Question

The one difficulty I see with LQG is that it requires an enormous number of degrees of freedom, e.g. these spin variables in the net. This is in contrast to stringy holographic theory where the fields in a space are equivalent to fields on a boundary or a horizon of one dimension lower. In this setting entropy of a black hole is the entanglement entropy of states interior and exterior to the black hole. This reduces the amount of data, and thus entropy, required.

Are there suggestions, conjectures or maybe serious theory which attempts to describe the spin variables of LQG according to such entanglements in string-brane theory?

Answer 1

Dear , an equivalence between LQG and string theory - or an LQG-like description of string theory physics - has surely been an attractive idea for many physicists (myself included) but it is impossible because of fundamental differences in virtually all general features and predictions of both frameworks.

As you correctly mentioned, the counting of the degrees of freedom disagrees. String theory respects the holographic principle. It also means that the entropy within a volume is always bounded by the surface in Planck units. On the other hand, LQG admits an arbitrary, volume-extensive, Planckian entropy density - in fact, it predicts a Planckian entropy density of the vacuum (the information about the details of the spin network). For a related class of examples, LQG always predicts a volume-extensive term in the black hole entropy, too. It can only be "cured away" by erasing it (together with the whole black hole interior) by hand and pretending it was never there. It's important to mention that the infinite multiplicity of "string fields" is just an artifact of a formalism - string field theory. One can't add an arbitrary number of excitations of all these kinds into a finite volume (because they would gravitationally collapse). The only truly "physically invariant" measure of the number of degrees of freedom boils down to entropy and ST - as a holographic theory - predicts a much smaller entropy than non-holographic theories such as LQG. In particular, string theory vacua are unique and carry no entropy density.

LQG breaks the local Lorentz symmetry while string theory exactly preserves is. Because the Fermi satellite has showed that there is no Lorentz violation at the Planck scale, LQG was falsified. (It was falsified in many other ways, too.) String theory remains compatible with the observations. The preservation of Lorentz symmetry in string theory may be seen e.g. perturbatively, by considering strings propagating on a target spacetime. The SO(d?1,1) symmetry of the spacetime directly arises from the SO(d?1,1) global symmetry rotating fields (representing spacetime coordinates) on the world sheet. The violation of Lorentz symmetry in LQG may be seen from the fact that the hypothetical solution - a spin network - picks a privileged reference frame, analogous to the luminiferous aether. In this frame, the entropy density is huge, essentially Planckian, and in all other reference frames, there would be a huge entropy flow in a direction, which would break the rotational symmetry. In the preferred frame, the motion of all objects instantly stops as their kinetic energy is dissipated to the thermal energy of the spin network which is a de facto infinite heat bath.

String theory implies that the space is smooth and almost flat at long distances. Although there is no fully-general "no-go theorem", all partial models and circumstantial evidence suggest that LQG in any form can never predict a smooth space at long distances. It gets crumbled. For this reason, it doesn't even make sense to ask whether LQG reproduces Einstein's equations at long distances - there are no long distances in LQG.

LQG implies that there can't be any forces and elementary particles aside from gravity. String theory predicts that gravity has to exist, much like non-gravitational forces and particle species. The absence of other forces in LQG isn't a cosmetic problem that can be fixed. The strength of other forces doesn't go to zero, not even at the Planck scale. In fact, there exist general arguments that gravity has to be the weakest force - just like it is in the real world - so a valid microscopic description can never start by neglecting the non-gravitational forces because it is really gravity that is a correction, not the other way around. String theory predicts the right "draft" of the world with spin-1/2 fermions, spin-1 gauge bosons, potential for gauge anomalies and a nontrivial anomaly cancellation, chiral fermions and chiral interactions, Higgs bosons, Higgs mechanism, confinement of non-Abelian gauge fields, running couplings and other phenomena related to the renormalization group, and so on, while LQG has nothing whatsoever to do with particle physics and is incompatible pretty much with all the basic concepts of particle physics I enumerated. The contrast becomes even stronger if we realize that string theory has also led to (or at least inspired) some of the most novel, explanatory, and important models of beyond-the-standard-model phenomenology such as supersymmetry, models with extra dimensions, deconstructions, and others that are currently studied by a big portion of phenomenologists, even those who don't consider themselves string theorists in any sense.

