Ok, so yesterday I put here the part 1 of the abstract about holography. Today I will go into part 2 of this idea. This relates quite a lot with something called quantum entanglement. In principle this is a purely statistical and topological effect. It means essentially that because one cannot in general separate a density matrix in any factorization of its algebra it may happen that in some cases, as Schrodinger would put it, speaking about “the whole” cannot be done while consistently speaking about the parts. Entanglement is in some sense a measure of how much we would lose in information when speaking about the parts distinctly in this way without considering “the whole”. The main observation however is that the factorization of the algebra is up to us. Of course, experimentally that would lead to changing the whole project but in purely mathematical sense there is always another way to factorize the algebra of the density matrices. In some cases this leads to more entanglement, in some cases to less but in principle it cannot be reduced to no entanglement at all, and this is a quantum property. Now, after having this in mind we also have to understand what critical points or systems at criticality mean: in principle they are described using some conformal theory and they are characterized by the divergence of some correlation length. In principle this means we have long range effects and this can in principle harm our area laws. So, the area laws that follow from the holographic principle are violated in fermion systems when long correlations are included but also some types of entanglement may produce violation of the area laws. Now we have to be careful at the terms used. While for me “area law with logarithmic divergence” is NOT an area law, for most of the authors that want to publish something this is “a weakly violated area law”… now, about the character of a logarithmic divergence one can discuss a bit… it is “weak” in some sense and it can be eliminated following some renormalization criteria when it appears in perturbative Quantum Field Theories. However, when it appears in the expression of an area law for entropy I don’t see why one should go towards quantum field theory when the whole idea was to avoid it because of its poor UV behavior and anyhow one relates terms defined differently in different areas of physics just because they involve “logarithmic divergencies”… Using the old proverb that not everything that flies is a bird one may be skeptical…

Now, there was an emphasis on D0-branes in the paper I quoted yesterday. Why D0-branes? Well, they are no strings, they are point like objects or lines in space-time… why these? Why of course, because area laws have been proved to be exact only in 1 dimensional systems so the clear choice of proving the area law is on a system that you are sure it possesses an area law. Chance makes it that in more than 1 dimension most of the ideas fail pretty badly… well, it doesn’t fail for free or almost free bosons, this is why our friends in Japan chose carefully to disregard anything related to fermionic statistics in their “geometric” approach. Whenever one has a gapped, local model, and hence a length scale provided by the correlation length, we are told to believe an area law may be something plausible. Fact is, it is almost never so mainly because we have also interactions between bosons and the temperature is generally not 0 (Zero) Kelvin…

So, what did we learn today? Mainly that the “area laws” implied by the holographic principle are far from being “universal” as claimed by this or that pop-science newspaper (science and nature included). They are mostly related to the presence of some kind of “horizon” and are representation dependent. They are not specific to black holes. Any system that has a well defined boundary that plays the role of a “horizon” in the sense of we just ignoring the inside, will have some sort of area law in it but this is due to a choice of a specific factorization of the algebra of states and the therein resulting entanglement. Area laws fail dramatically whenever the “true nature” comes into play i.e. interactions, long distance correlations, entanglement etc. The holographic principle is again a choice of a framework and nothing more. Its predictive power is the same as the power of (co)bordisms in topology but with a far narrower applicability in physics.