Solutions to three problems in quantum gravity
Quantum gravity is considered to be the Holy Grail of theoretical physics, the final merger of high energy particle physics and general relativity.
These two theories are frequently referred to as being “incompatible” or contradictory, but that isn’t entirely true. Standard methods of quantization, the procedure by which classical field theories are turned into quantum field theories, are incompatible with the general form of general relativity. But standard methods of quantization are actually incompatible with many theories. In fact, most theories cannot be quantized. That all the other fundamental forces and matter can be is a special feature of those theories.
There are, however, special cases of general relativity that can be quantized in a more or less standard way. That is because those cases are dual to theories that we do know how to quantize. Duality means that the predictions of one give you predictions of the other.
These special cases of general relativity live in a kind of spacetime called an asymptotically Anti-deSitter (AdS) spacetime.
An asymptotic space just means one that has certain characteristics at large distances within some region, so an asymptotically flat space is one that becomes flat at large distances. The Earth is, asymptotically, an oblate spheroid because it has a bulge in the middle that becomes apparent at large distances while variations in the surface like mountains become irrelevant by comparison.
An AdS space is one that has constant negative curvature. We also say that it is maximally symmetric, which just means that every point looks like every other and every direction is the same.
A space of negative curvature is often called a saddle shape but most Americans are probably familiar with another shape: a pringle.
Pringles, however, do not have constant negative curvature. The point at the origin is quite important compared to the other points. This is quite different from something with constant curvature, like a sphere, which has no special points or directions.
To understand AdS space, we have to understand more about hyperbolic geometry in general. Russian mathematician Nikolai Lobachevsky and Hungarian mathematician János Bolyai published the first non-Euclidean geometry in the early 1800s and one the geometries they proposed was a hyperbolic geometry. M. C. Escher, fasciated with non-Euclidean geometries drew his famous tesselation using a hyperbolic geometry:
An AdS space can be thought of as being a cylinder made of stacks of hyperbolic or Poincaré disks:
The Poincaré disk is a map of a hyperboloid, a two dimensional surface. Each of the red and white shapes above are triangles in a hyperbolic space and they are all supposed to be the same size. The reason they get smaller and smaller as we go towards the outer edge is a mathematical trick of projection, much like how we project the spherical Earth onto a flat geomap or the interior sphere of the sky onto a starmap. The edge of the map above is actually infinitely far away from the center. Thus, despite appearances, in this space, there is no special point or direction, much like the sphere.
These hyperboloids, when stacked up into a cylinder, form an AdS space.
Note, that this cylinder is like stacking a bunch of maps of the Earth on top of one another to represent the globe at different points in time. It is perfectly legitimate to look at the history of, say, continental drift that way but you will see distortions and you will be forced to, e.g., pick a center and a coordinate system as in this image below:
The reason we care about AdS spaces is because gravitational theories on them are dual to Conformal Field Theories (CFTs) on their boundaries.
A CFT is a special kind of theory that is scale invariant, and scale invariant theories are critically important to defining quantum field theories.
Quantum field theories are not all CFTs, but all the ones that are well defined, meaning we can use them to make general physical predictions, become CFTs at two important scales: one very tiny scale and one very large scale. These are called the UltraViolet (UV) and the InfraRed (IR) fixed points.
All this means is that all well defined QFTs run from a theory which is a CFT at the smallest scale where they are scale invariant all the way up to a CFT at the largest scale where they are also scale invariant. Take Quantum ChromoDynamics (QCD) for example, which is the study of quarks and the strong force which make up atomic nuclei. At the smallest scale, quarks are nearly free, meaning their interaction strength with one another is infinitesimal. The theory where they don’t interact is a CFT and so QCD at the smallest scale has a fixed point there. It also has an IR fixed point (called the Wilson-Fisher fixed point), which is not non-interacting but is a point where the strength of interaction stops changing.
Because the theory becomes scale invariant at these two finite points, we don’t have to worry about what happens outside those scales because no details about the theory can be communicated across those “fixed points”. That’s basically what a fixed point is.
