Black holes: quantum mechanics at macroscopic distances

SlavaM has figured out what is the best hep-th paper today. This work by a Benelux (or, more precisely, Bene) collaboration is just amazing and I am surely going to write about it:

Jan de Boer, Sheer El-Showk, Ilies Messamah, Dieter Van den Bleeken: Quantizing N=2 multicenter solutions

Let me begin.

Can quantum mechanics reveal itself at macroscopic distances? The first obvious answer is No. Quantum mechanics governs the microscopic world. However, the former dean of the Faculty of Mathematics and Physics in Prague, Prof Bedřich Sedlák, used to be working on low-temperature condensed matter physics.

Mr Sedlák also liked the legs of a female classmate of mine - a friend who married my diploma thesis adviser (who was also a co-author of our linear algebra textbook, "We Are Growing Linear Algebra" - his name is translated as Gardener) - but concerning his discipline, he said that it is the best discipline of physics because

quantum phenomena may exhibit themselves at macroscopic distances.

He was talking about superfluids and superconductors, among other things. But he was really cheating. The relevant fields that mimic the wave function of the BCS pairs, to pick an example, are not really wave functions. They are classical fields. So what we're looking at is just another classical limit (arising from a particular combination of a large number of electron pairs), not a case of "macroscopic quantum mechanics".




Back to general relativity

However, many people have been suspecting for years that quantum gravity may offer us a more "genuine" example of quantum phenomena at macroscopic distances: holography and the information loss puzzle are two hints among many. Where do you expect quantum phenomena to become "really important"?

The old procedure to find the answer would follow the semiclassical approximation, the same approach to quantum gravity that allowed Stephen Hawking to deduce that black holes emit a thermal radiation back in the 1970s. You calculate the first quantum corrections - the semiclassical corrections - to the classical equations and if they are much smaller than the classical starting point, you decide that you can rely on classical Einstein's equations as far as all kinds of qualitative and approximate quantitative conclusions go.

In reality, this comment means that whenever and wherever the curvature is small relatively to the Planck scale (whenever the curvature radius is greater than the tiny Planck length), Einstein's equations locally work. Also, if you construct a solution to Einstein's equations that has a low curvature everywhere (and the singularities in the highly curved regions have a good "character", which is purely a qualitative condition), there should exist a corresponding state in the Hilbert space of a quantum theory. And parameters that look continuous in the classical theory should remain quasi-continuous in the quantum theory, with the spacing uniformly going to zero in the classical limit.

But is it true? Can you really trust classical Einstein's equations if these conditions are satisfied? Or is it possible for the quantum phenomena to modify Einstein's classical conclusions even in cases where an innocent semiclassical physicist wouldn't expect it?

You may guess why I am asking these questions. The information loss paradox may be viewed as the first reason to believe that nonlocal phenomena (which are clearly incompatible with the principles of classical general relativity, including causality) may enter the scene at macroscopic distances, in order to preserve the information. But that's just a guess whose explicit description is not fully understood - which is why many people say that the information loss paradox hasn't been fully solved.

However, the present authors demonstrate that there exist fully controllable situations in which quantum mechanics influences physics at macroscopic distances. In some sense, they seem to confirm and illuminate the fuzzball paradigm by Samir Mathur. But their approach is slightly different and more explicit.

The setup

What black holes do we consider? It is still a kind of "general relativity in four dimensions coupled to other fields". But it is useful to consider highly supersymmetric low-energy theories resulting from highly supersymmetric compactifications because the supersymmetric nonrenormalization theorems allow us to describe physics in many ways (including configurations of D-branes and strings) and many physical quantities may be calculated exactly. The most helpful class of such compactifications is type IIA string theory on the four-dimensional Minkowski space multiplied by a six-dimensional Calabi-Yau manifold.

We have encountered these compactifications in the context of the Ooguri-Strominger-Vafa (OSV) conjecture. We're still interested in the same backgrounds and the same solutions.

