A “quantum gravity expert” is presumably
someone well acquainted with the details
of our immense ignorance of the subject.
I suppose I count.
I long ago promised that I would discuss some of my own research. Here’s the first post that makes good on that promise. Today I’ll discuss a theory of quantum gravity.
Why Quantum Gravity?
Without a doubt, the two greatest advances in physics in the last 120 years were the advent of general relativity and quantum mechanics. These two amazing theories have totally changed the way we see the world. Quantum mechanics describes the physics of the very small, while general relativity describes the physics of the very massive.
Usually we don’t encounter very small, very massive things. However, they do exist, and we’d like to understand them. Black holes are the quintessential quantum gravitational mystery. A black hole is so incredibly massive that it it pulls all matter within it down to a tiny—perhaps infinitesimally small—point. And this point is small enough that we need quantum mechanics to understand it.
We also need quantum gravity to describe the universe on large scales. Modern cosmology tells us that the universe is expanding, even accelerating. If we extrapolate back to right before the universe began with a bang, it was infinitesimally small. A point. And it was—in a sense—the most massive it is possible for anything to be. We need quantum mechanics to understand this. Indeed, the most successful story we have about the early universe, inflation, relies heavily on quantum mechanics.
The accelerating universe is also a mystery, and scientists hope that quantum gravity will be able to explain it.
Unfortunately, quantum mechanics and general relativity don’t agree. Not at all. Quantum mechanics assumes that quantum particles, described by their wavefunctions, evolve in a static, eternal universe. However, in general relativity, the background itself is a living thing. Space and time reshape themselves according to the stuff contained in the universe. The quantum particles are affected by the changes in shape of the universe and affect the universe in turn, forming a feedback loop. This makes combining quantum mechanics and general relativity extremely hard.
Later, people came up with ideas like string theory, loop quantum gravity, and causal sets, all of which attempt to solve the problem of quantum gravity. But so far, although each theory has its success stories, no one theory has proven itself to be correct… or even predicted anything we can test. The best we can say is that most of them can show they look like general relativity if you take out quantum mechanics.
Needless to say, this problem is hard.
One of the things I work on is a candidate theory of quantum gravity called Causal Dynamical Triangulations, or CDT for short. Here’s how it works.
Adding Up All Universes
I already described one way to handle quantum mechanics, called the Feynman path integral. Classically, a particle takes the path between two points that minimizes (technically extremizes) the energy cost for the particle. In quantum mechanics, the particle is wave and it takes all possible paths between the two points. Then the probability of the particle traveling from the first point to the second point is given by the sum of a function of the energy costs of all possible paths.
We can take this idea and apply it to quantum gravity. Roughly, a classical universe starts with some three-dimensional shape and ends with some three-dimensional shape. It will evolve from the initial shape to the final shape in a way that minimizes (extremizes) the energy cost of that transition. Since the universe is a single shape of spacetime, we think of this sort of like a soap bubble connecting two wire rings. The wire rings force a shape at the beginning and end of the bubble, but the middle of the bubble can be whatever it wants.
So what’s the quantum analog? In quantum gravity, the probability of the universe of evolving from some initial shape to some final shape is given by the sum of some function of the energies of all possible spacetimes that connect the two initial and final shapes. This is called a sum over histories, since we’re summing over all possible histories of the universe.
Unfortunately, this sum over histories is incredibly hard to compute, or even define. Given an initial shape of the universe and a final shape of the universe, there are uncountably many spacetimes that connect the two. How do we sum over all those histories? How do we even find all of those histories? We need some clever tricks to do it.
Adding Up Some Universes
Right now, we don’t know how to find all the spacetimes that should contribute to the sum over histories. As a next best thing, we want to find the spacetimes that contribute the most to the sum. Imagine you take the number 1. You add it to . Then you add that to , and then , add inifnitum. Your sum looks something like this:
But pretty quickly the number stops changing when you add more terms to your sum.
If each successive term in the sum shrinks quickly enough, the sum itself stops growing very quickly at all. If we added new terms ad infinitum, we’d get
But with only five terms we’re almost there! Although there are infinity more terms to add before we get to the final answer, five terms gives us a darn good approximate answer.
We’d like to do the same with quantum spacetime. We can approximate the sum over all histories by taking the sum only over the histories that add the most to the Feynman path integral. So for now, our goal is to find those histories.