String theory includes dualities. They're transformations that totally rearrange the degrees of freedom and change their interpretation. Quantum mechanics is totally crucial for those S-dualities, T-dualities, U-dualities, holographic dualities, and other dualities to work. On the other hand, LQG doesn't imply any dualities. More generally, it doesn't employ quantum mechanics in any deep way. It is just a variation of the ancient Greek models of atoms whose properties are promoted to operators - but this promotion never leads to anything interesting.

LQG doesn't admit supersymmetry, wants to avoid extra dimensions, strings, extended objects, etc. So LQG is unlikely to be a dual description of any aspect of string theory. It's been established in string theory that supersymmetry is an omnipresent, fundamental symmetry that has to appear in all semi-realistic models at some scale. Extra dimensions are needed for consistency. On the other hand, LQG starts by assuming that none of these things exist, and even though the appearance of extended objects etc. is generic in consistent field theories and vacua of string theory, the LQG research is based on the assumption that they must be avoided. This leads me to a much general point.

String theory is a natural theory based on objectively important mathematical structures and relationships. Physicists are discovering these features, much like Columbus was discovering America. They are learning new things - and they are identifying the previous errors in their reasoning. On the other hand, LQG is a man-made theory. It is being invented in a similar way as Edison was inventing the light bulb. Preconceptions are what ultimately decides about the shape of the theory. LQG is being constructed step by step. That's why one can never make any solid statements about anything - and one can make no statements that he wouldn't believe at the beginning. This strikingly differs from string theory that implies unique answers to many fundamental questions. For example, it implies that the equivalence principle, local Lorentz symmetry, and constancy of the universal constants have to hold in general. All these issues are permanently open in LQG because someone may always modify the theory in a different way tomorrow. One may learn new conceptual insights about physics - and mathematics - from string theory. That's different from LQG that is designed to depend on no nontrivial mathematics that would be difficult for average undergraduate students. Consequently, one can never learn anything about physics, space, time, or mathematics from LQG. The whole enterprise is meant to find justifications for a predetermined opinion that quantum gravity can be approached in this simple-minded way - a strategy that is not unsimilar to scientists proving the Intelligent Design or geocentrism. So far, however, no justifications have been found.

The multiplicities of possibilities how the vacuum may look like according to string theory boil down to solutions of objective equations that we pretty much understand: the rules of the game are constant. That's very different from LQG where new models are created at any point by arbitrarily changing the rules of the game. That's related to the previous point that string theory makes some general predictions, even when the right "vacuum" is unknown. LQG can never make any predictions of this kind.

Information is lost in LQG. Indeed, it is a local theory of a very naive type so even if space and black hole were possible in LQG, one could show that the assumptions of Hawking's original argument are satisfied which implies that the information cannot get out of the black hole for causal reasons even if the black hole could evaporate (which is established in string theory but surely not in LQG). On the other hand, string theory implies that there exist subtle nonlocalities in the bulk spacetime that imply that the information gets out. This answer is known to be valid because there often exist dual descriptions of the stringy physics where unitarity is manifest.

LQG tries to use ill-defined observables and ignore the well-defined ones. In particular, the area of a surface isn't well-defined at the Planckian accuracy in a theory where measurements cannot measure distances shorter than the Planck scale. This is why all statements of LQG about the "quantization of areas" cannot be operationally or otherwise defined. On the other hand, string theory implies that areas of small surfaces are not well-defined observables and automatically leads us to the physically meaningful observables such as the scattering amplitudes for gravitons - which can't be calculated in LQG. Scattering amplitudes may be measured experimentally and they satisfy important theoretical constraints such as unitarity, too: they're the right way to parameterize "all predictions" of a meaningful relativistic quantum theory. Quite generally, string theory automatically addresses quantities that are important in high-energy physics while LQG is disconnected from all the 20th century physics and its key concepts.

To summarize, the differences between the technical properties of the two frameworks as well as very philosophy what it means to do good science are completely insurmountable.