Problems with the Quantum Theory of Gravity

In the early twentieth century, the two frontiers of today’s Physics were born. The General Theory of Relativity (GR) by Einstein¹ describes gravity as the large-scale structure of the universe, and the quantum theory by some notable 20th-century physicists describes the physical phenomenon on a very small scale. By that time, Einstein realized the description of gravity within the mathematical framework of differential geometry can be regarded as a field, called the Gravitational field². There have been many experimental verifications of classical tests of general relativity over the years, including gravitational lensing of light rays by the sun during an eclipse, the measurement of “frame dragging” via satellites like Gravity Probe B, lunar ranging experiments, the detection of the decay of the Hulse-Taylor pulsar system, and the Pound-Rebka experiment. But, the LIGO detection is only the most recent and one of the most dramatic. The gravitational field has been experimentally verified by LIGO and Virgo teams on 11 February 2016. It was first observed on 14 September 2015 in two LIGO detectors situated at Handford, Washington, and Livingston, Louisiana then named the signal as GW150914 (which means “Gravitational Wave” and the date of observation 2015-09-14) with the merger of the binary black hole as its source³. On the other hand, quantum theorists were building the tools sufficient to explain the quantum theory in terms of a field-theoretic approach, called quantum field theory (QFT). This was developed to combine special relativity and quantum mechanics. This was necessary for understanding, for example, how light and matter interact. It is also important for studying large collections of particles. The physics of very, isolated particles, can be modeled very well with single-particle quantum mechanics (a beginner-level quantum theory). Note that quantum field theory wasn’t fully developed for many decades after quantum mechanics was discovered. Finally, this resulted in the famous Standard Model in Particle Physics and theorized Higgs Mechanism in the 1960s which helps to discover the Higgs boson with a mass of about 125.09$\pm$0.24 GeV ⁴ in July 2012 at Large Hadron Collider (LHC) and confirmed in March 2013 by ATLAS and CMS experimental groups at CERN⁵. From this experimental verification, we understood that Higgs Boson is responsible for giving mass to fundamental particles.

In the seminal work of Penrose and Hawking in the singularity theorems⁶, every blackhole exists singularities where the spacetime curvature becomes infinitely strong and unpredictable, and the starting point of the very early universe described by the well-known big bang model is a singularity, as well. The singularity theorems don’t simply imply that black holes possess singularities. They imply that no matter what the initial conditions of the universe were, singularities will form. This is profound because it implies general relativity drives itself into a regime where predictability fails. This may imply there’s a more complete theory to which we don’t have access that takes over such a small scale. When studying such very small, dense objects, we need both quantum mechanics and general relativity. We call the unknown theory required to study this combination “quantum gravity” (QG). This means we want to see gravitational properties by probing the universe directly to the Planck scale where one can find that the quantum effects are much more relevant. The first attempt to quantize general relativity date back to the early 1930s⁷ and physicists are still hoping that the gravitational field needs to be quantized. Because Einstein’s picture of gravity couples universally to all forms of energy.

One may expect that using the standard tools offered by QFT, one can able to quantize any fields from the Klein-Gordon field to the Higgs field. Unfortunately, this did not work out by proceeding with a naive quantization of GR, thus appearing to be perturbatively non-renormalizable⁸. Why is it so? If we look at Newton’s gravitational constant $G$ under Planck units⁹, it has a mass dimension to be -2 by dimensional analysis. This has a straightforward consequence in that it encounters divergence in the integral of the Feynman diagram of higher-order on large momenta where coupling for gravitation is perturbatively non-renormalizable. Such a situation is named “ultraviolet divergence”. In 1979, Steven Weinberg circumvented this issue by suggesting a stringent condition of renormalizability based on the Wilsonian renormalization group method, known as asymptotic safety^{8 10}. It states a theory is asymptotically safe if the essential coupling parameters approach a non-trivial (non-Gaussian) fixed point as the energy scale of their renormalization point goes to infinity⁸. Coupling parameters are considered essential if these parameters in Lagrangian are invariant under field re-definition. If they are not invariant then these are considered to be inessential coupling parameters. If a theory has a trivial (Gaussian) fixed point then the theory enjoys asymptotic freedom, discovered by David Gross and Frank Wilczek. A notable example is quantum chromodynamics. The renormalization group suggests the physical quantity should not depend on the arbitrary choice of renormalization point. These shreds hope that quantum gravity can be formulated in a non-perturbative way. We will come back to this point later, but let’s discuss some initial difficulties while formulating a quantum theory for gravity with the first principles of general relativity and quantum theory.