These theories have 8 supercharges, twice as many as needed for a viable phenomenology. So they are called N=2 vacua in four dimensions and they have the "ideal" amount of supersymmetry that is neither too constraining - which would make the results (such as the moduli spaces) trivial - nor too unconstrained - which would make them uncalculable. As Paul Aspinwall used to say, the real world only has one half of this ideal amount at most but it was not his fault (a mild criticism directed at his colleague God). But the solutions to these N=2 supergravity equations can also be interpreted in a five-dimensional language: the black holes can be "lifted" to M-theory which has one more dimension. The corresponding five-dimensional supergravity is called N=1 d=5 supergravity because 8 supercharges is the minimum supersymmetry you can get for five large spacetime dimensions.

The relevant black hole solutions are BPS: they preserve one half of the supercharges of the background, i.e. 4 supercharges.

This is a sufficient number of supercharges for the number of black hole microstates to be constant as you change the string coupling. In fact, the whole phase space is identical at weak coupling and strong coupling and the authors explicitly check this fact in this paper, too.

What phase space am I talking about? Well, you must first write down the black hole solutions. They are multi-centered solutions with many charges described in terms of split attractor flows by Frederik Denef. You may imagine that these black holes look like threshold (=zero binding energy) bound states of many mono-centered black holes with some charges so that the charges add up properly.

You might expect that the space of these static solutions is a configuration space of these objects and you have to add some velocities (or, equivalently, canonical momenta) to get a phase space. However, these black holes are not really static because they carry the angular momentum (a typical feature of objects with both electric and magnetic charges, i.e. dyons). And with a nonzero angular momentum, the space of solutions is actually not the configuration space but the phase space itself! You don't have to add anything. It's already moving.

This observation is related to the fact that "x" and "p" eigenstates are not normalized to unity (but rather to the delta-function) but "j, j_z" eigenstates usually are. Alternatively, you may say that different components of the angular momentum don't commute with each other so they're the canonical momenta of other components. Once again, the solution space is the phase space.

The near-horizon geometry of these black hole solutions is eleven-dimensional, namely "AdS_3 x S^2 x CY_3" with M-theory in the bulk. Via the AdS/CFT correspondence, this gravitational theory has a conformal dual which is called the Maldacena-Strominger-Witten (MSW) (0,4) superconformal theory, describing some kind of membranes living in the Calabi-Yau background whose near-horizon geometry is the "AdS_3"-type space. Note that MSW is symmetric under a rotation by pi. These are big names with a lot of symmetry, indeed. ;-)

OK, so the present authors are now able to find the exact form of the solution space i.e. the phase space. The resulting manifold happens to be a toric one - you can describe it by a toric diagram (where the coordinates correspond to some "radial" distances and toroidal fibers describing the complex phases are added at each point; various cycles of these toroidal fibers can shrink to zero at special points, the boundaries of the toric diagram, according to well-known rules of toric geometry). It also happens to be a Kähler manifold and the authors explicitly find the Kähler potential. Some links of the space to weighted projective spaces are mentioned, too.

Fine, so the solution space is exactly known. If you want to be accurate, there is a subtlety here because in order to find the microstates corresponding to the black hole, you should first construct the full quantum theory - by quantizing the full phase space of supergravity - and take the subspace of the Hilbert space, describing the black hole microstates, at the end. Instead, they truncate the classical theory at the beginning (into the space of solutions) and quantize it as a phase space. It's not quite the same thing because the operations don't commute but it's close. The dimension of the solution space is a linear function of the number of centers.

Shocking conclusions

So these difficult problems turn out to be exactly solvable although, using Barbie's observations, the math class is tough. But what are the surprising qualitative consequences of these results?

There have been many types of the Denef-like solutions with unusual qualitative properties. For example, some of them allowed the "throat" to be infinitely long. In such a long throat, you could have excitations with an arbitrary i.e. continuous momentum. That would lead you to believe that the spectrum of the CFT should include a continuous portion. That looks strange.

Fortunately, the Benelux authors find this not to be the case. Quantum mechanics actually imposes an upper bound on the length of the throat. Quantum mechanically, the throat is not infinite but finite! That's great because the spectrum of the dimensions in the MSW theory can stay finite, as every sane quantum person would always expect.

Similarly, they resolve another old paradox involving barely bound black holes. Using the classical equations, you may find solutions that look like barely bound states where the components are arbitrarily far from each other. That also looks paradoxical because there could be infinitely many such increasingly delocalized bound states. In this case, quantum mechanics cuts the maximum distance between the components of the bound states in the black hole.

