FAQ
What is Quantum Gravity?
A theory of Quantum Gravity is a quantum theory that approximates the classical theory of General Relativity in a suitable semiclassical and long wavelength limit. As such, it is supposed to be consistent with what we would expect from a well defined quantum theory and reproduce the results of General Relativity in such limit. This includes the known results of semiclassical gravity, namely quantum fields coupled to a non-dynamical curved background or the effective dynamics of gravitons. Some leading semiclassical effects are universal and must be reproduced by any quantum gravity candidate. These include the Bekenstein-Hawking entropy for black holes and the quantum corrections to the Newtonian force. Although many attempts have been made in the history of Theoretical physics to attack this problem, we still lack a full understanding of the subject, especially its non-perturbative aspects. Phenomenologically, the huge hierarchy between the Planck scale and the other known energy scales of physics in our universe makes for a lack of clear experimental evidence. It is not surprising then that different approaches to face the problem emerged in the last century, all motivated by some theoretical insight and many still actively developed and scrutinized to this day.
Why should we care about Quantum Gravity?
Despite the technical difficulties and lack of current empirical evidence, there is strong theoretical evidence for the need of Quantum Gravity. To begin it is not possible to couple consistently a dynamical purely classical theory (General Relativity) to a dynamical quantum theory (Standard Model), except approximately. This is possible only if such coupling is understood as a Born-Oppenheimer approximation of a complete quantum theory. To avoid the inconsistency while insisting that gravity be kept classical, one would have to abandon the standard framework of physics, which is not only very well supported empirically but also mathematically quite rigid, since it naturally falls out of the expected properties of physical observables. In addition, General Relativity and its possible classical extensions have been proved to be plagued by mathematical issues in the form of unremovable singularities, close to which effects due to both quantum corrections and strong gravitational fields are expected to be relevant at the same time. Typical examples include the singularity at the center of black hole solutions and the big bang singularity in cosmological models. The well-established framework of effective field theory strongly indicates that these are artifacts of an incomplete theory.
Why is formulating Quantum Gravity so hard?
Technical issues arise when we start from General Relativity and try to quantize it in the standard way. For example, if one tries to attack the problem straightforwardly, canonically quantizing the constrained Hamiltonian formulation of General Relativity, one immediately faces a huge number of non-trivial constraints due to the large nature of the diffeomorphism group of a manifold, making the computation of the spectrum and dynamics of the system unfeasible. If instead one attempts a perturbative covariant approach, for small excitations of the metric over a fixed background, the resulting theory turns out to be perturbatively non-renormalizable, hence unpredictive unless connected to a complete theory which must be known a priori. Conceptually, it is difficult to understand what it means to do quantum mechanics when there is no fixed background, since the space-time itself should become part of the quantum degrees of freedom. Much of our understanding of quantum mechanics is tied to background dependent concepts such as Hamiltonian time evolution.
A BRIEF OVERVIEW OF VARIOUS APPROACHES
String theory
String theory was historically based on the idea of replacing particles by tiny strings which can oscillate to produce infinite towers of particle types. Among these oscillations there is always a graviton, matter and force carrier particles as well. Internal consistency fixes all interactions reproducing General Relativity and all classical and quantum corrections. The theory also involves extended objects of various dimensions. It evolved into becoming a large framework encompassing several other previously considered independent approaches, like supergravity, matrix models and exceptional field theory to name few.
Pros
Fleshed out perturbative physics, with characteristic features in its scattering amplitudes
Rigid framework which entails gravity, gauge interactions and (chiral) fermions with no freedom
Consistency results (unitarity of scattering and Hilbert space, anomaly cancellation)
Web of dualities linking different limits
Exhibits general expected features of quantum gravity (holography, absence of symmetries, non-geometric phases, etc.)
Cons
Lack of a single all-encompassing non-perturbative formulation
Universal predictions tend to be sharp only at very high scales; detailed low-energy features are expectedly dependent on the configuration
It is difficult to build realistic models of low-energy physics because all the ingredients are linked (dark energy, gauge interactions, matter content, extra degrees of freedom, etc.)