Right now, it is not at all obvious which histories contribute the most to the sum. To find them, we take advantage of a correspondance between quantum mechanics and statistical mechanics, called the Wick rotation. I’ve discussed before how in relativity, distances in the time direction square to negative numbers. This means we can think of the time direction as imaginary. (See my previous post on imaginary numbers for more info on what that means.)
But what happens if we make time real again? If the spacetime is sufficiently well-behaved (and it doesn’t have to be!), we can rotate the time axis through the complex plane to make it real. This transforms our spacetime into something we’re more used to, where all the distances square to positive numbers. Later, once we’ve evaluated our sum over histories, we can undo the rotation to find the right answer. This is called a Wick rotation.
Why do we Wick rotate? By changing the time axis to a real axis, our quantum system becomes a classical exercise in probabilities. If we had a humongous bag of Wick-rotated spacetimes (also called Euclidean spacetimes), and we stuck our hand in the bag and pulled a universe out, we could use statistics to figure out how likely it is that we’d pull out a given universe. And better yet, the most likely universes are the ones that contribute the most to the sum over histories if we Wick-rotate back.
So all we need to do is make a bag of Wick-rotated universes and pull universes out of the bag at random. In other words, we need to randomly generate Euclidean universes. And we can do that on a computer.
The Universe On Your Laptop
Spacetime as we know it is continuous. If you take a cube of empty space, and you zoom in on it with your microscope, you could zoom in forever. No matter how close you look, no matter how small the things you’re looking at are, you can always zoom in further and look at smaller stuff. In other words, things can be infinitely small. (We don’t know whether this is actually true. People have proposed a quantum of distance called the Planck length, which might be as small as things get. But we usually treat things as continuous.)
A computer has finite precision, though. It can’t encode things that are infinitely small. Instead we have to stop somewhere. And this means that we have to transform the smooth continuous shape of the universe into something made up of points and lines of fixed, nonzero size.
(To make things easier to understand and visualize, I’m going to drop from four dimensions to three. Now we have two spatial dimensions and one time dimension. This isn’t as crazy as it sounds. A lot of my research has been on three-dimensional quantum gravity. The reason is that the physics is easier, but we can still learn something about the four-dimensional case. Here’s a whole article in Scientific American about why quantum gravity in flatland is a good idea.)
In Causal Dynamical Triangulations, we make a specific choice about how to encode information on the computer. This choice is motivated by making the spacetime “nice” enough to Wick-rotate. For that to be possible, we enforce that there is a well-defined time direction. This sounds obvious, but it’s not always true in general relativity. You can have arbitrarily crazy spacetimes where time loops on itself, or where the time direction depends on where you are in the spacetime. Indeed, in string theory, the basic spacetime, a Calabi-Yau manifold absolutely does not have a well-defined notion of time.
We also enforce that there are no wormholes or baby universes, which also add ambiguity to the notion of “time.”
We construct our computerized universe out of equilateral tetrahedra, each of which is a tiny piece of Minkowski space. Each tetrahedron spans two discrete times. The orientation of the tetrahedron determines the effect it has on spacetime. The three possible orientations are shown below. They’re labeled by the number of vertices they have on each time slice. So a -tetrahedron has three vertices on the lower time slice and 1 vertex on the upper time slice. And so on.
We put the tetrahedra together so that face meets face and edge meets edge—there can’t be any gaps. When we put it all together, the spacetime looks something like the image below. The image doesn’t quite capture what’s going on because the tetrahedra are all the same size and you can’t really pack them together in a flat spacetime. So when the edges look like they’re different sizes, they’re not. This is actually curvature of the spacetime the tetrahedra are supposed to make up.
To figure out which spacetimes are most probable, we need to be able to measure how curved they are. This is an integral piece of Einstein’s theory of general relativity. So let’s step back and think about how we can measure curvature. I’ve discussed before about how it’s possible to measure curvature by looking at angles. Basically, look at a triangle and measure the failure of the interior angles of the triangle to add up to 180 degrees. We can use a similar idea here. However, we look at the interior angles of all tetrahedra that meet at an edge and measure the failure of the sum of those angles to add up to 360 degrees.
Let’s look at an example in two dimensions. A single tetrahedron approximates a sphere. Now three triangles of the tetrahedron meet at a single vertex, as shown below. In flat space, if we rotated around that single vertex, we’d travel 360 degrees. However, the interior angle of each triangle at that vertex is less than 120 degrees and they add up to a smaller number. This tells us the curvature at that point. This is called Regge calculus.
