Friday, January 11, 2019

Quantum gravity from self-collisions of the configuration space

One of the seemingly quirky ideas that are waiting to revolutionize theoretical physics – at least your humble correspondent feels it should be the case – is the unified treatment of world sheets + world volumes + other auxiliary spaces, spacetimes, and configuration spaces. All these spaces may be used to encode the information about physical systems. All these spaces have some quantum gravity on them (I will discuss the last one). They are mapped to each other in various ways.

There isn't a self-evident metaphysical difference between them which is why I believe that the truly exact formulation of quantum gravity doesn't care whether you think that you live on the world sheets, in the spacetime, or on the configuration space. The laws of quantum gravity are certain consistency criteria that have to work on all the spaces above.

For more than 20 years, one of the reasons why I believed in the paradigm above were the Kutasov-Martinec (2,1) heterotic strings. When you combine the left-moving \(\NNN=2\) supersymmetric string theory with the normal 10-dimensional superstring, you apparently get a string theory whose target space dynamics looks like the world sheet dynamics of the superstring – or another world sheet. You may even recursively generate the (2,1) string in the target space of itself.

To say the least, I think that these observations encourage one to try to import all tricks from the world sheet to the spacetime and vice versa; and all the tricks from the spacetime to the fields' configuration space and vice versa.

Here's an exercise for you to warm up and show the point. Answer the following question:
The string world sheet seems to harbor a 2D quantum theory of gravity. Quantum gravity should imply the holographic principle. Does holography – and do the holographic/entropy bounds – apply to the world sheet?
Because I just said that the same laws should apply to the spacetime and the world sheet, it's not hard to guess that my prejudice is "Yes". Some kind of holography should apply to the perturbative string world sheet, too. Are there black holes over there? What do the holographic bounds say?

The normal entropy bounds says that the maximum entropy in the volume surrounded by the area \(A_{D-2}\) should be\[

S \leq \frac{A_{D-2}}{4G}.

\] What if the "spacetime" is two-dimensional? The surface is just a zero-dimensional sphere, i.e. two points. Well, the zero-dimensional area \(A_{D-2}\) should be just two – it's dimensionless in this case. (Or is it one?) But you still have the denominator, there's still the factor of \(1/4G\). What is (the equally dimensionless, in this case) Newton's constant on the perturbative string world sheet?

Well, I think that \(1/16\pi G\) should still be the coefficient in terms of the Einstein-Hilbert action,\[

S_{EH} = \frac{1}{16\pi G} \int d^2 \sigma \sqrt{|h|} R_{(2)}.

\] Do we have this term on the world sheet? We do. In two spacetime dimensions, it's a topological invariant, however, so it counts the Euler characteristic \(\chi=2-2h\). Do we use this term in the world sheet? We do, we use it to adjust the string coupling. The coefficient of \(R\) in this action is basically the dilaton \(\Phi\), the logarithm of the desired string coupling constant \(\log g_s\)! So roughly speaking,\[

-\log g_s = \frac{1}{16\pi G_{2D}}.

\] Please fix the errors in the factors of two. I hope the sign is as indicated. When the strings are weakly coupled, the logarithm should be a large negative number – and the entropy should be proportional to that. So the holographic entropy bound should say something like\[

S \leq - C \log g_s.

\] The information that you may encode in one string is infinite at \(g_s=0\) – the string may be arbitrarily excited and there's the whole infinite Hagedorn tower. But for a nonzero coupling, there's an effective cutoff here. A piece of the fundamental string – between the two points, parts of the boundary – should have a finite entropy that is bounded by \(\log (1/g_s)\).

The number of states indicated by the entropy is therefore\[

\exp(S) \leq \frac{1}{g_s^C}.

\] After this exponentiation, you may see that the previously neglected coefficient \(C\) is pretty influential and one should better know what this numerical constant is. If you link this \(\exp(S)\) to the degeneracy of excited string states, you may see that the holographic bound effectively tells you that only levels up to a level comparable to \(1/g_s^C\) are meaningful.

This isn't quite a new shocking law. At nonzero \(g_s\), the excited string levels are unstable and decay, and they effectively cease to exist. So I guess that this holographic bound could be another way to derive that too excited levels of the string are "unreal" for a small but nonzero string coupling \(g_s\). Great. I hope that you can complete this to a convincing story. You're invited to do so. But in your paper, you're supposed to acknowledge your sources – instead of following the unethical recommendations to steal by D.H.

But let me get to the ideas announced in the title.

Quantum field theory is subtle, has forced the people to learn things like the renormalization, the renormalization group, critical behavior, and many other things. However, in some sense, we think it's been mastered. We basically feel that we know what quantum field theories are. We think that we know a representative subset of them. We think that we know that the scale-invariant ones are a particularly important class. And we know how other are obtained by perturbing the scale-invariant ones.

