We always appreciate it when explanations add up. There is a sort of soothing and satisfying effect that comes along and suffuses us when things make sense.
Precisely for those reasons, what has irked many theoretical physicists for already several decades is the absence of the theory of quantum gravity, which intends to put gravitational and quantum behaviour under one coherent roof regardless of the energy scale.
This article sketches a brief overview of the quantum gravity landscape and a couple of its leading candidate theories, namely M-theory, loop quantum gravity, causal dynamical triangulations, and asymptotically safe gravity.
The Outline of the Marriage
Quantum mechanics is the ensemble of the laws of physics that accurately account for the kinetics of atomic and subatomic particles as well as radiation (light) at minute distance scales. The theory explains behaviour that we do not normally observe in our macroscopic world, for instance, establishing an instantaneous connection over large distances (quantum entanglement), crossing barriers otherwise impenetrable to larger objects (quantum tunnelling), and taking different paths at the same time (quantum superposition).
It is this theory — more correctly, it is quantum field theory, which is quantum mechanics intertwined with classical field theory and special relativity — that explicates the dynamics of the most fundamental building blocks of our Universe, collectively captured in the Standard Model of particle physics. In addition to fermions, such as the electron, the quarks, and the neutrinos, that model also contains bosons, including the photon and the Higgs boson.
In contrast, gravitation is not described within the contours of the Standard model, but by Albert Einstein’s theory of general relativity, which posits that the curvature of spacetime is what we experience as gravity. In the words of physicist John Wheeler: “Spacetime tells matter how to move; matter tells spacetime how to curve.”
Theories of quantum gravity look at regimes where both gravitational fields and quantum effects grow very strong. This usually occurs beyond the Planck scale — this is the smallest known physical distance scale — where the equivalent energies are much higher than those probed in the Standard Model — the energy region beyond the Planck scale is often figuratively labelled as the ultraviolet (UV) regime or UV limit. In particular, the said regime becomes relevant in the vicinity of the most inner part of a black hole and at the moment of the Big Bang (the early Universe).
The ultimate purpose of a theory of quantum gravity is to duly spell out the physics beyond the Planck scale in such a way that at lower energies the theory can still reproduce the core tenets of quantum mechanics plus general relativity.
Due to the fact that the energy scales associated with quantum gravity are much higher than the upper limits of our current experimental technologies, e.g., the Large Hadron Collider, which we use to test the Standard Model, the proposed theories of quantum gravity remain out of scope of such tests. Given the lack of direct observational data in those regimes, it is thorny to rule out some of the theories and thus to progress towards the theory of quantum gravity.
At the conceptual side, two difficulties stand out in the attempt to merge quantum mechanics with gravity in the UV limit, to which different theories have different answers.
First, when treating gravitation dynamically as a quantum field, an infinite number of terms show up in the calculations which makes it evidently hard for a theory to be predictable. Even so, a boson (underpinned by a quantum field) is expected to exist for the gravitational field, i.e., the graviton, although it has not yet been detected up to this point.
Bear in mind that if you would put an upper limit on the energy scale — a so-called UV cut-off — you would as a matter of fact be able to design a quantum field theory of gravitation (more specifically, this is referred to as an effective field theory approach to quantum gravity). Nonetheless, a full theory of quantum gravity should not need to be cut off at some energy level in order to be consistent but, rather, should be finite-dimensional, and thus predictive, within the entire UV regime. Once coherently describing the dynamics at high energies, such theory would be designated as the UV completion of quantized gravity.
The second difficulty is that spacetime in the Standard Model is a priori defined — that is, spacetime is fixed and the theory is categorized as background-dependent — whereas in general relativity the specific spacetime structure comes out of the equations — that is, spacetime is dynamic and the theory is considered background independent. This geometric difference causes an incompatibility between general relativity and quantum mechanics that quantum gravity aims to smooth out.
Let us now have an introductory glimpse at some of the competing theories within the field of quantum gravity.
M-Theory
Arguably, one of the top contenders is M-theory, which is an overarching theoretical framework that provides shelter not just to the five superstring theories — Type I, Type IIA, Type IIB, Heterotic E₈xE₈, and Heterotic SO(32) — but also to supergravity. The prefix ‘super’ hints at the idea of supersymmetry, a hypothesized symmetrical relationship between fermions and bosons. Supergravity is a kind of quantum theory that relies on supersymmetry and naturally includes gravity.
Various relations are discovered between all these theories. For one, at lower energy scales, the superstring theories and M-theory can be approximated by supergravity in 10 and 11 dimensions, respectively — in other words, 1 time dimension and 9 or 10 spatial dimensions. Not only that, the five superstring theories each represent a separate limiting case of M-theory and are coupled to one another through special symmetries called dualities (see Fig. 1).