A fixed point is basically a place where a system comes to a stop. In the case of quantum field theory, this means that couplings, how strong particles and fields interact, stop changing.
In order to predict the behavior of fields and particles, we need to understand how the behavior at both larger and smaller scales influences behavior at the scale we are probing. If there were no fixed points, then we would have effectively infinite influences at all scales. In order to deal with the infinite influences, we would have to make infinite measurements.
General Relativity is notorious for having no UV fixed point. It actually does have an IR fixed point because GR just becomes a non-interacting linear wave theory at large scales and non-interacting theories are scale invariant. At the smallest scales, such as the Planck length, GR just produces tiny black holes.
It turns out, however, that in AdS space GR behaves much more nicely at small scales. It still produces black holes but they are AdS black holes which behave like (in the way their energy scales) CFTs. That is a consequence of the AdS/CFT correspondence.
AdS/CFT is a general theory that shows how theories of gravity on AdS are dual to CFTs on the boundary of that AdS space (the “wall” of the cylinder above).
Because CFTs are UV fixed points, they are, themselves, UV complete and, hence, any theory dual to them is UV complete.
Does this mean that quantum gravity has actually been solved?
Not by a long shot.
Problem 1
Firstly, we don’t live in an (asymptotically) AdS space as far as we know. We live in a (asymptotically) deSitter (dS) space, a space of constant positive curvature given by the cosmological constant. There is no known duality between CFTs and dS space theories of gravity.
Problem 2
A second problem is that we don’t know what CFT actually corresponds to the correct theory of gravity. Famously, we know that a certain version of string theory (IIB) corresponds to a definite CFT (a supersymmetric N=4 Yang-Mills theory which I won’t go into). Most of the 1000s of papers on the AdS/CFT explore some aspect of string theory.
One of the issues that has to be dealt with is that the correspondence defines a relationship between a quantum CFT and a semi-classical theory of gravity. Usually you will read this as a correspondence between an “off-shell” and an “on-shell” theory. An off-shell theory is fully quantized and has virtual particles while an on-shell theory does not and so must be considered semi-classical.
String theory gets around this because we can connect the quantum CFT to specific elements of the string theory Scattering matrix (S-matrix). An S-matrix describes how incoming particles interact (i.e., scatter) and become outgoing particles. Each boundary state corresponds to one element of the S-matrix, so if we understand all the boundary states we can understand the whole quantum picture. Unlike string theory, however, GR plus the standard model of particle physics does not have a complete quantum description and so we can’t define its S-matrix.
In order to construct a relationship between a quantum CFT and a quantum theory of gravity, you need to have a UV completion for the theory of gravity too so you can construct the S-matrix. Since string theory has that completion, it has been used primarily for that. In order to use a CFT to make gravitational predictions and vice versa we have to know the precise correspondence, not just that one exists.
Problem 3
A third problem is that AdS/CFT doesn’t explain where space and time come from. It simply states that there is a duality between gravity on space and time and CFTs on a boundary space and time. That is at least philosophically unsatisfying and has led some to develop alternative theories of gravity such as Loop Quantum Gravity and spin networks.
I could go on about other technical problems, but these are the main issues.
Solution to Problem 1
The first problem, that we don’t live in an AdS space, can be addressed in a couple ways. Some physicists (such as Andrew Strominger) are working on extending the correspondence to dS spaces (and have so far made some progress but not succeeded). DeSitter spaces lack certain nice properties of AdS spaces and I feel that, at a fundamental level, the correspondence will fail.
I believe a better approach is to assume that we do live in an AdS spacetime that is five dimensional. Indeed, a great deal of the successes related to AdS/CFT are for the five dimensional AdS5. It is possible to slice AdS5 like a loaf of bread into four dimensional deSitter (dS4) slices which would correspond to our four dimensional spacetime with a cosmological constant. These slices would be concentric rings within the cylinder above, much like a cylindrical onion.
This additional dimension could be invisible to us if it behaves like time, and our dS4 spacetime is evolving perhaps away from the CFT from which it started. We might not notice these changes either because our history is simply evolving, and we are changing with it or because the different slices combine together to create our classical universe in a more complex way I will talk about below.