The first difficulty one may find during the quantization of general relativity is that a new degree of freedom of the theory is the geometry of the universe (i.e. spacetime), the object, which in most other approaches is considered to be fixed, or perhaps slowly changing. A hope is that the correct formulation is background-independent. Any fixed average (classical) geometric structure may appear only dynamically and it should not depend on a particular chosen background (solution of Einstein’s equations). The geometry by which the theory is defined is dynamically changing depending on how much matter and energy is contained in the region. It should always have a causal structure which means a meaningful direction of time and invariance under the spacetime diffeomorphism group that maps one spacetime parametrization into another. This suggests that the extension of general relativity to high energies should satisfy these principles.

The main reason why it is difficult is that general relativity and quantum mechanics have different, seemingly incompatible principles at their cores. General relativity uses the equivalence principle and locality. Quantum mechanics, on the hand, assumes the conservation of probability, or unitarity. It is not clear that these ideas are compatible. This results in difficulties when one tries to renormalize gravity using standard quantum field theoretical tools around a classical solution. Moreover, the quantum effect of gravity only appears on a length scale close to the Planck scale, about $10^{-35}$ meters. The blackhole firewall result is a demonstration of this. The scale is much smaller, so it can only be obtained at a much higher energy than the energy available in current high-energy particle accelerators.

Although the theory belongs to an extremely high energy regime, specifically, there are three directions to formulate a candidate theory of quantum gravity as pointed out by Carlo Rovelli⁷. i.e. covariant, canonical, and sum over histories. Even though some of the candidate theories cannot be included in any one of these. For example, string theory, twistor theory, non-commutative geometry, and so on. Carlo stated the three directions are because the majority of the candidates belong to (or, emerge from) these categories. Let me give you a short overview of these three directions.

The covariant line of research is the attempt to build the theory using a flat Minkowski or some other metric space $(\mathcal{X}, g)$ as a background geometry and then, let it fluctuate using a quantization algorithm over that metric. This means, the spacetime metric $g_{\mu\nu}(\mathcal{X})$ as a sum of $\eta_{\mu\nu} + h_{\mu\nu}$ where $\eta_{\mu\nu}$ is the Minkowski metric with signature $(-, +, +, +)$ and then, we quantize $h_{\mu\nu}$ using standard relativistic QFT⁷. This approach example includes high derivative theory and supergravity. The search converged successfully to string theory in the late eighties of the 20th century. It was initially inspired by the famous QCD flux tube. String theory (a more general version called M-theory) uses the approaches given by covariant research but it is not field theoretic because spacetime points are replaced by extended structures such as strings and branes. The main idea to construct this theory is a quantum theory of all the interactions and suggests graviton as a fundamental particle for the quantum gravitational field, and is due to the excitation of closed strings. It tries to complete the standard model of particle physics by adding a new dimension to spacetime. The theory is not yet experimentally verified. But, most of the quantum gravity nowadays focuses on understanding the behavior of quantum spacetimes, and the possible observables that can be extrapolated to phenomenology.

