## Thursday, July 30, 2009

### Mark Van Raamsdonk: the entanglement glue

Mark van Raamsdonk (UBC Canada) is quite a physicist. After dozens of very solid technical papers, he decided to write a rather deep conceptual essay (if we want to avoid the term "vague speculative fairy-tale") called
Comments on quantum gravity and entanglement (click).
SlavaM originally wanted to promote this article in his own review but finally it's up to me. ;-)

When a technically weak physicist decides to write an ambitious paper about big questions of quantum gravity, she or he usually ends up with a pile of gibberish and pap. You can't take much of it because it makes no sense.

With Mark's paper about qualitative conceptual questions, we face the opposite problem. When I am reading it, I pretty much agree with every word of it. So at the end, one inevitably has to ask what is it that he has written but is not yet appreciated as a part of the lore?

After the second reading, I tend to believe that there's something like that and it's very powerful but I am still not capable to articulate it crisply. :-)

The kind of questions he's asking

Let us begin with the general problems he would like to fully solve by following his program. In string theory, we have learned many quantum mechanical systems that are dual i.e. equivalent to quantum gravity in some appropriate large N limits.

The non-gravitational systems that display such an emergent gravitational behavior are pretty diverse. So the question is
What feature that they share is responsible for the magic of emergent gravity?
In other words, why does the gravitational description emerge? What is universal about the theories or states where a dynamical, weakly curved spacetime emerges?

And Mark would also like to answer all questions about the origin of complementarity, e.g.: Can we associate Hilbert space with limited patches or just "full" spacetimes? What is the link between the Hilbert spaces - and the space of density matrices - for different patches? Do we need to learn some quadratic maps between pure states and mixed states?

The kind of his gluing answer

Fine, so I have described some of his questions by now. What are his answers? If I strengthen and streamline his statements a little bit, he wants to argue e.g. that:
The primordial degrees of freedom may be viewed as non-geometric ones, and even referring to disconnected components of a not necessarily geometric spacetime. Whenever you want to connect or glue these disconnected components of the spacetime, you need a glue. Mark's glue is the entanglement between the pure states in the components.
Now, I can formulate these statements in a nearly equivalent way so that the propositions will look like obvious tautologies. Truly disconnected parts of spacetime are independent rather than entangled. Yes, indeed.

Whenever you want to talk about the distances between two points, which is pretty helpful in geometry, you want the components to be connected. That's right, too. Once you glue two parts of the spacetime, the degrees of freedom in these two parts get entangled as long as you know that the total energy is low. Yes, indeed: that's true even in quantum field theory. The more tightly they're connected (by thick and/or short necks, like if you're trying to minimize resistivity of a system of resistors), the more entangled the degrees of freedom are. Yes, that's true, too.

While it may be hard to find a clearly new or provocative statement, I think that they are written somewhere over there. It's just not easy to pinpoint the location. :-)

Maldacena's eternal black hole

It seems clear that Mark also wants to generalize Maldacena's insights about eternal black holes in AdS spaces that can be described as an entangled state in two different CFTs. The message is simply that:
What looks like a geometric interpolation between two boundaries in the gravitational picture is described as some entanglement in the non-gravitational description (CFTs).
I am sure that Juan did realize that his idea could have been more general but in order to present as solid evidence as possible, he remained modest and talked about the particular eternal black hole geometry only. But it is true that a qualitatively similar phenomenon exists in other contexts, too.

Stephen Hawking's happily lost bet

At any rate, it is fascinating to realize that an eternal black hole in a connected spacetime may be expressed as a linear superposition of many states, each of which describes two disconnected spacetimes. Stephen Hawking was surely thrilled by this observation, too, before he presented his own argument why the information was preserved: at the level of microstates (or microscopic contributions to the path integral), there's no topology change and no event horizon: all these things must arise from the sum over many microstates or contributions (but Hawking wasn't terribly specific about the way how the approximate horizon actually arises!).