One of the reasons that M-theory is branded as a theory of quantum gravity is because the superstring theories spontaneously supply us with the graviton, the quantum-mechanical particle description of the gravitational field. In these theories, the fundamental entities of the Universe are 1-dimensional vibrating strings (not particles), whereby the different modes of vibration give birth to the different particles.
Fig. 1. Overview of M-theory, supergravity, and the five superstring theories as well as their connecting dualities. (Source: adapted from universereview).
Superstring theory is coherent only when operating in 10 dimensions. In order to make sure that we still live in a macroscopic 4-dimensional Universe (1 time dimension and 3 spatial dimensions), physicists have conjectured the possibility that every point in our 3-dimensional space consists at the smallest of scales — and therefore invisible to us — of a 6-dimensional topological space in which the strings dwell and which is entirely curled up on itself. The way in which this microscopic space is folded affects the strings’ vibrations which in turn determine the type of particle that arises.
By postulating strings instead of particles as elementary building blocks, M-theory is able to solve the conceptual issue of insurmountable infinities when quantizing gravity in the UV regime (see above section “The Outline of the Marriage”). The strength of M-theory lies furthermore in its ability to account for the Bekenstein-Hawking entropy of black holes, to present a solution to the black hole information problem with the assistance of the holographic principle, and to embrace the most favourite candidate for dark matter.
Then again, M-theory is struggling with becoming background independent, with exclusively predicting our Standard Model since it leaves the door open for the existence of a gigantic number of other possible universes, with explaining the accelerated rate of expansion of our Universe, and with formulating an answer to questions related to the origin of the Big Bang and the nature of the black hole singularity.
Loop Quantum Gravity
The theory of loop quantum gravity, which is often cited as one of M-theory’s main opponents, encompasses the dynamic properties of Einstein’s general relativity: it is background independent. Moreover, it hitches quantum mechanics’ essential feature of quantum uncertainty to that background. That is, loop quantum gravity treats spacetime itself as quantum mechanical. This ultimately means that, within the context of this theory, spacetime is fundamentally discrete and looks granular at the smallest of scales.
At these scales, loops are quantum-mechanical excitations of the gravitational field — which delineates the actual physical space — and combined they form a network, baptized as the spin network. Put differently, this network is a means to describe the quantum geometry of space.
The most basic components in this theory are quanta of space — the grains of space — which sit precisely at the points where the loops intersect — the nodes of the network. A loop subsequently appears when taking a sequence of various links between the nodes — the links are called connections.
Both the concept of volume and surface area, which are affiliated with the nodes and connections, respectively, are discrete in nature (see Fig. 2) — this entails that the nodes and connections have quantum numbers attached to them. As a result, a certain region of physical space finds itself in a quantum superposition of the spin network states, which are expressed by these quantum numbers.
Fig. 2. Each node in a spin network determines a cell, or an elementary grain of space. (a) Nodes are represented by small black spheres and the links as black lines, while cells are separated by elementary surfaces shown in purple. Each surface corresponds to one link, and the structure builds up a 3D space. (b) When the surfaces are pulled away, we can see that the sequence of links form a loop. These are the “loops” of loop quantum gravity. (Source: Carlo Rovelli).
Given that in loop quantum gravity the concept of volume is quantized, the expansion of the Universe therefore suggests that this process should occur in discrete time steps. In other words, the existence of spacetime implies a history of spaces. As space in loop quantum gravity exists of spin networks, the discrete (quantum) geometry of spacetime is then constructed through a temporal evolution of these networks, a structure that in its entirety is named causal spin foam. In so doing, spacetime can similarly be regarded as a quantum superposition of the spin foam states.
In terms of the theory’s advantages, not only does it get rid of the lingering infinities that rise to the surface when handling gravity as a quantum field in the UV limit (see above section “The Outline of the Marriage”), but loop quantum gravity can also do without supersymmetry or extra spatial dimensions — as yet undetected traits of M-theory — to design a mathematically coherent theory of quantum gravity.
What is more, the theory ensures that, at lower energy approximations, the collective behaviour of loops at larger scales gives rise to gravitons. In addition to that, loop quantum gravity dissolves aside from the Big Bang singularity (it is replaced by a Big Bounce) also the black hole singularity (the black hole shrinks to a Planck star and bounces back as a white hole). Also, it predicts the Bekenstein-Hawking entropy of black holes (after fixing the Immirzi parameter) as well as the phenomenon of Hawking radiation.