Quantum gravity is much harder. It looks deceptively similar to quantum field theories. And theories of quantum gravity may apparently be approximated by effective quantum field theories. But they're different. The spacetime is dynamical and should even be allowed to change the topology. And the theory therefore loses the usual notions of locality. Quantum gravity has to be non-local to one extent or another, at least in some variables. It's needed for the information to be liberated from the evaporating black hole's interior. It's needed for other things.

But how does exactly a quantum field theory – an effective field theory – ceases to be a local theory and how does it get deformed to a theory of quantum gravity? Decades ago, people would naively think that the non-locality is just some effect that kicks in at very short distances comparable to the Planck length where the quantum gravitational theory ceases to be local.

However, I believe that we know better today – or we feel better. The non-locality of quantum gravity isn't confined to the Planckian distances. The reconstruction of the information from the Hawking radiation is hard but it indicates a conspiracy between all (or a big fraction) of the degrees of freedom near the event horizon. Generic observables – those that may be measured by cheap apparatuses in practice – think that the probability of a non-local effect drops to tiny values unless the distances are Planckian. But in principle, some non-local relationships have to be able to operate over arbitrarily long distances.

Another reason to think in this way is the topology change and the presence of wormholes. A wormhole may connect arbitarily distant places, right? And in quantum gravity, within John Wheeler's "quantum foam", there should be some probability that it actually happens over longer distances. What happens with the dying locality when the formerly "effective local quantum field theory" suddenly starts to be "able" to produce wormholes with their paired throats? Do you know the answer? And if you don't, have you ever asked this obvious question?

My point is that the quantum field theories – at least those admitting a path integral – have some configuration space for the fields. And the disappearing non-locality should have some at least qualitative description on that configuration space, too. What is it?

Let's borrow some ideas from the world sheet. The configuration space of the quantum field theory on the world sheet is the spacetime, or some copies of it for each point on the world sheet. The world sheet carries a quantum theory of gravity and one of the consequences of the gravity is that the topology of the world sheet is allowed to change – in other words, strings may split and join.

Now we discuss the spacetime which also carries a quantum theory of gravity so the spacetime topology should also be variable – wormholes may be created and destroyed in that spacetime. It should be possible to create wormholes – the newborn wormholes must be short, however. Can we see what happens on the configuration space of the effective quantum field theory when a wormhole is born?

According to the ER-EPR correspondence, an Einstein-Rosen bridge is equivalent to the maximally entangled pair of black holes. Roughly speaking, you sum \(\ket i \otimes \ket i\) products over all black hole microstates \(\ket i\) in a certain class. But if there marginally exists a quantum field theory that describes the microstates (think about some fuzzball geometries), the tensor product structure indicates a correlation between the fields near the two throats, too.

So at least in some language that is marginally applicable, the existence of a wormhole should be equivalent to the agreement between the profiles of the quantum fields in two regions. When two people have the same brains and think about the same things, they should also be "geometrically close to each other" through some wormhole.

For those reasons, I think that when a quantum field theory "starts to gravitate", something special must happen when fields in two regions almost exactly coincide with each other. When \(x^\mu(\sigma)\) on one string happens to have the same value (spacetime position) as the same quantity on another string, the strings may meet in the spacetime and split or join – rearrange the topology of the string. I do think it's possible to describe the configuration space for 1+1 split strings or 1 merged string in terms of "quantum mechanics on graphs", lines with crossroads, and I think that matrix string theory adds some gauge-field degrees of freedom to turn this vision into a full story. This should also be possible for the agreement between the fields in two regions of a gravitating theory in the spacetime.

OK, we may be getting to an answer (if not "the answer") to the question why the locality of the quantum field theory may start to disappear when gravity is turned on. It disappears because the fields \(\Phi_i(x,y,z,t)\) at two different points cease to be fully independent variables. When these configurations – not just at two points but, optimally, at two larger regions – are very close to each other (informally: when two very similar "localized objects" exist at two places of the spacetime), the spacetime is allowed to change its topology and all these states with "copies of regions" admit some simpler description.

The stronger the Newton's gravitational constant is, the less precisely you have to adjust the fields in the two regions for the theory to know that the topology change becomes possible. The wormholes become more common. You may obviously make the (dimensionful) gravitational constant stronger by focusing on the physics at shorter distances – in that way, you should see Wheeler's quantum foam with your naked eyes.

The birth of a wormhole is a topology-changing event in the spacetime but it should also be possible to describe it as a collision in the configuration space – much like the joining of strings is equivalent to a spacetime collision. It means that the hint for the further research is to study the possible changes of the topology of the infinite-dimensional configuration spaces of field theories with the gravity turned on. There should be some shortcuts or identifications on that configuration space as well and when counted correctly, they could also explain why the number of degrees of freedom cease to be volume-extensive.

These collisions could even be derivable if you parameterized the "no longer local" field theory with gravity turned on in terms of vertex-like operators that create black hole microstates. I believe that the best way to deal with the black hole interior is to cut it entirely. Imagine that you use coordinates for a black hole where the whole horizon is shrunk to the origin of \(\RR^3\).

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