(a) Forget for a moment that this is supposed to be describing our known universe. Is there a universe that it could describe? In mathematical terms, does this theory have a model, that is, is one pretty confident that it is free of self-contradiction (that is, assuming the consistency of, say, Peano arithmetic, or ZFC+ there exists a measurable cardinal if you wish)?

(b) Coming back to our known universe, are there any experimental results which contradict a prediction of M-theory (if we ignore any positive answer to (a)).

The two questions together ask: although M-theory is not yet experimentally verified, and possibly is not even possible to be experimentally verified, is there anything else that would disqualify it, in its present state, from being a theory of quantum gravity?

 

[haushofer]

How about knowing what M-theory is in the first place? :P

 

[nomadreid]

Yes, obviously it would be ideal for me to understand M-theory completely, in which case I would not need to pose this question. But as I stated in my question, I do not, only the kind of vague ideas that one gets from popular explanations (say, Wikipedia or Greene). I am not asking for anyone to make me into a string theorist, but rather appealing to physics experts as I might appeal to a doctor to tell me whether a disease is fatal in most cases, without myself knowing much about medicine.

 

[julian]

Yes, obviously it would be ideal for me to understand M-theory completely, in which case I would not need to pose this question. But as I stated in my question, I do not, only the kind of vague ideas that one gets from popular explanations (say, Wikipedia or Greene). I am not asking for anyone to make me into a string theorist, but rather appealing to physics experts as I might appeal to a doctor to tell me whether a disease is fatal in most cases, without myself knowing much about medicine.

I think @haushofer is saying that nobody knows what M-theory is! M-theory is supposed to unify all consistent versions of string theory but at the present moment M-theory is ONLY conjectured to exist - nobody knows how to formulate it, not even Witten. The "M" is often said to stand for mystery - in that nobody knows how to formulate it or even if it actually exists!

 

[nomadreid]

Ah, thanks, @julian. I had indeed misunderstood @haushofer's remark. So, your clarification was interesting in itself, but let me then modify my question in light of your answer. You mentioned "consistent versions of string theory" -- presuming "consistent" here is a combination of (a) and (b) together (not self-contradictory and not yet contradicted by experiment), is it known if any presently understood string theories fit this description, and if so, which ones?

My question might seem silly, in that it might be supposed that a theory that was not consistent might not even be considered, but there have been theories in the past that have been provisionally used because they are useful in areas where the contradictions don't appear, in the hope that the contradictions will be cleaned up later.

Or maybe my question really is silly. It is not meant to be, but I would accept that criticism in good grace.

 

[haushofer]

Yes, obviously it would be ideal for me to understand M-theory completely, in which case I would not need to pose this question. But as I stated in my question, I do not, only the kind of vague ideas that one gets from popular explanations (say, Wikipedia or Greene). I am not asking for anyone to make me into a string theorist, but rather appealing to physics experts as I might appeal to a doctor to tell me whether a disease is fatal in most cases, without myself knowing much about medicine.

Let me be a bit more elaborate. At a certain point we had 5 different kinds of string theory. They looked all quite different, e.g. in the amount of supersymmetry and the particle content, but they are all allowed by the rules we think we understand now. But then people like Witten, Townsend and others discovered that these string theories are related by dualities, a key term here.

Sometimes differently-looking theories are secretly the "same", in the sense that there is a one-to-one mapping. Sometimes this mapping is a bit trivial. Take e.g. the fact that the equations of certain electronic circuits are the same as the equations governing free fall in space. But sometimes these dualities are much more useful, when they tend to map situations in one theory which are easily calculated to situations in the dual theory which (in that dual theory) is horribly complicated.

So people discovered all kinds of dualities between the different string theories, suggesting that maybe we don't have a complete picture yet. The hypothesis now is that for string theory, this "complete picture" is given by M-theory. But we only know what this M-theory reduces to in certain limits. In the point-like limit M-theory reduces to a theory called maximal supergravity, in 11 dimensions. That's more or less all we (at least, I) know.

This supergravity theory is related to M-theory like Newtonian gravity is related to general relativity; Newtonian gravity is a little "corner" of general relativity in a certain limit where a lot of phenomena are thrown away (bending of light, black holes, gravitational waves, etc). Just as we cannot find general relativity uniquely from Newtonian gravity, we don't know the rules how to find the completion of the supergravity theory called M-theory.

To be concrete about your questions:

a) We expect M-theory to be consistent, but how it relates to purely mathematical theorems like the existence of Peano arithmetic I don't know. I regard math as a language, and not related to the consistency of physical theories which can describe humans who can invent math. But maybe I'm overlooking something.

b) About predictions: you really should look then at that maximal supergravity theory in 11 dimensions. Well, this theory lives in 11 dimensions, so we have to compactify it to our 4 known dimensions. This can be done in a lot of different ways, which makes it (as we know understand it) impossible to falsify.

Hope this helps.

 

[haushofer]

You mentioned "consistent versions of string theory" -- presuming "consistent" here is a combination of (a) and (b) together (not self-contradictory and not yet contradicted by experiment).

Well, no. With "consistency" string theorists often mean in this context a mathematical consistent theory after applying the Rules of Quantum Mechanics. You see, a quantum mechanical string is very picky about certain aspects of the theory. It is by far not trivial that even such a theory could exist. When it turned out it does, this was huge news. Especially because it contained all kinds of ingredients we need to describe the standard model (which describes fundamental physics, which describes chemistry, which describes whether life can form, especially in the form of humans inventing certain kinds of math).

 

[nomadreid]

Sorry for the delayed reply, what with the end-of-the-year-and-beginning -of-a-new-one shenanigans. Thanks, haushofer, I had read about these dualities (although I am not sure whether they are true isomorphisms), so that if one of the string theories is consistent, then the others should be, right?

What I do not understand is your "Well, no" to my question about consistency, because you then follow it up with telling me that the theory is mathematically consistent, and not mentioning any observation that contradicts it (which is different from having an observation that confirms it, of course), which would seem like a "yes" to the question asked. So I think there is still something I do not understand here.

 

[haushofer]

Yes, sorry, I read your post as if String Theory would say anything sensible about Peano arithmetic, but I now see you put it under the assumptions. My bad. So, indeed, it's a "well, yes". Happy 2020 ;)