However, loop quantum gravity also comes with a couple of unresolved issues. First, it is not clear yet how the theory can recover general relativity in the low energy limit. Second, there are some complications with background independence with respect to topology (the field that studies the properties of geometric objects that remain invariant under deformations). Finally, it struggles with the issue of time; it grapples with reconciling two seemingly mutually exclusive notions: in quantum mechanics, time is addressed in an absolute fashion, whilst in general relativity it is considered a relative phenomenon.
Causal Dynamical Triangulations
This background-independent (dynamical) theory takes discrete surfaces as the building blocks for the assembly of its quantum spacetime, whereby the background itself is subjected to quantum fluctuations — as in the case of loop quantum gravity. Sewing together the surfaces at the smallest of scales leads to the emergence of our four-dimensional, continuous, curved spacetime structure at macroscopic dimensions (simply put, it gives us back general relativity).
Such a discrete, fundamental surface is called an n-simplex, which is a generalization of a triangle to higher dimensions (in the case of an n-simplex, the number of dimensions are n). For example, a triangle is a 2-dimensional simplex (2-simplex) that consists of three vertices (0-simplices) and three links or edges (1-simplices). Triangulation then means subdividing a geometric object into n-simplices.
In this theory, space comes in a foliation structure, whereby each triangulated spatial leave, i.e., a slice, is assigned a discrete time step and successive slices are connected by n-simplices (see Fig. 3). Stitching these n-simplices together in such a way that the accompanying arrow of time of the edges (links between vertices) corresponds in direction eventually results in the emergence of quantum spacetime (see Fig. 4).
Fig. 3. A visualization of fundamental building blocks of four-dimensional Causal Dynamical Triangulations (4-simplices in solid line). The simplices join two successive slices t and t+1 and are divided in to two types: {4,1} and {3,2}. The simplices are equipped with the flat Minkowski metric imposing the light-cone structure (blue-coloured image). The dotted line illustrates the cross section of a simplex by a three-dimensional hyperplane placed between the slices, at time t+1/2. (Source: Thesis by Andrzej Görlich).
Importantly, because of this built-in discrete temporal system, the premise of causality is embedded and conserved within this geometric structure. This imposed causality condition subsequently prohibits either wormholes (a sort of cosmic shortcut to other locations in spacetime) or baby universes (parts of the universe that have branched off of it) to ever arise in this specific spacetime.
What is more, the resultant quantum spacetime resembles our physical spacetime; the causal dynamical triangulation algorithm generates a so-called de Sitter universe, which, much like ours, is characterized by a positive cosmological constant — this constant refers to the energy density of space itself and a positive constant signals the accelerated expansion of the Universe — unlike superstring theory, whose fixed background is an anti-de Sitter spacetime containing a negative cosmological constant.
Note that, even though space is locally flat within each discrete simplex, the entire spacetime in its continuum, macroscopic limit is globally curved (see Fig. 4). Furthermore, the curvature of spacetime always resides on the n-2 dimensional sub-simplices for an n-simplex. That is, the location of curvature can be traced back to vertices for 2-simplices and to triangles for 4-simplices.
Fig. 4. Left: n-simplices for different numbers of n. For example, the 4-simplex contains 5 vertices and 10 edges or links. Right: A lattice geometry simplicial approximation. (Source: Marco Beria).
The fact that the quantum super-positioned elementary simplices arrange themselves dynamically (by an as yet unknown underlying mechanism) into a de Sitter-shaped spacetime has led Jurkiewicz et al. to allude to this triangulated modelled spacetime as a ‘self-organizing quantum universe’.
So, the causal dynamic triangulations theory’s strong suit is its ability to design a quantum (discrete) geometry which develops into a smooth classical spacetime at cosmic scales by falling back on a minimum of principles from quantum field theory and general relativity.
Moreover, this approach provides insight into the physics beyond the Planck scale: Calculations suggest that at these high-energy scales physics takes place in a two-dimensional spacetime (rather than our familiar four dimensions at macroscopic scales). Other approaches to quantum gravity, such as asymptotically safe gravity and Lifshitz gravity, have come to the same conclusion.
Not only that, additional information about the quantum dynamics of the system is made accessible through the possibility of running numerical computer simulations, which is a unique aspect of the theory within the whole pool of candidate theories.
Nevertheless, some physicists argue that, by integrating macroscopic qualities (e.g., curvature) into the simplices, information about the microstructure might be obscured. Further, it is not yet entirely clear how matter, black holes, and the Big Bang cosmology fit into this theory. Finally, a point of friction may be that causal dynamical triangulations treats space and time in a different manner whereas Einstein’s general relativity puts them both on equal footing.
