The Holographic Principle

what is the “holographic principle”? Well, it is the result of an observation related to how one can encode information inside the… “universe”… where by “universe” I mean essentially spacetime. Let me start it in a different way: One of the finest and most complicated theories nowadays is… no, not string theory… it is Quantum Field Theory. In principle quantum field theory takes advantage of linear superposition of “fields” in order to encode the quantum “substrate” of nature. “Quantum” means mainly “topological”, or, otherwise stated, whatever is accessible in a statistical sense while having to deal with the full topological structure of your problem. But one knows pretty well that a field theoretical description has redundancies. There was never a true mystery in this. Not only that this theory has redundancies (gauge redundancies) but also it does have a strange property: it generally over-counts degrees of freedom. This is also well known and gravity just makes the issue more acute. If you want to excite the degrees of freedom of a field theory in a volume of the universe you start adding energy. While adding energy, after some limit you will obtain a black hole. Now, a black hole has a horizon… a horizon is that damn thing that doesn’t allow you to access whatever is inside it. Once something goes beyond the horizon it is unaccessible to you. In order to preserve whatever made sense out of thermodynamics (which is, nota bene, again a statistical theory) you have to consider the way in which information about the in-falling matter  is encoded in the area that surrounds the region. So, information encoded on an area gives away whatever is in the “volume”… now, that doesn’t mean the volume ceases to exist. There are several pretty good arguments that tell us that whatever is on the other side of the horizon is a “volume”… Ok, call it a “belief system” because I won’t sit here to explain it to you in detail but I do believe that once passing the horizon you won’t hit the “end of the universe”… So, we have volume behind the horizon so we have information there too… If we encode it on the boundary of that region it’s ok but it is just a way of representing it. In some situations this representation is useful. Now, I told you already that string theory is a theory of “worldsheets” i.e. paths described by strings (and branes in the end) while moving in some space-time. In the string case there is no surprise one has some holographic dualities: the theory has been constructed from the very beginning such that it is “critical” i.e. it generates conformal invariance in the critical dimension on the worldsheet. So, the whole construction is fundamentally “bordant”. According to this there can be no surprise of string theory being “holographic” in one sense or another. For whoever understands topology, the “holographic principle” is just the theory of (co)-bordisms.

Now, major statements are being made by some… first, I hear things about “the universe being intrinsically holographic”… that would mean that the universe (and whatever may be in it) must obey an area law… like the entropy S going like L^{d-1} where d is the dimension of the “universe”… This is not so, and it is not so for fermions especially, where corrections to the area law in form of logarithms must be introduced (S going like L^{d-1}*Log(L) ). This means that the entropy diverges logarithmically. This is a good indication of why Density Matrix Renormalization methods CAN NOT WORK, and also a good reason why I usually don’t even consider going to conferences or summer schools on DMRG… but this is not the scope of this post. The scope here is to show why the holographic principle, while interesting from some points of view is not so “fundamental” as one might believe… also the ER-EPR conjecture is also an interesting observation but it is by no means “fundamental” or “new” in any way… at least not to someone with a decent knowledge of Morse Surgery in topology… You may see that the real problem is the fact that for fermionic systems there is a “small” issue with the area law. Unsurprisingly so, what we read in the latest over-quoted paper by some Japanese authors is that higher derivative terms in the graviton 4-point amplitude have an important role in the description of the entropy of black holes but the “supersymmetrization” of them is not “yet” “fully” understood… what means that? Well… essentially it means that if you ask for local supersymmetry of your full action you don’t really know what you get. It is not necessarily bad because it just adds some points to the idea that while you want to keep the holographic principle and its area law you have a problem with the supersymmetric partners of scalar fields and while you want to give up the supersymmetrization you end up with trivial cases… however you do it, the “theory” is incomplete… but since the authors (like all the authors nowadays) are interested only in the things they can do we see that they ignore gravitino contributions, gluino contributions and whatever cannot be calculated in a pertinent way for the paper… After bravely imposing cancellations of terms in order to respect local supersymmetry wherever they consider it fit and neglect the places where they cannot do this trick we are rapidly said that in order to have an explicit and “controlled” behavior of quantum corrections we need to consider the supergravity (low energy effective theory) of M-theory in 11 dimensions. The only problem is that the fundamental object (the membrane) of M-theory cannot be quantized (for now)… however we can obtain some sort of quantum corrections by requiring the local supersymmetry of M-theory. Now, M-theory and the original type IIA superstring theory are related via a dimensional reduction and string theorists believe that IIA is some sort of lower energy effective theory of M-theory in the sense that the effective theory of M-theory should include some aspects of the effective theory of IIA theory. In principle this is because of the “dimensional reduction” relation between the two theories…  However, we know that the most important terms that should contribute to the quantum corrections in IIA cannot be supersymmetrized… these terms should alter the contributions to the area laws that they describe and falsify any claim of “holography”. However, after ignoring almost anything fermionic (Majorans, gravitinos, etc.) nobody will notice this “detail”… and anyway nobody can do anything better either…

So, I hope I explained here pretty clearly why the last paper presented in the previous sense is technically correct but essentially pointless and with no contributions to any of the problems it raises… what a typical paper in these times…

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2 thoughts on “The Holographic Principle

  1. Pingback: Time to ditch falsifiability? | The Leather Library

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