Monday, September 28, 2015

No-transmission principle, antiholographic LQG

I want to mention two recent conceptual, not too technical papers about the connection of holography to seemingly local theories.

Today, Netta Engelhardt and Gary T. Horowitz released their paper
Holographic Consequences of a No Transmission Principle
which argues that whenever a gravitational spacetime dynamics is described by quantum field theories, their background spacetimes have to overlap for them to be able to influence each other and transfer energy. This assumption seems to imply
  1. ban on certain bounces
  2. ban on the resolution of certain black hole singularities
  3. ban on traversable wormholes
Here, the third ban seems to be right and desirable and the first two are "plausible" so if you judged their principle by its selected consequences, you could be tempted to say that it should be a correct one. However, this is not a logically valid method to decide about the validity of a claim. If a claim implies some correct implications, it doesn't mean that the original claim is correct, too. ;-)

General principles of this sort often turn out to be very important. But is their no-transmission principle actually right? They claim that whenever you find a holographic description of some gravitational situation and it is composed of disconnected non-gravitational boundaries, there has to exist some (conformal) transformation from one of these non-gravitational spacetimes to others (to some regions of the other non-gravitational world volumes) for any interaction to be possible at all.

More than 20 years ago, as soon as I began to learn string theory at a technical level, I became enthusiastic about a very similar hypothesis of mine, although one in a more limited context. I believed that when two strings were joining, i.e. "O" and "O" (two closed strings) become an "8", if you understand me, the right Hilbert space shouldn't actually contain separate basis vectors for the pair of strings, "OO", and for a single complex string, "8".

Instead, I believed, the Hilbert space should have a more complicated structure analogous to "quantum mechanics on graphs". There is a point at which "OO" looks the same as an "8" – and there should be a single vector in the Hilbert space that represents both of these configurations at the same time. Locally, this may be described as the identity between a degenerate hyperbola and two lines that happen to intersect.

Locally in the spacetime, two closed strings may be described as straight lines given by\[

x=0, \,\,z=a \quad {\rm and} \quad y=0, \,\,z=-a.

\] They're two lines in the 3D space whose directions are orthogonal but which only intersect for \(a=0\), OK? Meanwhile, a hyperbola may be given by\[

z=0, \,\,xy=b.

\] This hyperbola – which has two components – is what the complicated single string "8" locally looks like. Funnily, the pair of lines for \(a=0\) and the hyperbola for \(b=0\) give you the same thing – a pair of intersecting lines. In any string field theory, we still describe these two states as two different states: one may always distinguish states with one string excitation and a state with two string excitations. However, I believed that the points with \(a=0\) and \(b=0\) correspond to the same basis vector in the Hilbert space. The continuous basis vectors of this Hilbert space should look like a cross and the intersection in the middle of the cross should only represent one basis vector and not two. In a quantum mechanical wave function, there should be some more general boundary conditions at the intersection – the sum of the derivatives \(\psi'\) calculated in all the four directions should probably vanish.

This belief of mine – unrealized in any known formalism in string theory – helped me to discover matrix string theory a few years later. Matrix string theory does naturally incorporate the single-string and double-string states into the same Hilbert space of a \(U(N)\) gauge theory. One may continuously change the configurations and connect one-string states with two-string states. They're no longer sharply separated. They no longer live in two different components of the Hilbert space. In this sense, matrix string theory did prove my older but more vague principle.

However, you may see that the "overlap" between the Hilbert spaces – which I postulated to be necessary for the stringy joining/splitting interactions – was very small. Only \(a=0\) and \(b=0\) points of the "two parts" of the Hilbert space overlapped. Engelhardt and Horowitz are proposing that a much bigger overlap is needed for interactions.

Despite the plausible implications, I have some doubts about their no-transmission principle. It just doesn't seem to agree with how I understand the glue-entanglement and ER-EPR correspondences. I think that the point is that there may exist entangled states in the two a priori disconnected CFT world volumes and it is this entanglement that creates the "bridges" between the boundaries through the gravitational bulk. But the entanglement doesn't mean that the two CFTs become generally interacting or overlapping. They may still be described by independent degrees of freedom. The two CFTs may even be non-isomorphic which seems possible if the two asymptotic regions have different vevs. But in the vicinity of an entangled state of the two QFTs, the effective degrees of freedom needed to describe the two QFTs basically cease to be independent.

Maybe I misunderstand something but it seems to me that Horowitz et al. are insisting on some "more kinematic" overlap which isn't necessary for the gravitational glue, I think.

LQG contradicts holography

The second paper I want to mention was written by less famous authors, Ozan Sargın and Mir Faizal, and it has a large number of typos etc. But the main conclusions of their paper
Violation of the Holographic Principle in the Loop Quantum Gravity
seem to be self-evidently correct. Loop quantum gravity – and any other approach that starts to build the quantum gravitating spacetime as a local LEGO-like construction ("polymer quantization" – basically a lattice that isn't regular in general) – obviously does violate the holographic principle. Such constructions assume \(\O(1)\) LEGO pieces per Planck volume of the bulk spacetime so it is not hard to estimate that the entropy will be proportional to the volume. But the holographic principle demands otherwise: the maximum entropy should be proportional to the surface area.

The holographic principle does imply that no old-fashioned "local" description of the regions in the bulk spacetime may exist. Any exact description must blur the locality properties of regions – they are emergent (especially the locality in the new holographic dimension must be emergent). The polymer quantization is more naive than a theory of an average kid in the kindergarten so it obviously doesn't have what is needed for holography.

Their paper making the same conclusion contains lots of rather complicated formulae. It seems to me that they actually ignore the leading-order behavior and they want to show that some corrections to the entropy behave differently than the holographic principle demands. The details of their paper are not quite clear but the conclusion is clear and obviously right.

A writer has argued that the conclusion can't be right because it makes the writer feel sick. Well, if you read that blog post carefully, you will see that the text contains exactly three arguments:
  1. The author is a vegan of a sort and has some digestive problems
  2. "LQG is a discretization approach. It reduces the number of states, it doesn’t increase them."
  3. "So if Loop Quantum Gravity would violate the Holographic Principle that would be a pretty big deal, making the theory inconsistent with all that’s known about black hole physics!"
Concerning (1), I think that most readers understand that the writer's unhealthy diet and the problems resulting from that can't be considered a valid argument.

Concerning (2), it is nice that a discretization approach reduces the number of degrees of freedom but it doesn't reduce them enough. On the contrary, a discretization approach that demands \(\O(1)\) degrees of freedom per Planck volume makes it more unavoidable, and not less unavoidable, that the total number of degrees of freedom will scale with the volume which holography forbids.

Concerning (3), well, that's not a new fact. Every credible physicist knows that loop quantum gravity contradicts everything we know about the quantum dynamics of black holes. It contradicts pretty much everything else that is known about quantum gravity, too.

Her actual argument, one that she didn't write explicitly, is the following one:
4. Any paper making it more obvious that loop quantum gravity is pure and worthless garbage is inconvenient because just like some other third-class would-be physicists, she has been associated with this garbage. And inconvenient insights should be attacked on the Internet.
However, the writer's links with loop quantum gravity don't make these subpar opinions about science any better. And the scientific attitude leads one to attack demonstrably wrong claims, such as loop quantum gravity itself, and not the inconvenient ones.

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