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 Laplaceoperator, 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?
arXiv:1302.3680
The basic idea, which is still in a very early phase, is the following.
The spacetime 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 spacetime to carry information about the geometry.
Concretely, consider a spacetime 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 spacetime itself.
The Greensfunction however is the inverse of the Laplaceoperator, so its eigenvalues are the inverses of the eigenvalues of the Laplaceoperator. 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 Laplaceoperator'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 2tensor 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 spacetime one is dealing with; it might not be necessary for lowerdimensional 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 spacetime, 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 spacetime by these eigenvalues, which are invariant properties of spacetime, one is in a promising new starting position for quantizing spacetime.
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.
24 comments:
Without having looked into Achim's work in detail, I wonder: how does it relate to Connes' reconstruction theorem, which proves that a Riemannian manifold can be recovered from its underlying spectral triple?
Yes I like this approach Bee.
The conversion process still has to have specifics and the theoretic involved in terms of it's construction would have to have some association for the correlation to work.
For example:
Often, an increase or decrease in some level in this information is indicated by an increase or decrease in pitch, amplitude or tempo, but could also be indicated by varying other less commonly used components. See: Sonification
This conversion process is very important.
Another example would be:
HiggsJetEnergyProfileCrotale and HiggsJetEnergyProfilePiano use only the energy of the cells in the jet to modulate the pitch, volume, duration and spatial position of each note. The sounds being modulated in these examples are crotales (baby cymbals) and a piano string struck with a soft beater, then shifted up in pitch by 1000 Hz and `dovetailed'.
In HiggsJetRythSig we are simply travelling steadily along the axis of the jet of particles and hearing a ping of crotales for each point at which there is a significant energy deposit somewhere in the jet.
HiggsJetEnergyGate uses just the energy deposited in the jet's cells. At each time point (defined by the distance from the point of collision) the energy is used to define the number of channels used from the piano sound file. So high energy can be heard as thick, burbly sound whilst low energy has a thinner sound. See: Listen to the decay of a god particle
It is exciting for me to see your demonstration in concert with the approach of quantum gravity.
Best,
You might like this link below as well.
We think of space as a silent place. But physicist Janna Levin says the universe has a soundtrack  a sonic composition that records some of the most dramatic events in outer space. (Black holes, for instance, bang on spacetime like a drum.) An accessible and mindexpanding soundwalk through the universe. See: Janna Levin: The sound the universe makes
@saibod
Indeed the paper appears to describe techniques dual to those used by Connes in his noncommutative geometrical studies of the standard model and gravity. The two approaches are related through the fact that the Dirac operator can be thought of as the squareroot of the Laplacian. Kempf prefers to use the Laplacian while Connes uses the Dirac operator in his spectral triples (A,H,D) to encode the spectral geometry. In Connes spectral triple, A is the operator algebra of functions over the given manifold, H is the Hilbert space on which it acts on and D is the Dirac operator whose spectrum is used to recover the structure of the manifold, much like Kempf uses the spectrum of the Laplacian to recover the "shape" of the manifold.
In Connes approach to the standard model and gravity, to recover the gauge group of the standard model he considers a product space M x F, where F is a finite geometry, related to the "sprinkling of points" mentioned in Kempf's paper that has a matrix interpretation. Specifically, Connes considers the algebra of functions A over the 6point space in his model, where A=C+H+M_3(C). Here, C is the set of complex numbers, H the algebra of quaternions (transforming in M_2(C)) and M_3(C) the set of 3x3 matrices over the complex numbers, acting on the one, two and 3point spaces respectively. Classically, the manifolds which these encode are the unit circle, CP^1 and CP^2, each discretized by the eigenvalues of the matrix operators in the algebra of functions over the finite geometry F.
In string theory, such a finite geometry also arises in the guise of internal worldvolume degrees of freedom. In this framework, gauge groups can be seen as the internal degree of freedom at every point on the worldvolume of Ncoincident branes. The gauge symmetry is the freedom that a fundamental string has in deciding which of the N identical branes it can end on. In Connes' model, there would be a total of six branes encoded by the spectral triple of his finite geometry F, giving the U(1), SU(2) and SU(3) symmetry groups of the standard model.
