The only finite mathematical framework that incorporates both the standard model of particle physics and gravity under one umbrella that I am aware of is string theory. I would like to know whether there are any other mathematical possibilities exist which do not depend on supersymmetry and still consistent with the standard model and gravity and produce finite answers. In a nutshell my question is: can there be any alternative to string theory? (Remember, I am not talking about only gravity. I am talking about gravity as well as other phenomena).
[Physics] Other possible theories (other than string theory) which are generalizations of the standard model with incorporation of gravity
beyond-the-standard-modelquantum-gravitystandard-modelstring-theorytheory-of-everything
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The idea which is being challenged, though certainly not disproved yet, is that there are new particles, other than the Higgs boson, that the LHC will be able to detect. It was very widely supposed that supersymmetric partners of some known particles would show up, because they could stabilize the mass of the Higgs boson.
The simplest framework for this is just to add supersymmetry to the standard model, and so most string models of the real world were built around this "minimal supersymmetric standard model" (MSSM). It's really the particle physicists who will decide whether the MSSM should lose its status as the leading idea for new physics. If they switch to some "new standard model", then the string theorists will switch too.
Whether they are aiming for the SM, the MSSM, or something else, the challenge for string theorists is, first, to find a shape for the extra dimensions which will make the strings behave roughly like the observed particles, and then second, use that model to predict something new. But as things stand, we still only have string models that qualitatively resemble reality.
Here is an example from a year ago - "Heterotic Line Bundle Standard Models". You'll see that the authors talk about constructing "standard models" within string theory. That means that the low-energy states in these string models resemble the particles of the standard model - with the same charges, symmetries, etc.
But that's still just the beginning. Then you have to check for finer details. In this paper they concern themselves with further properties like proton decay, the relative heaviness of the different particle generations, and neutrino masses. That already involves a lot of analysis. The ultimate test would be to calculate the exact masses and couplings predicted by a particular model, but that is still too hard for the current state of theory, and there's still work to do just in converging on a set of models which might be right.
So if supersymmetry doesn't show at the LHC, string theorists would change some of these intermediate criteria by which they judge the plausibility of a model, e.g. if particle physics opinion changed from expecting supersymmetry to show up at LHC energies, to expecting supersymmetry only to show up at the Planck scale. It would mean starting over on certain aspects of these model analyses, because now you have changed the details of your ultimate destination.
Because the "theory" you write down doesn't exist. It's just a logically incoherent mixture of apples and oranges, using a well-known metaphor.
One can't construct a theory by simply throwing random pieces of Lagrangians taken from different theories as if we were throwing different things to the trash bin.
For numerous reasons, loop quantum gravity has problems with consistency (and ability to produce any large, nearly smooth space at all), but even if it implied the semi-realistic picture of gravity we hear in the most favorable appraisals by its champions, it has many properties that make it incompatible with the Standard Model, for example its Lorentz symmetry violation. This is a serious problem because the terms of the Standard Model are those terms that are renormalizable, Lorentz-invariant, and gauge-invariant. The Lorentz breaking imposed upon us by loop quantum gravity would force us to relax the requirement of the Lorentz invariance for the Standard Model terms as well, so we would have to deal with a much broader theory containing many other terms, not just the Lorentz-invariant ones, and it would simply not be the Standard Model anymore (and if would be infinitely underdetermined, too).
And even if these incompatible properties weren't there, adding up several disconnected Lagrangians just isn't a unified theory of anything.
Two paragraphs above, the incompatibility was presented from the Standard Model's viewpoint – the addition of the dynamical geometry described by loop quantum gravity destroys some important properties of the quantum field theory which prevents us from constructing it. But we may also describe the incompatibility from the – far less reliable – viewpoint of loop quantum gravity. In loop quantum gravity, one describes the spacetime geometry in terms of some other variables you wrote down and one may derive that the areas etc. are effectively quantized so the space – geometrical quantities describing it – are "localized" in some regions of the space (the spin network, spin foam, etc.). This really means that the metric tensor that is needed to write the kinetic and other terms in the Standard Model is singular almost everywhere and can't be differentiated. The Standard Model does depend on the continuous character of the spacetime which loop quantum gravity claims to be violated in Nature. So even if we're neutral about the question whether the space is continuous to allow us to talk about all the derivatives etc., it's true that the two frameworks require contradictory answers to this question.
Best Answer
This is the proverbial sixty-four thousand dollar question for fundamental physics. It may be helpful to split it down into steps.
Once we have answered these questions the theoretical program to understand the foundations of physics is essentially complete and the rest is stamp collecting and experiment. That is not going to happen today but let's see where we are.
String Theory works very well as a perturbative theory of gravitons that appears to be finite at all orders, but there is no full proof that it is a complete theory of quantum gravity. It requires matter and gauge fields with supersymmetry to avoid anomalies. The size of gauge groups suggests that it could potentially include the standard model. It is too strong a claim to say that it does incorporate the standard model. A popular view is that it has a vast landscape of solutions which is sufficiently diverse to suggest that the standard model is covered, but crucial elements such as supersymmetry breaking and the cosmological constant problem are not yet resolved.
Supergravity theories are potentially alternative non-string theories that could provide a perturbative theory of quantum gravity. Indications are that they are finite up to about seven loops due to hidden E7 symmetry but they are likely to have problems at higher loops unless there are further hidden symmetries. These theories have multiplets of gauge groups and matter. The 4D theories do not have sufficiently large gauge groups for the standard model but compactified higher dimensional supergravity does. A more subtle problem is to include the right chiral structure and this may be possible only with the methods of M-theory.
It has long been the conventional wisdom that supergravity theories can only be made complete by adding strings. Recent work using twistor methods on 4D supergravity seems to support this idea (e.g. Skinner etc.)
Loop Quantum Gravity is an attempt to quantise gravity using the canonical formualism and it leads to a description of quantum gravity in terms of loops and spin network states which evolve in time. Although this is regarded as an alternative to string theory and supergravity it does not give a picture of a purtabative limit which would make it possible to compare with these approaches. It is possible that ST/SUGRA and LQG are looking at similar things from a different angle. In fact the recent progress on N=8 supergravity as a twistor string theory has some features that are similar to LQG. Both involve 2D worldsheet objects and network like objects.
The main distinctions are that LQG does not have supersymmetry and N=8 SUGRA does not use knots. Even then there has been some progress on a supersymmetric version of LQG and the Yangian symmetries used in N=8 SUGRA should be amenable to a q-deformation that brings in knots. It remains to be seen if these theories can be unified.
It is worth saying that all these approaches involve trying to quantise gravity in different ways. Although quantisation is not a completely unique procedure it is normal to expect that different ways of quantising the same thing should lead to related results, If something like supersymmetry or strings or knots are needed to get consistency in one approach the chances are that they will be needed in another.
I have not mentioned other approaches to quantum gravity such as spin foams, group field theory, random graphs, causal sets, shape dynamics, non-commutative geometry, ultra-violet fixed points etc. Some of these are related to the other main approaches but are less well developed. It should also be mentioned that there are always attempts to unify gravity and the standard model classically e.g. Garrett's E8 TOE, Weinstein's Geometric Unity etc. These may tell us something interesting or not, but it is only when you try to quantise gravity that strong constraints apply so there is no reason to think they should be related to the attempts to quantise gravity.
So in conclusion all approaches that have made any kind if real progress with quantising gravity look like they may be related. Much more has been revealed so far from this need to quantise consistently than from directly trying to unify gravity with the standard model. This may not be so surprising when you consider the enormous difference in energy scales between the two.