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": $${{{\cal L}}_{\operatorname{complete}}} = - \frac{1}{4}{H^{\mu \nu \rho }}{H_{\mu \nu \rho }} + i\hbar {c_0}\bar \psi \not \nabla \psi + {c_0}\bar \psi \phi \psi + \operatorname{h.c.} + {\left\| {\not \nabla \phi } \right\|^2} - U\left( \phi \right){\rm{ }}+\Re \left( {\frac{1}{{4\kappa }}\mbox{}^ \pm\Sigma _{IJ}^\mu {{\rm{ }}^ \pm }F_{IJ}^\mu} \right) $$


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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    $\begingroup$ 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). $\endgroup$ – Prof. Legolasov Dec 18 '15 at 8:11
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    $\begingroup$ 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. $\endgroup$ – Luboš Motl Dec 18 '15 at 15:23
  • $\begingroup$ 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. $\endgroup$ – Prof. Legolasov Dec 19 '15 at 15:41
  • $\begingroup$ 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. $\endgroup$ – Luboš Motl Dec 20 '15 at 20:14
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    $\begingroup$ 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. $\endgroup$ – Nogueira Jan 3 '16 at 23:31

One can pinpoint the technical error in LQG explicitly:

To recall, the starting point of LQG is to encode the Riemannian metric in terms of the parallel transport of the affine connection that it induces. This parallel transport is an assignment to each smooth curve in the manifold between points $x$ and $y$ of a linear isomorphism $T_x X \to T_y Y$ between the tangent spaces over these points.

This assignment is itself smooth, as a function on the smooth space of smooth curves, suitably defined. Moreover, it satisfies the evident functoriality conditions, in that it respects composition of paths and identity paths.

It is a theorem that smooth (affine) connections on smooth manifolds are indeed equivalent to such smooth functorial assignments of parallel transport isomorphisms to smooth curves. This theorem goes back to Barrett, who considered it for the case that all paths are taken to be loops. For the general case it is discussed in arxiv.org/0705.0452, following suggestion by John Baez.

So far so good. The idea of LQG is now to use this equivalence to equivalently regard the configuration space of gravity as a space of parallell transport/holonomy assignments to paths (in particular loops, whence the name "LQG").

But now in the next step in LQG, the smoothness condition on these parallel transport assignments is dropped. Instead, what is considered are general functions from paths to group elements, which are not required to be smooth or even to be continuous, hence plain set-theoretic functions. In the LQG literature these assignments are then called "generalized connections". It is the space of these "generalized connections" which is then being quantized.

The trouble is that there is no relation left between "generalized connections" and the actual (smooth) affine connections of Riemanniann geometry. The passage from smooth to "generalized connections" is an ad hoc step that is not justified by any established rule of quantization. It effectively changes the nature of the system that is being quantized.

Removing the smoothness and even the continuity condition on the assignment of parallel transport to paths loses all contact with how the points in the original spacetime manifold "cohere", as it were, smoothly or even continuously. The passage to "generalized connections" amounts to regarding spacetime as just a dust of disconnected points.

Much of the apparent discretization that is subsequently found in the LQG quantization is but an artifact of this dustification. Since it is unclear what (and implausible that) the generalized connections have to do with actual Riemannian geometry, it is of little surprise that a key problem that LQG faces is to recover smooth spacetime geometry in some limit in the resulting quantization. This is due to the dustification of spacetime that happened even before quantization is applied.

When we were discussing this problem a few years back, conciousness in the LQG community grew that the step to "generalized connections" is far from being part of a "conservative quantization" as it used to be advertized. As a result, some members of the community started to investigate the result of applying similar non-standard steps to the quantization of very simple physical systems, for which the correct quantization is well understood. For instance when applied to the free particle, one obtains the same non-separable Hilbert spaces that also appear in LQG, and which are not part of any (other) quantization scheme. Ashtekar tried to make sense of this in terms of a concept he called "shadow states" arXiv:gr-qc/0207106. But the examples considered only seemed to show how very different this shadowy world is from anything ever seen elsewhere.

Some authors argued that it is all right to radically change the rules of quantization when it comes to gravity, since after all gravity is special. That may be true. But what is troubling is that there is little to no motivation for the non-standard step from actual connections to "generalized connections" beyond the fact that it admits a naive quantization.

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  • $\begingroup$ @ Urs Schreiber, nice answer, +1, and thanks for giving Refs $\endgroup$ – wonderich Sep 29 '17 at 15:41

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