The problem of describing the quantum regime of the gravitational field is still open. There are tentative theories and competing research directions. For an overview, see [258, 155]. The book  presents a large and interesting spectrum of viewpoints and opinions. The two largest research programs are string theory and loop gravity. Examples of other directions explored are noncommutative geometry , causal dynamical triangulations , causal sets , twistor theory , doubly-special relativity , and Euclidean quantum gravity [141, 144]. Research directions are variously related; in particular, formalisms such as spin foams (Section 6.7), group field theory (Section 6.8), or uniform discretizations (Section 6.10) are variously viewed as strictly related to loop gravity or independent research directions.
String theory and loop gravity differ not only because they explore distinct physical hypotheses, but also because they are expressions of two separate communities of scientists, which have sharply distinct prejudices, and who view the problem of quantum gravity in surprisingly different manners.2
High-energy physics has obtained spectacular successes during the last century, culminating with the laborious establishment of quantum field theory as the general form of dynamics and with the extraordinary and unexpected success of the standard model. This success is now several decades old. Thanks to it, physics is in a position in which it has been rarely: there are virtually no experimental results that clearly challenge, or clearly escape, the present fundamental theory of the world. The theory we have encompasses virtually everything – except gravitational phenomena.3 From the point of view of a particle physicist, gravity is then simply the last and the weakest of the interactions. The problem of quantum gravity is perceived as a last step in the path towards unification. It is then natural to try to understand the quantum properties of gravity using the strategy that has been so successful for the rest of microphysics, or variants of this strategy.
The search for a conventional quantum field theory capable of embracing gravity has spanned several decades and, through an adventurous sequence of twists, moments of excitement and bitter disappointments, has lead to string theory. The foundations of string theory are not yet well understood; and it is not entirely clear how the current versions of the theory, which are supersymmetric and formulated in 10 or 11 dimensions, can be concretely used for deriving comprehensive univocal predictions about our world. But string theory may claim remarkable theoretical successes and is today the most widely studied candidate theory of quantum gravity.
In string theory, gravity is just one of the excitations of a string or other extended object, living on some metric space. The existence of such background spaces, in which a theory is defined, is the key technical tool for the formulation and the interpretation of the theory, at least in the case of the perturbative definition of the theory. In tentative nonperturbative definitions, such as aiming to define the physical theory indirectly via a boundary quantum field theory, the theory relies only on the background “at infinity”, needed for the definition of the boundary quantum field theory.
In all cases, for a physicist with a high-energy background, the central problem of quantum gravity is reduced to an aspect of the problem of understanding the still mysterious nonperturbative theory that has the various perturbative theories as its perturbation expansion.
For a relativist, on the other hand, the idea of a fundamental description of gravity in terms of physical excitations over a background space sounds physically wrong. The key lesson learned from general relativity is that there is no background metric space over which physics happens (except, of course, in approximations). The world is more complicated, or perhaps simpler, than that. For a relativist, in fact, general relativity is much more than the field theory of one particular force. Rather, it is the discovery that certain classical notions about space and time are inadequate at the fundamental level: they require modifications, which are possibly as basic as those introduced by quantum mechanics. One of these inadequate notions is precisely the notion of a background space (flat or curved), in which physics happens. This profound conceptual shift, which has led to the understanding of relativistic gravity, the discovery of black holes, relativistic astrophysics and modern cosmology, is now considered by relativists to be acquired knowledge about the world.
From Newton to the beginning of the last century, physics has had a solid foundation in a small number of key notions such as space, time, causality and matter. In spite of substantial evolution, these notions have remained rather stable and self-consistent. In the first quarter of the last century, quantum theory and general relativity have deeply modified this foundation. The two theories have obtained solid success and vast experimental corroboration, and can now be considered well established. Each of the two theories modifies the conceptual foundation of classical physics in a (more or less) internally-consistent manner, but we do not have a novel conceptual foundation capable of supporting both theories. This is why we do not yet have a theory capable of predicting what happens in the physical regime in which both theories are relevant, the regime of Planck-scale phenomena, 10–33 cm.
General relativity has taught us not only that space and time share the property of being dynamical with the rest of the physical entities, but also – more crucially – that spacetime location is relational (see Section 5.3). Quantum mechanics has taught us that any dynamical entity is subject to Heisenberg’s uncertainty at small scale. Therefore, we need a relational notion of a quantum spacetime in order to understand Planck-scale physics.
Thus, for a relativist, the problem of quantum gravity is the problem of bringing a vast conceptual revolution, begun with quantum mechanics and general relativity, to a conclusion and to a new synthesis.4 In this synthesis the notions of space and time need to be deeply reshaped in order to take into account what we have learned with both our present “fundamental” theories.