The canonical line of research is the attempt to build the theory using the Hamiltonian formalism that carries an appropriate Hilbert space that is a representation of the operators corresponding to the full metric without consideration of the background metric to be fixed. Then, we identify the canonical variables and conjugate momenta¹¹. This means one will start with the standard Einstein’s field equation along with natural units (i.e. Newton’s gravitational constant, the velocity of light, and Planck constant) to be unity, $G_{\mu\nu} = 8 \pi T_{\mu\nu}$ such that the left-hand side is the geometry of spacetime which is purely classical. It means it’s just an ordinary function of the spacetime points indicating no fixed background metric. The right-hand side is a stress-energy tensor that has quantum operators. Thus, we can think of it in this form: $G_{\mu\nu} \ket{\Psi} = 8 \pi \hat{T}_{\mu\nu} \ket{\Psi}$ which we can interpret as an eigenvalue equation. But this equation no longer makes sense because the ten operators components of stress-energy tensor ($T_{\mu\nu}$) do not commute with each other¹² thus suggesting simultaneous eigenstates will not exist. It simply means it has no any solutions. One way one can replace it with the expectation¹² of this tensor which was suggested by Møller (in 1962) and Rosenfield (in 1963) as $G_{\mu\nu} = 8\pi \braket{\Psi \vert \hat{T}_{\mu\nu} \vert \Psi}$ which lead to introduce “semiclassical gravity”. But the main difficulty is that $T_{\mu\nu}$ is a function of the classical field $g_{\mu\nu}$ and thus relates to $G_{\mu\nu}$. This suggest to have an uncertainty relation¹² of the form $\Delta g_{\mu\nu} \Delta G^{\mu\nu} V^{(4)} \ge 1$. This relation arises from the commutation relation between the operators in the stress-energy tensor thus, results in a non-unique set of solutions. To resolve this issue, observing the ordinary Hamiltonian procedure that there are commutation relations between the dynamical variable and its first-time derivative, we need to find those variables to have unique sets of solutions. Thus, this rose to the approaches like Wheeler’s quantum geometrodynamics and loop quantum gravity.

The sum over histories line of research uses the “Feynman quantization of general relativity” approach¹³ which was first suggested by John Archibald Wheeler and introduced by his doctoral student Charles William Misner in 1957 as $$ \int \exp(i/\hbar) ~(\text{Einstein action})~ \text{d}(\text{field histories}). $$

Instead of using perturbative approaches that turn into non-renormalizable for gravity, a new window has been opened on the non-perturbative approach using Feynman path integral for gravitation. In Feynman’s path integral, we have a resultant path to be a superposition of all the possible paths. We can define in terms of resultant spacetime as a superposition of all possible spacetimes. Thus, new approaches were born. For example, Hawking’s Euclidean quantum gravity, quantum Regge calculus, Sorkin’s Causal sets theory, and Causal Dynamical Triangulations. This line of research bags with its own problem. The main problem is in the starting point at Feynman geometric path integral where we not only have to sum over all possible spacetimes up to diffeomorphism but also accounting the sum of all possible topologies¹⁴. This is because, during the superposition, the topology of the spacetime can change. This is a very non-trivial problem.

Due to the enormous efforts taken for building a complete theory of QG, some believe there’s no way to quantize GR, considering the possibility that gravity is purely classical interaction. Thus this class of classical fields is incompatible with quantum mechanics. This is due to the lack of experimental evidence and signifies the difficulty of physical measurement. To verify gravity has quantum entities in short-length scales, proposed tests have been invented under the phenomenology of specific QG models and cosmological observations. But, none of these yet provide conclusive shreds of evidence. Still, there is another way one can suggest by just looking at the Cavendish experiment but taking the two masses to be very small and emphasizing the gravitational interaction between them via graviton such that we can study superposition between them¹⁵. However, this approach does not clarify how quantum coherence will be observed due to gravity. Thus, results in different modified interferometry techniques (like molecular interferometer¹⁶, Stern-Gerlach interferometer¹⁷, Mach-Zehnder interferometer¹⁸, and so on) to understand this question. But, these techniques are not yet able to provide conclusive evidence. Thus, the question is still open.

In the nutshell, quantum gravity is the most puzzling problem in the field of theoretical physics.

Footnotes & References