Solution to Problem 2
The second problem is difficult to address since we have no data to tell us what the true theory of gravity is nor what the true CFT is. It is still possible that it is string theory but so far no luck in proving anything about it. It is also possible however that the gravitational theory is fundamentally semi-classical, meaning it is not a quantum theory and needs no UV completion. Instead, the universe only appears to be quantum because spacetime is evolving in a 5th dimension.
This is effectively the same as a multiverse approach to quantum physics but a little more restrictive. Each dS4 slice is a different probable world in a multitude of worlds represented by the many slices. (You could call this the Many Slices Interpretation of quantum physics.) Every possible reality does not, however, have a corresponding world, only those that are actual slices of the AdS5 space. Highly improbable slices would not exist.
The UV completion of the theory, is, therefore, achieved via its correspondence to the CFT which can be considered to be the initial condition of the evolution of the dS4 slices.
This approach is appealing but has a downside in that it suggests a classical AdS5 space, and we have to now explain where quantum mechanics comes from. How are theories actually quantized on this space? And I don’t mean how do we do it mathematically. I mean why do we have to invent concepts like virtual particles, wave-particle duality, and quantum fluctuations in our theories if we live in a semi-classical spacetime?
One option that I favor is that the AdS5 space actually contains a propagating wave rather than a classically evolving set of slices. The wave is still semi-classical within each slice, but, if you sum all the waves over all the slices, you get a complete Schroedinger-type wavefunction. Such a wave would include virtual particles, fluctuations, and wave-particle duality by its nature. Classical reality, the reality we know, would then only appear when many slightly different waves cohere together in a process called constructive interference. There could still be multiple worlds arising from this constructive interference, but at the classical level they would be effectively cut off from one another by destructive interference rather than evolving into one another.
Still, in order to define the wave, you have to define it on an AdS5 spacetime, that includes having to define the gravitational field as a, at its simplest, spin-2 graviton field propagating against that spacetime which is a bit ugly.
Solution to Problem 3
That brings me to the third issue: where does that spacetime come from?
On the face of it, and this is a frequent criticism of string theory, it seems as if we are trying to destroy Einstein’s beautiful theory by proposing that gravity operates against a background spacetime. It turns out, however, that AdS/CFT does not require this to be true.
Instead, we can turn AdS/CFT on its head and say that we don’t have an AdS space at all but simply a definition for a quantum theory that can be interpreted as being on AdS space. The quantum theory itself, however, represents spacetime rather than being defined on it.
In order for a theory to have a background spacetime we must fix that spacetime independently of the theory. We have to choose it and fix it. But we do no such thing here. We aren’t fixing the spacetime at all.
Instead we have a wave we use to define gravity and spacetime. That wave is a function over spacetime slices, but we don’t know what those slices are! We can’t fix them as we would in a fixed background theory. The assumption that we have a wave on an AdS space, on the other hand, is merely a reflection of how the wave behaves under rescaling of the slices. This is the beauty of the AdS spacetime representation and why it works. The 5th dimension is merely the scaling dimension, and the boundary on which the CFT lives corresponds to an infinite rescaling. Hence, our quantum gravity wavefunction is the truth while the AdS is merely a way to represent how the wave behaves under scaling. There is, in fact, no background spacetime at all.
Conclusion
These are far from the only problems that have to be solved to create a complete theory of quantum gravity. Despite the heavy focus on string theory, the AdS/CFT correspondence can be applied to more general quantum gravity theories as we have seen. It is possible that in doing so we may gain not only an insight into gravity but the origin of quantum theory itself.
Zaffaroni, Alberto. "Introduction to the AdS-CFT correspondence." Classical and Quantum Gravity 17.17 (2000): 3571.
Kaplan, Jared. "Lectures on AdS/CFT from the bottom up." (2016).
Freidel, Laurent. "Reconstructing AdS/CFT." arXiv preprint arXiv:0804.0632 (2008).