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  Why is Standard Model + Loop Quantum Gravity usually not listed as a theory of everything

+ 9 like - 0 dislike

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

asked Jun 6, 2013 in Theoretical Physics by dimension10 (1,985 points) [ revision history ]
edited Apr 25, 2014 by dimension10
Most voted comments show all comments
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.

This post imported from StackExchange Physics at 2014-03-07 13:39 (UCT), posted by SE-user anna v
Possible duplicate: physics.stackexchange.com/q/4340/2451 Related: physics.stackexchange.com/q/3967/2451

This post imported from StackExchange Physics at 2014-03-07 13:39 (UCT), posted by SE-user Qmechanic
Related question physics.stackexchange.com/q/55358

This post imported from StackExchange Physics at 2014-03-07 13:39 (UCT), posted by SE-user twistor59
I personally would like to know why string theory counts, not why other theories don't.

This post imported from StackExchange Physics at 2014-03-07 13:39 (UCT), posted by SE-user MBN
@Qmechanic and closevoters: I think the as duplicate suggested question is much broader, and that one very specific so they are in my opinion not the same. In addition, such a specific theoretical question should be allowed here, even if the answer to the question might be a nicely explained why from a physics point of view negative one and such a composite Lagrangian does not work. This is no reason to close the question.

This post imported from StackExchange Physics at 2014-03-07 13:39 (UCT), posted by SE-user Dilaton
Most recent comments show all comments
@Lubos Motl: Thanks.

This post imported from StackExchange Physics at 2014-03-07 13:39 (UCT), posted by SE-user Dimensio1n0
@Luboš that should be an answer

This post imported from StackExchange Physics at 2014-03-07 13:39 (UCT), posted by SE-user David Z

2 Answers

+ 9 like - 0 dislike

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.

This post imported from StackExchange Physics at 2014-03-07 13:39 (UCT), posted by SE-user Luboš Motl
answered Jun 6, 2013 by Luboš Motl (10,278 points) [ no revision ]
+ 4 like - 0 dislike

One can pinpoint the technical error in LQG explicity:

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/abs/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" https://arxiv.org/abs/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.

answered Sep 29, 2017 by Urs Schreiber (6,095 points) [ no revision ]

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