Asymptotically Safe Gravity
As introduced in the above section “The Outline of the Marriage”, the approach to establish a theory of quantum gravity beyond the Planck scale by quantizing the theory of general relativity traditionally leads to the inability to come up with a finite number of physical parameters to describe the theory — the technical term for this applied method is perturbative renormalization.
Asymptotically safe gravity still wants to quantize general relativity, but instead of resorting to perturbative renormalization it uses a slightly distinct method, known as non-perturbative renormalization — more specifically, it relies on the functional renormalization group scheme. The concept of theory space, i.e., the theoretical space of all possible dynamics of a physical system, along with dimensionless couplings, i.e., the parameter that reflects the strength of particle or field interactions, take a central stage in this technique.
A theory is considered asymptotically free, when all the dimensionless couplings become zero in the high-energy UV limit. Stated in another way, such a theory must contain a UV fixed point in theory space — a fixed point can be interpreted as a consistent microscopic initial condition of the physical theory. One example includes the theory of quantum chromodynamics, which accounts for the strong nuclear force between quarks and gluons inside the nucleus of atoms, whereby the respective particles are non-interacting in the UV limit.
However, when the couplings are non-zero but still finite in the UV limit, a theory is deemed asymptotically safe. The corresponding fixed point is then referred to as a non-Gaussian fixed point. That is to say, the asymptotic safety-based approach to quantum gravity aims to be a quantum theory that accommodates a non-Gaussian UV fixed point in theory space and reduces to general relativity in the low-energy regime.
Asymptotically safe gravity indeed manages to identify such fixed point for one specific finite-dimensional subspace — called a truncation — of theory space, namely the Einstein-Hilbert truncation (see Fig. 5).
In terms of the geometric characteristics beyond the Planck scale, spacetime under the asymptotic safety scenario appears to reveal fractal-like features (a fractal is a geometric shape whose constituent parts are self-similar to the whole shape) and come in two dimensions (as in the theory of causal dynamical triangulations).
Fig. 5. Diagram of the flow of the dimensionless Newton’s constant g and cosmological constant λ based on the Einstein-Hilbert truncation. Arrows flow from UV to the infrared (IR). The origin corresponds to an infrared-attractive Gaussian fixed point and the non-Gaussian UV fixed point sits in the centre of the spiralling arrows. The green trajectory connects both fixed points and separates regions with positive and negative cosmological constant λ at low energy scales from each other. The candidate trajectory to be realized in nature is extremely close to the right of the green trajectory. (Source: Paper Simon Friederich).
To its advantage, the theory of asymptotically safe gravity is background-independent, does not appeal to new hypothetical objects (unlike superstring theory) and builds on existing theories (field theories and renormalization), seems to incorporate a procedure that explains the origin of the Big Bang as well as cosmic inflation, has predicted both the mass of the Higgs boson and the value of fundamental constants, finds a relationship between the questions of why there is a classical regime at all and why the cosmological constant in that low-energy limit is so small, and might provide an explanation for the gravitational behaviour of rotating flat galaxies without invoking dark matter.
Concerning its points of improvement, this theory requires more robust mathematical groundwork, inasmuch as there is no math-based proof available yet for the presence of a non-Gaussian UV fixed point in any truncation under observation, not just the Einstein-Hilbert truncation. Also, some physicists highlight some inconsistencies between this approach to quantum gravity and the Bekenstein-Hawking formula for black hole entropy (others refute this criticism, pointing out that the inconsistency makes sense, since it is not expected that this formula should hold in the high-energy limit of the theory in the first place).
In addition, asymptotically safe gravity does not offer any solution to the measurement problem in quantum mechanics and must show — except for the Einstein-Hilbert truncation, as it has done so already — that the abovementioned results continue to apply in a spacetime that carries a causal structure in it (technically, it has to demonstrate that the results are valid in a spacetime not only with a Euclidean signature but also with a Lorentzian one). In other words, it is not immediately obvious that asymptotically safe gravity constitutes a quantum theory in the UV regime.
Closing the Deal
Apart from the four main contenders for assembling the theory of quantum gravity discussed in this article, still many other attempts are out there, such as emergent gravity, infinite derivative gravity, causal set theory, background independent versions of string theory, non-commutative geometry, topos theory, and new approaches that mix various theories.
Perhaps our recent memory about the importance of trust in science might give a new impetus to bringing closure to this almost century-old challenge. Be that as it may, time will surely tell which one of the candidates will eventually succeed in officially marrying quantum mechanics to gravity all the way to UV completion.
In the meantime, all bets are open.
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