Classical mechanics is a good approximation to special relativity, which is a good approximation to general relativity etc. I have heard that if string theory/M-theory is right, then it is not just an approximation to a more accurate theory, but represents the end of the line as a TOE, why is this so? Can this be proved in string theory/M-theory?
[Physics] What comes after string theory/M-theory
string-theorytheory-of-everything
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Notice:
Perturbative string theory is defined to be the asymptotic perturbation series which are obtained by summing correlators/n-point functions of a 2d superconformal field theory of central charge -15 over all genera and moduli of (punctured) Riemann surfaces.
Perturbative quantum field theory is defined to be the asymptotic perturbation series which are obtained by applying the Feynman rules to a local Lagrangian -- which equivalently, by worldline formalism, means: obtained by summing the correlators/n-point functions of 1d field theories (of particles) over all loop orders of Feynman graphs.
So the two are different. But for any perturbation series one can ask if there is a local non-renormlizable Lagrangian such that its Feynman-rules reproduce the given perturbation series at sufficiently low energy. If so, one says this Lagrangian is the effective field theory of the theory defined by the original perturbation series (which, if renormalized, is conversely then a "UV-completion" of the given effective field theory).
Now one can ask which effective quantum field theories arise this way as approximations to string perturbation series. It turns out that only rather special ones do. For instance those that arise all look like anomaly-free Einstein-Yang-Mills-Dirac theory (consistent quantum gravity plus gauge fields plus minimally-coupled fermions). Not like $\phi^4$, not like the Ising model, etc.
(Sometimes these days it is forgotten that QFT is much more general than the gauge theory plus gravity plus fermions that is seen in what is just the standard model. QFT alone has no reason to single out gauge theories coupled to gravity and spinors in the vast space of all possible anomaly-free local Lagrangians.)
On the other hand now, within the restricted area of Einstein-Yang-Mills-Dirac type theories, it currently seems that by choosing suitable worldsheet CFTs one can obtain a large portion of the possible flavors of these theories in the low energy effective approximation. Lots of kinds of gauge groups, lots of kinds of particle content, lots of kinds of couplings. There are still constraints as to which such QFTs are effective QFTs of a string perturbation series, but they are not well understood. (Sometimes people forget what it takes to define a full 2d CFT. It's more than just conformal invariance and modular invariance, and even that is often just checked in low order in those "landscape" surveys.) In any case, one can come up with heuristic arguments that exclude some Einstein-Yang-Mills-Dirac theories as possible candidates for low energy effective quantum field theories approximating a string perturbation series. The space of them has been given a name (before really being understood, in good tradition...) and that name is, for better or worse, the "Swampland".
For this text with more cross-links, see here:
http://ncatlab.org/nlab/show/string+theory+FAQ#RelationshipBetweenQuantumFieldTheoryAndStringTheory
As with many discussions about string theory, it is sometimes good to recall some reality:
It was over 50 years ago that the Higgs mechanism was proposed. Compared to fully-fledged theories such as string theory, the Higgs mechanism is a tiny add-on to the observed standard model (as it was then). It took 50 years for experiment to get to the point of seeing it, and in fact so far just a first glimpse of it. For 50 years, the Higgs mechanism was speculation not confirmed by experiment. It had all theoretical backing behind it, theory all pointed to it being true, but it couldn't be checked experimentally for 50 long years. For 50 years, you were free to make TV documentaries about particle physics without mentioning the Higgs mechanism, if you thought it was too outlandish a proposal to have a chance of being confirmed. Then finally experiment reached its energy scale and there it was. 50 years later.
As you all know, there are scenarios thinkable where more beyond-the-standard-model-physics is right around the corner, but nothing to rule out that it takes another 50 years to see the next piece of "new physics". That's just a fact of our short life.
But that's not necessarily as bad as it may sound. While "new physics" may remain specuative for a long time to come, here is a well-kept secret to take note of: even old physics isn't fully understood yet. And string theory can help here, and theorists know (though TV stations may not yet have gotten the message).
For instance, computation of scattering amplitudes even in the known and confirmed standard model is still a challenge, if only you are ambitious enough. String theory has helped with understanding some subtle points in plain Yang-Mills perturbation theory. See the links at string theory applied elsewhere -- QCD scattering amplitudes. In particular check out the remarkable story linked to there, told by Matthew Strassler in his post From string theory to the large hadron collider, which is about how string theory insights into QCD scattering amplitudes helped raise the precision of loop computations to the level that it was possible in the first place to separate signal from background in the LHC. He cites people who were involved as saying that without these string theory insights the Higgs might have been produced, but not identified at the LHC. Have a look, it's an interesting story.
Another thing may be worthwhile to remember from time to time: while we are all fond of proclaiming that we understand fundamental particle physics via quantum Yang-Mills theory, fact is that quantum Yang-Mills theory is still an open theoretical problem. We know that we don't understand some very fundamental facts about qauntum Yang-Mills. It's a "Millennium problem" Yang-Mills existence and the mass gap.
Now, one thing that string theory has become after its "second revolution" is something like a map of the space of Yang-Mills like-field theories and various "dual" theories. Via D-brane physics, KK-reduction, AdS/CFT, etc. Yang-Mills like theories appear in various guises in various corners of string theory, and their embedding into string theory geometrically explains subtle equivalences between these, such as electric/magnetic duality, etc. If you haven't seen it before, check out at http://ncatlab.org/nlab/show/gauge+theory+from+AdS-CFT+--+table at least part of this string-theoretic "map" of the space of quantum field theories related to Yang-Mills theory. While this hasn't solved the mass gap problem yet, clearly, one may start to feel that the deeper nature of Yang-Mills theory is slowly but surely being probed here.
The punchline here is the following: besides being a framework for models of quantum gravity and gauge unification, string theory is a piece of theoretical physics that sheds light on the nature of quantum field theory as such. While experimentalists and public media are busy with indulging in the Higgs physics now that they waited for half a century, maybe theoreticians can use the time before the next accelerator to step back and think a bit more about the still open more fundamental issues of quantum physics. That's where string theory has already helped, and I think will help in the future. Of course you won't see this on public TV.
(Generally, it is surprising these days how not only the public media but also the broad community's attention is consistently attracted to the shallow and ignoring the deep advances that do happen in fundamental physics. For instance there is loads of excitement about, say, the firewall essay contest, but the really interesting advances, such as for instance in genuine mathematical characterization of string theory vacua here remains a topic among a tiny group of specialists. At the same time everybody has an opinion about the "landscape", and everbody else has the opposite opinion. What is needed instead is more decent theoretical work on the foundations of quantum field theory and, inevitably then, string theory.)
Best Answer
I would say that the questions that String/M-theory try to answer are the last ones that our current knowledge of reality allows us to ask.
One may think they are the last ones because they are already indirect, as no obvious experimental fact contradicts the current theories (General relativity and the Standard model of particle physics). Instead of experimental problems, M/String theory addresses the theoretical inconstencies between those theories. This could be seen as far-fetched (not to me), but for sure it will be difficult to imagine more questions afterwards.
Now, one should be careful with the idea of theory of everything for two reasons.
M/string theory progresses, but has not reached yet the point of predicting new facts to allow testing it. It is more in the state of 'consistent set of ideas' than in the state of a complete theory. It might occur that to produce predictions, it has to lower its ambitions to "theory of almost everything".
Past history tells that it already occurred that the Physics community thought collectively that "all was known (but a couple of details)" and actually the "details" led to complete changes (e.g. the creation of quantum mechanics & relativity).
Time will tell!