No, Loop Quantum Gravity has not been shown to violate the Holographic Principle

Didn't fly.

Tl;dr: The claim in the paper is just wrong. Read on if you want to know why it matters.

Several people asked me for comments on a recent paper that appeared on the arxiv, “Violation of the Holographic Principle in the Loop Quantum Gravity” by Ozan Sargın and Mir Faizal. We have met Mir Faizal before; he is the one who explained that the LHC would make contact to parallel universes [spoiler alert: it won’t]. Now, I have recently decided to adapt a strict diet of intellectual veganism: I’ll refuse to read anything produced by making science suffer. So I wouldn’t normally have touched the paper, not even with a fork. But since you asked, I gave it a look.

The claim in the paper is that Loop Quantum Gravity (LQG), the most popular approach to quantum gravity after string theory, must be wrong because it violates the Holographic Principle. The Holographic Principle requires that the number of different states inside a volume is bounded by the surface of the volume. That sounds like a rather innocuous and academic constraint, but once you start thinking about it it’s totally mindboggling.

All our intuition tells us that the number of different states in a volume is bounded by the volume, not the surface. Try stuffing the Legos back into your kid’s toy box, and you will think it’s the volume that bounds what you can cram inside. But the Holographic Principle says that this is only approximately so. If you would try to pack more and more, smaller and smaller Legos into the box, you would eventually fail to get anything more inside. And if you would measure what bounds the success of your stuffing of the tiniest Legos, it would be the surface area of the box. In more detail, the amount of different states has to be less then a quarter of the surface area measured in Planck units. That’s a huge number and so far off our daily experience that we never notice this limit. What we notice in practice is only the bound by the volume.

The Holographic Principle is a consequence of black hole physics, which does not depend on the details of quantizing gravity, and it is therefore generally expected that the entropy bound must be obeyed by all approaches to quantum gravity.

Physicists have tried, of course, to see whether they can find a way to violate this bound. You can consider various types of systems, pack them as tightly as possible, and then calculate the number of degrees of freedom. In this, it is essential that you take into account quantum behavior, because it’s the uncertainty principle that ultimately prevents arbitrarily tight packing. In all known cases however, it was found that the system will collapse to a black hole before the bound is saturated. And black holes themselves saturate the bound. So whatever physicists tried, they only confirmed that the bound holds indeed. With every such thought-experiment, and with every failure of violating the entropy bound, they have grown more convinced that the holographic principle captures a deep truth about nature.

The only known exception that violates the holographic entropy bound are the super-entropic monster-states constructed by Hsu and collaborators. These states however are pathological in that not only will they inevitably go on to collapse to a black hole, they also must have come out of a white hole in the past. They are thus mathematically possible, but not physically realistic. (Aside: That the states come out of a white hole and vanish into a black hole also means you can’t create these super-entropic configurations by throwing in stuff from infinity, which should come as a relief to anybody who believes in the AdS/CFT correspondence.)

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!

In the paper, the authors redo the calculation for the entropy of a particular quantum system. With the usual quantization, this system obeys the holographic principle. With the quantization technique from Loop Quantum Gravity, the authors get an additional term but the system still obeys the holographic entropy bound, since the additional term is subdominant to the first. They conclude “We have demonstrated that the holographic principle is violated due to the effects coming from LQG.” It’s a plain non-sequitur.

I suspect that the authors mistook the maximum entropy of the quantum system under consideration, previously calculated by ‘t Hooft, for the holographic bound. This is strange because in the introduction they have the correct definition for the holographic bound. Besides this, the claim that in LQG it should be more difficult to obey the holographic bound is highly implausible to begin with. LQG is a discretization approach. It reduces the number of states, it doesn’t increase them. Clearly, if you go down to the discretization scale, the number of states should drop to zero. This makes me think that not only did the authors misinterpret the result, they probably also got the sign of the additional term wrong.

(To prevent confusion, please note that in the paper they calculated corrections to the entropy of the matter, not corrections to the black hole entropy, which would go onto the other side of the equation.)

You might get away with the impression that we have here two unfortunate researchers who were confused about some terminology, and I’m being an ass for highlighting their mistakes. And you would be right, of course, they were confused, and I’m an ass. But let me add that after having read the paper I did contact the authors and explained that their statement that the LQG violates the Holographic Principle is wrong and does not follow from their calculation. After some back and forth, they agreed with me, but refused to change anything about their paper, claiming that it’s a matter of phrasing and in their opinion it’s all okay even though it might confuse some people. And so I am posting this explanation here because then it will show up as an arxiv trackback. Just to avoid that it confuses some people.

In summary: Loop Quantum Gravity is alive and well. If you feed me papers in the future, could you please take into account my dietary preferences?