|Quantum gravity researcher at work.|
But there’s a different way to think about geometry, which goes back about a century to Hermann Weyl. Instead of measuring distances between points, we could measure the way a geometric shape vibrates if we bang it. From the frequencies of the resulting tones, we could then extract information about the geometry. In maths speech we would ask for the spectrum of the Laplace-operator, which is why the approach is known as “spectral geometry”. Under which circumstances the spectrum contains the full information about the geometry is today still an active area of research. This central question of spectral geometry has been aptly captured in Mark Kac's question “Can one hear the shape of a drum?”
Achim Kempf from the University of Waterloo recently put forward a new way to think about spectral geometry, one that has a novel physical interpretation which makes it possibly relevant for quantum gravity
Quantum Gravity on a Quantum Computer?
The basic idea, which is still in a very early phase, is the following.
The space-time that we live in isn’t just a classical geometric object. There are fields living on it that are quantized, and the quantization of the fluctuations of the geometry themselves are what physicists are trying to develop under the name of quantum gravity. It is a peculiar, but well established, property of the quantum vacuum that what happens at one point is not entirely independent from what happens at another point because the quantum vacuum is a spatially entangled state. In other words, the quantum vacuum has correlations.
The correlations of the quantum vacuum are encoded in the Greensfunction which is a function of pairs of points, and the correlations that this function measures are weaker the further away two points are. Thus, we expect the Greensfunction for all pairs in a set of points on space-time to carry information about the geometry.
Concretely, consider a space-time of finite volume (because infinities make everything much more complicated), and randomly sprinkle a finite number of points on it. Then measure the field's fluctuating amplitudes at these points, and measure them again and again to obtain an ensemble of data. From this set of amplitudes at any two of the points you then calculate their correlators. The size of the correlators is the quantum substitute for knowing the distance between the two chosen points.
Achim calls it “a quantum version of yard sticks.”
Now the Greensfunction is an operator and has eigenvalues. These eigenvalues, importantly, do not depend on the chosen set of points, though the number of eigenvalues that one obtains does. For N points, there are N eigenvalues. If one sprinkles fewer points, one loses the information of structures at short distances. But the eigenvalues that one has are properties of the space-time itself.
The Greensfunction however is the inverse of the Laplace-operator, so its eigenvalues are the inverses of the eigenvalues of the Laplace-operator. And here Achim’s quantum yard sticks connect to spectral geometry, though he arrived there from a completely different starting point. This way one rederives the conjecture of (one branch of) spectral geometry, namely that the specrum of a curved manifold encodes its shape.
That is neat, really neat. But it’s better than that.
There exist counter examples for the central conjecture of spectral geometry, where the shape reconstruction was attempted from the scalar Laplace-operator's spectrum alone but the attempt failed. Achim makes the observation that the correlations in quantum fluctuations can be calculated for different fields and argues that to reconstruct the geometry it is necessary to not only consider scalar fields, but also vector and symmetric covariant 2-tensor fields. (Much like one decomposes fluctuations of the metric into these different types.) Whether taking into account also the vector and tensor fields is relevant or not depends on the dimension of the space-time one is dealing with; it might not be necessary for lower-dimensional examples.
In his paper, Achim suggests that to study whether the reconstruction can be achieved one may use a perturbative approach in which one makes small changes to the geometry and then tries to recover these small changes in the change of correlators. Look how nicely the physicists’ approach interlocks with thorny mathematical problems.
What does this have to do with quantum gravity? It is a way to rewrite an old problem. Instead of trying to quantize space-time, one could discretize it by sprinkling the points and encode its properties in the eigenvalues of the Greensfunctions. And once one can describe the curvature of space-time by these eigenvalues, which are invariant properties of space-time, one is in a promising new starting position for quantizing space-time.
I’ve heard Achim giving talks about the topic a couple of times during the years, and he has developed this line of thought in a series of papers. I have no clue if his approach is going to lead anywhere. But I am quite impressed how he has pushed forward the subject and I am curious to see how this research progresses.