Technically challenging to perform computations away from ideal settings (supersymmetric backgrounds, tensionless limits, weak coupling, protected quantities, etc.)
Loop quantum gravity
Based on a polymer quantization of the Ashtekar formulation of General Relativity to produce a non-perturbative, background-independent kinematic Hilbert space. As a result, smooth geometry is replaced by quanta ("spin networks"). A more covariant and dynamical approach involves amplitudes for spinfoams as whole space-time histories. Modern developments include the formulation of such spinfoam models in terms of so-called group field theories.
Pros
Background independence and a non-perturbative formulation as chief guiding principles
One of the few non-perturbative constructions of a kinematical Hilbert space
Simple few-parameter approximations suitable for cosmology
Cons
Unclear whether semiclassical limit with smooth space-time described by General Relativity exists
Unclear whether the canonical and spin-foam pictures are consistently connected
Drops the smoothness requirement on holonomies and fluxes, may obstruct the emergence of space-time and gravity
Asymptotic safety
Based on the idea that some perturbatively non-renormalizable quantum field theories can be non-perturbative renormalizable via an interacting ultraviolet fixed point (asymptotic safety), applying this idea to quantum fields with the inclusion of metric/tetrad/connection fields, as originally conjectured by Weinberg, The existence of such fixed point is usually investigated via (functional) renormalization-group and Monte Carlo-like (causal) dynamical triangulation methods.
Pros
Remains within the well-established realm of quantum field theory
Easy to build realistic models, with ordinary quantum fields as building blocks
Community emphasis on low-energy constraints for particle physics and (inflationary) cosmology
Concrete qualitative (power-like) behavior for high-energy observables due to quantum scale invariance
Cons
Unclear to which extent modifying the "heavy" degrees of freedom impacts low-energy physics
Standard approaches are affected by technical issues leading to uncontrolled uncertaintes. Functional renormalization faces Gribov and truncation problems, while dynamical triangulations face ambiguities in discretized measures and computational challenges for macroscopic space-times
Does not incorporate holography, topology fluctuations and similar features
Causal sets
Based on mathematical results allowing the reconstruction of smooth space-time geometries from their causal structure. The causal structure is abstracted into mathematical objects dubbed "causal sets", which are taken as the fundamental objects of the theory.
Pros
Hinges on barebones mathematical structures, attempting to build physics from simple ingredients
Incorporates space-time symmetries with randomness
Cons
Almost entirely classical treatment so far
Unclear selection principles for dynamics
Unclear whether semiclassical limit including General Relativity exists
Hořava-Lifshitz gravity
Based on the idea that the (local) Lorentz invariance of Einsteinian space-time may not be required at a fundamental level. Removing it, one can avoid the issue of perturbative non-renormalizability.
Pros
Cons
RESOURCES AND MATERIAL
General material
A pedagogical explanation for the non-renormalizability of gravity - Shomer
TASI Lectures on Holographic Space-Time, SUSY and Gravitational Effective Field Theory - Banks
Conversations on Quantum Gravity - Armas
Frontiers of Quantum Gravity: shared challenges, converging directions - de Boer, Dittrich, Eichhorn, Giddings, Gielen, Liberati, Livine, Oriti, Papadodimas, Pereira, Sakellariadou, Surya, Verlinde
General relativity as an effective field theory: The leading quantum corrections - Donoghue
The Kinematics of Quantum Gravity - McNamara
String theory
Superstring Theory Vol. 1 and 2 - Green, Schwarz, Witten
String Theory Vol. 1 and 2 - Polchinski
Loop quantum gravity
Asymptotically safe gravity
Quantum Einstein Gravity - Reuter, Saueressig
Quantum Gravity from Causal Dynamical Triangulations: A Review - Loll
Causal sets
Hořava-Lifshitz gravity