I still find it conceivable that a few years ago, when he gave up the bet, Hawking has actually understood something about the information loss paradox that no one else has. But I wish I knew exactly what it was and what is the complete picture.

My universal black hole entropy derivation

In my favorite idiosyncratic universal derivation of the Bekenstein-Hawking (or Wald's) formula, one can obtain the whole black hole entropy from some topology change of the (Euclidean) spacetime. Imagine that you switch from a description where the microstates can be distinguished to an effective description where they can't. This procedure modifies the topology near the event horizon. It caps and/or glues each point at the event horizon: the procedure is similar to in Juan's and Mark's trace over the microstates.

With the deficit angle "2.pi" removed from every point of the event horizon, the Einstein-Hilbert action gets shifted by "2.2.pi/16.pi.G" integrated over the horizon area, i.e. by "A/4G". The path integral gets rescaled by "exp(A/4G)" which can be interpreted as the result of a trace over the "unmeasurable", macroscopically indistinguishable microstates (i.e. indistinguishable in the effective description). That means that their number must be close to "exp(A/4G)", giving us the right entropy.

The argument can actually be generalized to the whole Wald's formula, including the higher-derivative terms in the gravitational action.

Similar issues have been relevant for various discussions of quantum gravity of eternal inflation: does the complementarity principle apply to the bubbles?

Large entanglement must be possible

Mark says that if a gravitational description with a large enough geometry emerges, the quantum mechanical system must be able to develop highly entangled low-energy states. Well, that's surely the case. We essentially require the CFT to have a large central charge and related criteria in order to have a lot of states with low energies (normalized as the dimensions of the corresponding operators in the AdS/CFT context).

But does it tell us anything else? What does it really mean to require that these states may be highly entangled? This is the part of the story I don't understand. Isn't entanglement a universal skill of all quantum systems, a construction that can be used for any Hilbert space? In other words, shouldn't your ability to construct "highly entangled states", whatever it exactly means, depend on the density of states only?

After all, entanglement doesn't care about the interpretation of the states, does it? Moreover, you shouldn't really start with any a priori interpretation of the states in your CFT or a similar theory because an interpretation - the holographic geometric one - is your goal, isn't it?

So I agree that the entanglement may be there and that there are many states whenever new holographic dimensions emerge: but I don't quite understand what is the actual role played by the entanglement.

Pure-mixed maps

On page 3 (PDF: 4 of 30), Mark offers a picture that argues that pure states of a CFT in AdS/CFT may be interpreted as pure states in the full Poincaré patch but also as mixed states in smaller causal patches inside the AdS space. Well, the first statement is rather uncontroversial, except for possible issues about the normalizability of the states (which might however be a damn important technicality for a proper discussion of the relation between different patches!).

The second statement, involving the mixed states, is much more controversial. At least, the nature of the map is controversial. In some sense, one can always calculate a mixed state for a region by tracing over all the degrees of freedom outside the region. At least at the moral level, this is the case. But is such an operation "canonical"? Can we possibly learn something out of it?

In other words, is it legitimate to expect an exact theory of quantum gravity for limited regions of spacetime - or at least for those with the null boundaries of some kind? The answer can be Yes and No. But I have some problems to understand what the Yes answer could mean.

After all, whenever you deal with density matrices (or with tracing over some degrees of freedom, which is a mundane way to obtain density matrices), you talk about situations with incomplete knowledge. A density matrix is no "objective state of affairs". It is a description of the system you're interested in - one in which you assume a certain kind of "ignorance" about everything else.

If you're only interested in a smaller system - e.g. events in a spacetime region - you're always allowed to ignore the rest. So a density matrix for this region calculated from a pure (or mixed) state of a bigger spacetime does the same job.

But isn't it true that you must always be able to see that the world continues even after you cross the null boundaries (either in the past, or in the future) and that your ignorance - i.e. the very reason why you used density matrices - is just your "psychological" problem (and a mathematical trick to overcome the ignorance), not an objective feature of reality?