"The correlations of the quantum vacuum are encoded in the Greensfunction which is a function of pairs of points." Green’s function opens Newton (e.g., terrain gravitometer sweeps to reconstruct buried dense ore or low density petroleum). To my knowledge, Green functions are not validated for general relativity. Green functions are all coordinate squares, removing chirality (versus Ashtekar). Green functions are defective if they uncreate fermionic matter parity violations.
Quantum gravitation and SUSY will founder until somebody discovers why persuasive maths do not empirically apply. Euclid plus perturbation is terrestrial cartography, and still fails to navigate the high seas, because rigorously derived Euclid is wrong in context. Green functions for linearized theory are established. Green functions describe complete nonlinear theory to any required accuracy. An odd polynomial to any number of terms is not a sine wave. It fails at boundaries.
Hi Saibod,
Achim submits the following: "Connes' spectral triple has much more information than just the spectrum of the Dirac operator. Namely, to know the spectral triple is also to know how the Dirac operator acts on concrete spinor fields. Having this much more information makes it way easier to reconstruct a manifold. The difficult part is to show under which conditions the spectrum (or spectra) *alone* suffice(s) to determine a manifold."
Best,
B.
Greensfunction > Green function.
The first form is not correct and never was. The second is the preferred form now, e.g. SchrÃ¶dinger equation, Maxwell equations, not SchrÃ¶dinger's equation, not Maxwell's equation (though the possessive forms are grammatically correct).
I remember that Max Tegmark commented in a talk at the 1994 Texas Symposium in Munich that he had looked up the official recommendations and "Green function" is correct, though he found that rather funny.
Hi Phillip,
I also find that rather funny, but I'll keep it in mind. Though I'm afraid that if I would write "Green function" nobody would know what I mean, which somewhat defeats the purpose of language. It's like that, after some years of complaining about the way the Swedes write dates that nobody knows how to read, I found out that it's the "international standard" for dates they're using... Best,
B.
I'm pretty sure that there is noone who knows what a Greensfunction is but doesn't know what a Green function is. The fact that it is capitalized hints that it is a proper name.
Awesome post Sabine! Should we call you The Quantum Gravity Doctor? ;)
Nice picture Sabine...
I guess finally your mother's dream become true. You are a 'real' doctor now with a stethoscope:)
Oops, sorry for my badly written comments. Anyway, "Green function" or "Green's function" are the terms that I know. Never heard of Greensfunctions...
Hi Christine,
He's only considering manifolds without boundary. I've been a little brief on the details for the sake of readability, but it arguably goes on the expenses of clarity, sorry about that. I can recommend Achim's paper though, I found it very well written and understandable. It's also not very long. Best,
Sabine
Hi Giotis,
There are some Dr med's in our family. I don't think it ever was my mother's dream I join them. My younger brother and I, we'd sometimes sneak into the doctor's office on weekends and play with the equipment. I've always been more interested in basic research though. And my younger brother, he's a mechanical engineer now. Best,
B.
Juan,
Yes, you can call me the Quantum Gravity Doctor. The patient is noncompliant :p Best,
B.
Hearing the Shape of the Drum
Thanks Bee
"Connes' spectral triple has much more information than just the spectrum of the Dirac operator. Namely, to know the spectral triple is also to know how
the Dirac operator acts on concrete spinor fields."
This sounds like an interesting mathematical question but in terms of physics one needs the spinors anyway to have fermions and to be able to reconstruct the Standard Model.
The Laplacian alone will not do. One just gets the bosonic part of the spectral action.
Moreover, to do serious physics at least an almost commutative spectral triple is required
anyway, rendering the overall manifold noncommutative.
Also, if just considering the Laplacian, I don't think one gets the gauge fields which are part of the bosonic action.
Regarding spacetime in isolation I regard as a major step backwards and completely against the very spirit of unification (of spacetime and matter), in particular given the sheer success of the noncommutative standard model.
Well, that's all based on my limited understanding of the subject, so please correct me if I am wrong.
You go, girl and get that Quantum Gravity!
Quantum gravity is the theory supposed to bridge the dimensional scales of quantum mechanics and general relativity, i.e. the human observer scales. I can see no reason, why the common chemistry and biology couldn't fall into subject of quantum gravity as well.
Last line from Alan Lightman's review of Smolin's new book in the NY Times Book Review section of the Sunday paper.
"For if we must appeal to the existence of other universes  unknown and unknowable  to explain out universe, then science has progressed into a culdesac with no scientific escape."
Science 1 ; pseudoscience 0
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