3 History of Loop Quantum 2 Quantum Gravity: Where are 2.2 What is the problem?

2.3 Strings or loops? 

Above I have emphasized the radically distinct cultural paths leading to string theory and loop quantum gravity. Here I attempt to compare the actual achievements the two theories have obtained so far regarding the description of Planck scale physics.

Once more, however, I want to emphasize that, whatever prejudices this or that physicist may have, both theories are tentative : As far as we really know, either, or both, theories could very well turn out to be physically entirely wrong. And I do not mean that they could be superseded: I mean that all their specific predictions could be disproved by experiments. Nature does not always share our aesthetic judgments, and the history of theoretical physics is full of enthusiasm for strange theories turned into disappointment. The arbiters in science are experiments, and not a single experimental result supports, not even very indirectly, any of the current theories that go beyond the Standard Model and general relativity . To the contrary, all the predictions made so far by theories that go beyond the Standard Model and general relativity (proton decay, supersymmetric particles, exotic particles, solar system dynamics) have for the moment been punctually falsified by experiments. Comparing this situation with the astonishing experimental success of the Standard Model and classical general relativity should make us very cautious, I believe. Lacking experiments, theories can only be compared on completeness and aesthetic criteria - criteria, one should not forget, that according to many favored Ptolemy over Copernicus at some point.

The main merits of string theory are that it provides a superbly elegant unification of known fundamental physics, and that it has a well defined perturbation expansion, finite order by order. Its main incompletenesses are that its non-perturbative regime is poorly understood, and that we do not have a background-independent formulation of the theory. In a sense, we do not really know what the theory we are talking about is. Because of this poor understanding of the non perturbative regime of the theory, Planck scale physics and genuine quantum gravitational phenomena are not easily controlled: Except for a few computations, there has not been much Planck scale physics derived from string theory so far. There are, however, two sets of remarkable physical results. The first is given by some very high energy scattering amplitudes that have been computed (see for instance [2, 3, 4, 5, 209, 199]). An intriguing aspect of these results is that they indirectly suggest that geometry below the Planck scale cannot be probed -and thus in a sense does not exist- in string theory. The second physical achievement of string theory (which followed the d-branes revolution) is the recent derivation of the Bekenstein-Hawking black hole entropy formula for certain kinds of black holes [198, 111, 110, 109].

The main merit of loop quantum gravity, on the other hand, is that it provides a well-defined and mathematically rigorous formulation of a background-independent, non-perturbative generally covariant quantum field theory. The theory provides a physical picture and quantitative predictions of the world at the Planck scale. The main incompleteness of the theory is regarding the dynamics, formulated in several variants. So far, the theory has lead to two main sets of physical results. The first is the derivation of the (Planck scale) eigenvalues of geometrical quantities such as areas and volumes. The second is the derivation of black hole entropy for ``normal'' black holes (but only up to the precise numerical factor).

Finally, strings and loop gravity may not necessarily be competing theories: There might be a sort of complementarity, at least methodological, between the two. This is due to the fact that the open problems of string theory are with respect to its background-independent formulation, and loop quantum gravity is precisely a set of techniques for dealing non-perturbatively with background independent theories. Perhaps the two approaches might even, to some extent, converge. Undoubtedly, there are similarities between the two theories: first of all the obvious fact that both theories start with the idea that the relevant excitations at the Planck scale are one dimensional objects - call them loops or strings. I understand that in another living review to be published in this journal Lee Smolin explores the possible relations between string theory and loop gravity [191].

3 History of Loop Quantum 2 Quantum Gravity: Where are 2.2 What is the problem?

image Loop Quantum Gravity
Carlo Rovelli
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