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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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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). $\endgroup$ Dec 18, 2015 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. $\endgroup$ Dec 18, 2015 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. $\endgroup$ Dec 19, 2015 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. $\endgroup$ Dec 20, 2015 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, 2016 at 23:31
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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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  • $\begingroup$ Just a nitpick, but should this be Ashtekar et al. or Ashtekar and colleagues if other people were involved in the work? $\endgroup$
    – Tom
    Dec 6, 2021 at 20:04
  • $\begingroup$ Also, one thing I have wondered for a while, is there a reason that CQG often favours articles on loop quantum gravity as a possible quantum gravity approach, whereas JHEP tends to favour articles on string theory? Is this some cultural difference, or is it due mainly to the preferences of the editors of those journals? $\endgroup$
    – Tom
    Dec 6, 2021 at 22:07
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    $\begingroup$ I think this answer is backwards. We shouldn't be judging the legitimacy of the heuristic procedure of quantization, but rather we should be judging the resulting quantum theory (its ability to reproduce GR in the classical limit being one of the key desired properties). Afaik canonical LQG fails this test, spinfoam LQG passes at least the classical limit test. $\endgroup$ Dec 27, 2021 at 18:08
  • $\begingroup$ @Urs Schreiber, what a point, why is smoothness dropped? smoothness is a nice property, why LQG drop it? For what? $\endgroup$
    – Arian
    Feb 12 at 22:29
  • $\begingroup$ @Urs Schreiber As I understand it, in covariant LQG we choose a triangulation of space with $L$ links and $N$ nodes, and get the (separable) spin-network Hilbert space $L^2(SU(2)^L/SU(2)^N)$. Is your answer claiming that something is lost in this discretisation, since we throw out the smooth structure of the manifold we started with? Why is the situation different to lattice QCD, which seems to apply essentially the same quantisation procedure to holonomies on a lattice? $\endgroup$ Apr 10 at 12:04
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This "Complete Lagrangian Density" might actually be the long after sought Theory of Everything! This theory you described, is a Quantum Field Theory on a curved spacetime, which obeys the known laws of physics.

To answer your first question, "Why exactly does physics.SE think that the only consistent theory of everything which we know of to date is string theory?" There are only two possible answers. Either string theory is right, and it is the theory of everything. Or string theory is wrong, and some other theory is the theory of everything. String Theory, uses a gauge group that incorporates SU(5) symmetry on a flat background with global O(3,1) symmetry. Internal SU(5) symmetry predicts proton decay, and we have not ever detected a single proton decay. All you need to prove or disprove proton decay is to get many tons of liquid hydrogen, and wait for a single proton to turn into a positron which annihilates the bound electron. This has never happened, which means that string theory is wrong. Other experiments have also disproved string theory, such as supersymmetry and extra dimensions. If you want to check out the Grand Unified Theory, below is a short lagragian, which predicts the incorrect value of the proton decay rate. So to answer your question, string theorists still believe in their theories, after they have already proven wrong. The point is to reject a theory that fails the experiment. Not only have string theorists done that, but they keep pushing the theory to it's limits and say that supersymmetry and extra dimensions, exist but are invisible, and proton decay happens too slowly to be detected. The reason why string theorists continue to do so, is because they find their theory elegant and beautiful. When trying to find a theory that explains reality, it is a much better idea to merge the existing conflicting theories into one theory. Sabine Hossenfelder goes into much more detail than I do on this question, so be sure to check out her video explaining how to test a theory of everything.

To answer you second question, "is LQG-SM still a complete description of nature". I would arrogantly answer "Yes Absolutely". Einstein merged his own special relativity and newtons universal gravitation to produce the general theory of relativity. Many of it's predictions were verified, such as the perihelion shift of planet mercury, and the bending of light around the sun. Later Dirac merged special relativity and non relativistic quantum mechanics, to produce the Dirac equation, which predicted antimatter. When Dirac stated that his theory is beautiful, what he meant was not that he just came up with it for artistic purposes, but that it explains all the symmetries of special relativity and quantum mechanics. Today, we know of four symmetries, which correspond to the four fundamental forces. O(3,1) symmetry for general relativity, and the SU(3) SU(2) U(1) symmetries of the standard model. Loop Quantum Gravity is basically the Canonical Quantization of General Relativity. It merges the physics of today, just like how Einstein and Dirac merged the physics in their day. Unlike the bad reputation of LQG, it does actually correspond to general relativity in the classical limit. Also the problem of time, has been completely solved in recent versions of LQG. The proof is given by equation(8.109) on page 191, of the LQG text. Since Loop Quantum Gravity merges universal gravitation, special relativity, and quantum mechanics, means that it will most likely pass all of the experimental tests, just like general relativity and the Dirac equation. LQG is the quantum field theory of gravity and has a gauge symmetry of O(3,1), which is the same global symmetry of special relativity, and also explains the properties of elementary particles such as spin. In short Loop Quantum Gravity is a complete theory of gravity, and when merged with the Standard Model of Particle Physics, will give a complete theory of the universe. We just have to wait for it to be confirmed like all the other theories that have been confirmed.

Sabine Hossenfelder explains, how can we test a Theory of Everything: https://www.youtube.com/watch?v=aUj6vEQkHt8&t=234s

Why Grand Unification does not work: https://einstein-schrodinger.com/Minimal_SU(5)_GUT.pdf

Covariant Loop Quantum Gravity Textbook: http://www.cpt.univ-mrs.fr/~rovelli/IntroductionLQG.pdf

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