The difference between the formulation of the problem of quantum gravity given by a high-energy physicist and a relativist derives therefore from a different evaluation of general relativity. For the first, it is just one additional field theory with a funny gauge invariance; for the second, it is a complete modification in the way we think about space and time.
This issue is often confused with the issue of whether the Einstein equations are low-energy equations that need to be corrected at high energy. But the two issues are not related: many relativists expect that the Einstein equations may very well require corrections at high energy. However, they do not expect that the corrected theory will mean a return to the old pre-general-relativistic notions of space and time.
Unlike string theory, loop quantum gravity has a direct fundamental formulation, in which the degrees of freedom are clear, and which does not rely on a background spacetime. Loop quantum gravity is thus a genuine attempt to grasp what quantum spacetime is at the fundamental level. Accordingly, the notion of spacetime that emerges from the theory is profoundly different from the one on which conventional quantum field theory or string theory is based.
Above I have pointed out the distinct cultural paths leading to string theory and loop gravity. Here I attempt to compare the actual achievements that the two theories have obtained so far in describing 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 truly know, either, or both, could very well turn out to be physically 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 great enthusiasms turned into disappointment. The arbiters in science are experiments, and not a single experimental result supports directly any of the current theories that go beyond the standard model and general relativity (say with neutrino mass and a cosmological constant).
On the contrary, a fact, which is perhaps not sufficiently emphasized, is that all predictions made so far by theories that go beyond the standard model and general relativity (proton decay, supersymmetric particles, exotic particles, anomalous solar-system dynamics, short-scale corrections to Newton’s law…) have for the moment been regularly falsified by experiment!
Comparing this situation with the astonishing experimental success of the standard model and classical general relativity should make us very cautious, I believe. The possibility that a large part of the current theoretical research is following a wrong direction is very concrete5. Lacking experiments, theories can only be compared on completeness and aesthetic criteria, but these criteria may be misleading. One should not forget that, according to many, for quite some time these criteria favored Ptolemy over Copernicus.
In this situation, the existence of competing ideas, competing prejudices and competing research programs is not a weakness of theoretical physics; to the contrary, it is a genuine strength. Science grows in debates and confrontation of ideas.
The main merits of string theory are (i) its elegant unification of different theories used in known fundamental physics, (ii) its well-defined perturbation expansion, expected to be finite order-by-order, and (iii) its theoretical and mathematical richness and complexity. The main incompleteness is that its nonperturbative regime is very poorly understood, and we do not know the background-independent formulation: in a sense, we do not really know what the theory is, and how to describe its basics degrees of freedom. Thus, we control the theory only in sectors that (because of the numbers of dimensions or the unbroken super-symmetry) are neither Planck-scale physics, nor low-energy physics. More precisely: (i) There is not much Planck-scale physics derived from string theory so far. Exceptions are the investigation of the Bekenstein–Hawking entropy, including Hawking radiation spectrum and greybody factors, for certain peculiar kinds of black holes (classical references are [282, 148, 147, 146]; see a string review for references on recent developments in this topic), and some very-high-energy scattering amplitudes [9, 10, 11, 12, 295, 283]. An intriguing aspect of these scattering amplitudes is that they appear to indicate that geometry below the Planck scale cannot be probed – and thus in a sense does not exist – in the theory. (ii) We are not able to recover the correct low-energy physics, namely the full standard model in 4D, without unbroken supersymmetry, three generations, and the full standard-model phenomenology, from string theory. We do not even know for sure if correct low-energy physics is really predicted by the theory, and, if so, if it is predicted uniquely or as one out of many possibilities.
The main merit of loop quantum gravity is that it provides a mathematically-rigorous formulation of a background-independent, nonperturbative generally-covariant quantum field theory. It provides a physical picture of the world, and quantitative predictions, at the Planck scale. This has allowed, for instance, explicit investigations of the physics of the Big Bang, and the derivation of black-hole entropy for physical black holes. The main incompleteness of the theory regards the formulation of the dynamics, which is studied along different directions, and in several variants. In particular, the recovery of low-energy physics is under investigation, but no convincing derivation of classical GR from loop gravity is yet available. Finally, recall that the aim of loop quantum gravity is to unify gravity and quantum theory, and not to achieve a complete unified theory of all interactions.
Strings and loop gravity may not necessarily be competing theories: there might be a sort of complementarity, at least methodological, between the two. Indeed, the open problems of string theory mostly concern its background-independent formulation, while loop quantum gravity is precisely a set of techniques for dealing with background-independent theories. Perhaps the two approaches might even, to some extent, converge. The possibility has been explored, for instance, in . Undoubtedly, there are similarities between the two theories: first of all the obvious fact that both theories utilize the idea that the relevant excitations at the Planck scale are one-dimensional objects – loops and strings.
But there are also key differences: in an image, strings are one-dimensional objects moving in space, while loops are one-dimensional objects forming space.
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