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I have often seeen statements on physics.SE such as,

The only consistent theory of everything which we know of to date (2013) is string theory.

Why exactly is this so? Adding the Loop Quantum Gravity Lagrangian Density (the Einstein-Hilbert-Palatini-Ashtekar lagrangian density) to the Standard Model Lagrnagian Density should be able to describe all the interactions and fermions, in my opinion. Maybe it isn't as elegant as string theory since it doesn't really unify all the forces/interactions and fermions but it is still a complet description, right? Because once the Lagrangian Densities are added, one obtains the following "Complete Lagrangian Density": $${{{\cal L}}_{\operatorname{complete}}} = - \frac{1}{4}{H^{\mu \nu \rho }}{H_{\mu \nu \rho }} + i\hbar {c_0}\bar \psi \not \nabla \psi + {c_0}\bar \psi \phi \psi + \operatorname{h.c.} + {\left\| {\not \nabla \phi } \right\|^2} - U\left( \phi \right){\rm{ }}+\Re \left( {\frac{1}{{4\kappa }}\mbox{}^ \pm\Sigma _{IJ}^\mu {{\rm{ }}^ \pm }F_{IJ}^\mu} \right) $$

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I a not a theorist so can only comment that my impression is that LQG does not respect Lorenz invariance in local spaces. I do not know whether this is evident in the lagrangian you have written. Lets hope somebody in the know will reply to you. Have a look at… – anna v Jun 6 '13 at 4:22
The arguments on either side are just going to be propaganda and cheerleading. Neither LQG nor ST has made a prediction that can be tested with any known or foreseeable technology, which means that neither is actually a theory yet. Since they're not theories of anything, they're not theories of everything. I'm pretty sure it's false that you can add all the particles of the SM into LQG just by adding Lagrangians; AFAIK this is an unsolved problem in LQG. – Ben Crowell Jun 6 '13 at 4:35
Because the "theory" you write down doesn't exist. It's just an incoherent mixing of apples and oranges. You can't construct a theory by simply throwing random pieces of Lagrangians from different theories as if you throw different things to the trash bin. LQG is inconsistent by itself, for millions of reasons, but even if it weren't, it has many properties that make it incompatible with the SM, for example its Lorentz symmetry violation. And even if these incompatible properties weren't there, adding up several disconnected Lagrangians just isn't a unified theory of anything. – Luboš Motl Jun 6 '13 at 4:47
@Luboš that should be an answer – David Z Jun 6 '13 at 6:20
In my opinion a necessary condition for any theory of everything is the demonstrated ability to embed the Standard Model without contradictions.The SM is the mathematical shorthand for thousands of experimental data. That is the first step that a TOE has to fulfill. I think that string theories have many possibilities of embedding the SM . Falsifieable predictions from a TOE would be great, but if it does not even explain the existing data it has lost at the start, imo. – anna v Jun 6 '13 at 6:36
up vote 27 down vote accepted

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.

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Excuse me for commenting on an old answer, but why are you trying to use the Standard Model beyond its domain of validity? If I am not mistaken, renormalizable QFT models (and Standard Model in particular) are considered nowadays infrared approximations of whatever fundamental degrees of freedom there are at the Planck scale (strings, loops, etc). – Solenodon Paradoxus Dec 18 '15 at 8:11
I haven't used the SM beyond its range of validity. Quite on the contrary, my answer was a more detailed version of your point. The Standard Model must be considered just an approximate, effective theory at long distances, and the complete theory is different e.g. because it includes gravity at the Planck scale. But a theory isn't just a collection of ingredients and properties you "demand" to be present in the theory. In particular, there can't be any theory (and there surely isn't any known theory) that would reduce to the SM and loop quantum gravity in the two limits. – Luboš Motl Dec 18 '15 at 15:23
I see. I suppose, I misunderstood you the first time I read your answer. You were writing about the violation of the Lorentz symmetry and how it affects the QFT formalism, but it seems like the domains where this effect can't be considered negligible are far beyond those where renormalizable QFT models are to be trusted. Do you agree with this statement? Anyways, thanks a lot for your time. – Solenodon Paradoxus Dec 19 '15 at 15:41
Dear @Hindsight, if I understand the statement well, I don't agree with it. The violation of the Lorentz symmetry, if it's nonzero and at least slightly natural, just never becomes negligible. A necessary condition for the Lorentz symmetry is that the maximum speed that any particle species (or composite objects) may converge to is the same, we call it the speed of light. If your theory fundamentally violates the Lorentz symmetry, the maximum speeds will differ for particle species and this difference in no way disappears at shorter or longer length scales. – Luboš Motl Dec 20 '15 at 20:14
Is right to thinking that what you are talking about translates to the fact that exist a bunch of marginal and relevant terms that do not obey Lorenzo symmetry?... and this would get a lot of trash in the IR lagrangiana, deviating from the Standar Model. – Nogueira Jan 3 at 23:31

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