A Universe Factory
Now we know how to put a universe on our computer. But we still haven’t got a likely universe. The way we get one of those is to take a universe, any universe at all, put it on our computer, and then make random changes to it. Each time we make a change, we measure whether the new, modified universe is more or less likely than the previous universe. If the new one is more likely, we keep the change. Otherwise, we reject it a fraction of the time and keep it a fraction of the time, based on how much less likely it is than the previous universe. This generates a single probable spacetime.
A typical simulation of a single Wick-rotated quantum universe is shown below. The long direction is the time axis and the other two directions show the size of the universe at a given discrete time. The movie is showing the universe evolve from an arbitrary initial configuration to a likely final configuration. About halfway through the simulation, we get to a probable configuration. After that, the changes are just quantum fluctuations around the mean.
This type of simulation is called a Monte Carlo simulation. The set of decisions the program uses to make the simulation go is called the Metropolis-Hastings algorithm.
The Average Universe
Unfortunately, it’s not enough to generate a single likely universe. To perform a sum over histories, we have to average over lots of them. If we do this, we can generate the average Wick-rotated, quantum universe. (We don’t know how to Wick-rotate back, so for now we have to do everything with real time.)
The expected quantum universe in the ground (or lowest energy) state is shown below. (Lowest energy means that the universe is empty and that it evolves from a big bang to a big crunch: nothing to nothing.) I’ve plotted spatial area of the universe as a function of discrete Euclidean time. Don’t worry about the details. I just wanted to show you that the plot is smooth after you average it out, even though each individual universe is pretty bumpy. The error bars show the quantum fluctuations. If we Wick-rotated the universe we lived in now, it would look a lot like this plot… which tells us that causal dynamical triangulations reduces to general relativity when we take away quantum mechanics.
So on large scales the universe of causal dynamical triangulations looks like Einstein’s universe. What about on small scales? Something very weird happens if you zoom in close enough… the universe begins to look like a spider web. In the past I’ve talked about the idea of fractional dimension and a way to measure it, called spectral dimension. We can measure the dimension of the universe of causal dynamical triangulations, and we see that it’s not what we expect.
The scale dependence of the dimension is plotted below. On large scales, the dimension is four, like we expect. But as we move to small scales, the dimension drops dramatically… all the way down to 2.8! We don’t really know what’s going on here, but it’s a hint of truly quantum behavior. We expect there to be a “quantum foam” and this might be what we’re seeing.
The State of the Art
Now you know the basics of Causal Dynamical Triangulations. Understanding the theory is an ongoing effort by less than fifty people around the world. So far, we can only simulate empty universes. Some people are working on putting matter into the model. I’m working on studying the probabilities of the universe evolving between different initial and final shapes. Others are working on testing how strict we have to be with the Wick rotation. It’s an ongoing story, so I hope you’ll keep your eyes peeled!
There isn’t much on causal dynamical triangulations. So here’s some further reading on that and on quantum gravity in general.
- The inventors of causal dynamical triangulations wrote an article for Scientific American. The article is here, but it’s behind a paywall. You can find it for free here.
- For a more technical introduction to causal dynamical triangulations, I recommend CDT founder Renate Loll‘s article, “The Emergence of Spacetime, or Quantum Gravity on Your Desktop.”
- For a perspective on Loop Quantum gravity, check out the community’s website.
- The string theorist Sean Carrol often talks about quantum gravity and physics in general in a very accessible manner.
- Lee Smolin is a popularist author on quantum gravity and physics in general. You might want to look at his website.
- I could hardly leave you without pointing you to my mentor in all things quantum gravity, Steve Carlip.
- And here’s a whole blog on quantum gravity!
Play With it Yourself
If you’re especially excited about quantum gravity, and especially brave, you might want to try and run a simulation yourself. We are planning on open sourcing the code in the near future. So, for reference, here’s a link to a (currently locked) github repository.
Here’s the code: https://github.com/ucdavis/CDT
And the documentation I’ve written is here:
If you use the code, please cite it as ours. The original author is Rajesh Kommu. However other authors include myself, Steve Carlip, Joshua Cooperman, Christian Anderson, David Kamensky, Kyle Lee, and Adam Getchell.
Alas, the majority of the code is in #LISP . You might like that, but most likely, you resent it. Sorry about that.
EDIT: Unfortunately, we have had trouble open-sourcing our code through the bureaucracy of the University. We still plan to release our code, but it is unavailable at the moment. I’m sorry about that.
Questions? Comments? Insults?
I’m afraid that this post might have been less clear than previous posts. It’s certainly longer! So if you have any questions, please let me know so I can clear up the confusion!