You may slice your spacetime by lightlike slices and all physical questions depend on the way how objects get from one slice to the next one (think about the light cone gauge). The physical systems on the opposite sides of a null slice may be entangled or related but they're also slightly different, because of the evolution in the other lightlike direction. All the dynamics is hidden in this dependence on the side, so dynamics can't be just about the entanglement.

Spacetime vs worldsheet and flops

It seems that Mark is intrigued by similar fascinating qualitative properties of our functioning descriptions of quantum gravity and sees a similar new general story as your humble correspondent does. But I am afraid that he hasn't still managed to write down the crisp answer that could be followed to answer all the questions about the origin of spacetime, its connectedness, entanglement, complementarity, and so on. And neither have I.

Maybe David Berenstein will? He claims to be completing a super duper important paper. ;-)

There is one more point that Mark doesn't discuss: the application of these ideas to worldsheets. After all, the stringy worldsheet is a consistent (two-dimensional) theory of quantum gravity, so many statements about quantum gravity should apply to the worldsheet, too.

If you open Polchinski's book, volume I, around Figure 9.7, you will see that higher-genus Riemann surfaces can be obtained by gluing two lower-genus surfaces that are connected with a tube. The tube may be represented by the sum of insertions of a complete set of operators at both places.

The similarity with Maldacena's rule for the eternal black hole is obvious: after all, we're solving a problem from the same universality class. But whatever Mark's ideas about the role of entanglement are, do they apply to this picture of the glued worldsheets?

Another question is whether the entanglement of two disconnected portions of spacetime may apply to some "ordinary" types of topology change. I have been thinking about a possible new kind of instability of the non-supersymmetric landscape - a decay of Calabi-Yau spaces with fluxes and branes into pieces (that would guarantee that only the "simplest" compactifications are viable). The entanglement story would clearly apply here, if this decay is possible at all.

But can't the gluing tubes have a different topology than a sphere times an interval? Cannot similar traces - and entangled states - be used to interpret the new branch in the case of well-known spatial topology-changing processes such as the flops and the conifold transitions?

Can Mark answer the question whether a stringy spacetime can actually decay into pieces, and/or absorb a new component that has been disconnected so far? Shouldn't baby universes remain forbidden? Is the process just "infinitely long", as measured by the logarithm of the "length over thickness" of the necks?

Summary

While we know many things and we can smell the flavor of many intriguing ideas, the full general story of the emergent spacetime patches remains an enigma. It may be helpful if every physicist with at least 500 citations for papers linked to holography or microstates in quantum gravity tried to write a preprint with her or his own attack on these questions, just like Mark did.

Maybe, the Nth paper from the sequence, with a finite N, could contain the (almost) complete answer. ;-)

1. Note to all the math fans out there:

That construct looks like Rene Thom's cobordism theory, which was applied by Milnor to find the (equivalence classes) of n-manifolds that can admit a (global) homeomorphism - but not a diffeomorphism - structure with the n-spheres

[The lowest n is seven]

2. Stephen Paul KingJun 6, 2013, 4:37:00 AM

ISTM that the gluing of pairs of space-times to each other is equivalent to the computation of a diffeomorphism between them.

3. Brian G ValentineJun 6, 2013, 7:10:00 AM

If they are manifolds, and dimension n <7, the diffeomorphisms so constructed are in fact homeomorphisms.

4. Stephen Paul KingJun 6, 2013, 7:52:00 AM

Would there be any condition where the space-times being considered here would not be manifolds?

5. Brian G ValentineJun 6, 2013, 1:22:00 PM

Insufficient matter or energy present to close it, for one.

6. Stephen Paul KingJun 6, 2013, 4:52:00 PM

Could we just borrow the energy required from the vacuum?

7. Brian G ValentineJun 6, 2013, 6:30:00 PM

I honestly don't know. It's easier for me to contemplate the phase space that is a Banach space that give rise to these operations so that the unit ball in it is not compact at all!

8. Stephen Paul KingJun 6, 2013, 6:38:00 PM

Ummmm, maybe we should stick to the minimum that is required to represent a finite pair of "disconnected portions of spacetime".