They see these things by simply looking at the geometry (and measure) induced by the symplectic structure on their solution spaces because they know all these things explicitly.

What is the general lesson? The general lesson is that in quantum gravity, the number of degrees of freedom is often much lower than what you would need to realize all of your fantasies based on classical physics. The entropy bounds and the holographic principle were the old moral examples why it is so.

The new Benelux paper gives you a new and, in some optics, more concrete picture why not all of your classical fantasies are allowed. Why is it so? Simply because you often don't have enough quantum phase space (not even one Planck volume, up to powers of (2 pi), necessary for one quantum microstate) to realize them.

A priori, this comment could sound crazy to you. If you have large objects, the corresponding phase spaces - parameterizing things like the distances between the components of a bound state - are also large. And if they are large, these phase spaces will have a large enough volume in all of their regions to represent all the classical geometries rather faithfully.

But the vague argument above is actually incorrect. Long distances on the phase space actually don't imply large volumes. ;-) Picky mathematicians would always know that they don't but most physicists would suspect that the mathematicians' counter-examples are inevitably pathological. But they can't really be that pathological because they appear in the description of some of the most canonical black hole solutions in supergravity and string theory.

A toy model of the small volume

For example, imagine that the phase space has two coordinates, "x" and "y". The coordinate "x" is a real number greater than a positive number "epsilon" while the coordinate "y" is real and periodic with period "1/x^2" (a fiber). The phase space is a tube that is getting thinner. For too high values of "x", you won't simply have enough volume in the phase space: the integral of "1/x^2" from "X" to infinity is equal to "1/X".

If "X" is substantially greater than one, the integral is smaller than one (and than 2 pi) and you will simply not find a single microstate that corresponds to the points of the classical configuration space, even though classically, "x" could have been arbitrarily high. For arbitrarily high values of "x", the corresponding classical configuration looks smooth and fine. Nevertheless, when you actually try to create a quantum microstate exactly at this point, with a high value of "x", you will obtain a violently fluctuating monster that doesn't resemble the classical smooth geometry at all.

The microstate is actually smeared out along a huge distance in the phase space. A few hours ago, I was explaining Benjamin in the fast comments that he could always imagine that the uncertainties of canonical coordinates and canonical momenta both go like the square root of Planck's constant which obeys the uncertainty relation while all the uncertainties still go to zero in the classical limit. Well, it's no longer the case here. If the canonical momenta have a very short circumference, the uncertainty of the corresponding canonical coordinates must be macroscopically large!

I believe that this major conclusion may be imported into other contexts in quantum gravity.

Once people look at them carefully, they could also understand the origin of holography (and the Bekenstein-Hawking entropy for general backgrounds) somewhat more constructively. What do I mean? The surprising feature of holography is "how do all those numerous degrees of freedom - that a priori seem to be associated with the large black hole interior - disappear?" And the answer could be that they don't really disappear but if you construct the corresponding phase space (whose cells form a basis of the Hilbert space), you find out that the volume of the phase space is much lower than you expected and the calculation reduces to those simple phase spaces that lead to the finite "S=A/4G" entropy.

Also, this result perhaps clarifies some of the confusion that many of us have had about the fuzzball proposal by Samir Mathur. We were asking him: shouldn't the black hole really look like the empty black hole solution we learned as undergrads, because of locality? How could all the mess of horizon-free smooth solutions replace the black hole geometry, including a singularity and a horizon, that seems to be weakly curved everywhere and that should be reproduced by general relativity?

The partial answer is that the black hole microstates can perhaps be still parameterized by a phase space - ideally, a subspace of the phase space of supergravity (even though some other, stringy coordinates could be needed in the general case) - and all the classical fuzzball solutions are still in it. But none of them gives rise to a well-behaved, stationary black hole microstate because the quantum microstates can only be obtained by combining many classical configurations that macroscopically differ because the states supported on a phase space region with a "small diameter" simply have a lower volume than required for one microstate. It doesn't quite explain why an observer will see the "boring", old-fashioned black hole interior with the singularity. But it does explain why he will never see the sharp fuzzball geometries inside.

This effect could be important for our understanding what really happens when you cross a black hole horizon. I guess that some previously confusing things start to make sense.