Wednesday, February 13, 2019

Matrix theory: objects' entanglement entropy from local curvature tensor

I want to mention two papers that were released today. A Czech one and an Armenian one. In the Czech paper,
Hierarchy and decoupling,
Michal Malinský (senior co-author) and Matěj Hudec (also from a building where I spent a significant part of my undergrad years) exploit the new relaxed atmosphere in which everyone can write things about naturalness that would be agreed to be very dumb just some years ago. ;-) OK, so they don't see a problem with the unnaturalness of the Higgs potential in the Standard Model.

Harvey Mudd College, CA

If they nicely ban all the high-energy parameters and efforts to express physics as their functions, they may apply the perturbation theory to prove things like\[

m_H^2 \sim \lambda v^2

\] to all orders. The Higgs mass is always linked to the Higgs vev and no one can damage this relationship, assuming that you ban all the players that could damage it. ;-) OK, it's nice, I am probably missing something but their claim seems vacuous or circular. Of course if you avoid studying the dependence of the theory on the more fundamental parameters, e.g. the parameters of a quantum field theory expressed relatively to a high energy scale, you won't see a problematic unnatural dependence or fine-tuning. But such a ban of the high-energy independent parameters is tantamount to the denial of reductionism.

I believe them that they don't have a psychological problem with naturalness of the Higgs potential but I still have one.

That was a hep-ph paper. On hep-th, I regularly search for the words "string", "entan", "swamp", and "matrix" (although the list is sometimes undergoing revisions), not to overlook some papers whose existence should be known to me. So today, "matrix" and "entan" converged to the same paper by an author whom I am fortunate to know, Vače Sahakian (or Vatche Սահակեան, if you find it more comprehensible):
On a new relation between entanglement and geometry from M(atrix) theory
He has sent the preprint from a muddy college in California which might immediately become one of the interesting places in fundamental physics. ;-)

OK, Vače assumes we have two objects in our beloved BFSS matrix model which are, as the matrix paradigm dictates, described by a block diagonal matrix. The upper left block describes the structure of the first object, the lower right block describes the second object, and the off-diagonal (generally rectangular) blocks are almost zero but these degrees of freedom are responsible for the interactions between the two objects.

Vače allows the non-center-of-mass degrees of freedom of both blocks to optimize to the situation, he sort of traces over them, and wants to calculate the entanglement entropy of the center-of-mass degrees of freedom (which are the coefficients in front of the two blocks' identity matrices). He finds out that the von Neumann entropy depends of the derivatives of the gravitational potential, \(\partial_i \partial_j V\).

By a process of "covariantization", he translates the gravitational potential and its derivatives to the variables that are more natural in Einstein's general relativity, such as the Riemann tensor, which leads him to a somewhat hypothetical form of the entanglement entropy \[

S_{ent} = -\gamma^2 {\rm Tr} \zav { \frac{{\mathcal R}^2}{4} \ln \frac{{\mathcal R}^2}{4} }

\] which is finite, concise, and elegant. Here, the \({\mathcal R}\) object is the Riemann tensor contracted with some expressions (partly involving matrices) that are either necessary for kinematic or geometric reasons or because of the embedding into the matrix model.

Aside from the finiteness, conciseness, and elegance, I still don't understand why this particular – not quite trivial – form of the result should make us happy or why it should look trustworthy or confirming some expectations that may be obtained by independent methods. "Something log something" is the usual form of the von Neumann entropy which has terms like \(-p_i\ln p_i\), as you know, but the probabilities should be replaced by a squared Riemann tensor. If it is true, I don't know what it means.

At the end, if a result like that were right, it could be possible to determine some entropy of black holes or wormholes or holographic deviations from locality (from the independence of regions) or something like that in Matrix theory but I have no idea why. It may be because I don't have a sufficient intuitive understanding of the entanglement entropy in general. At any rate, this is a kind of a combination of Matrix theory and the entanglement-is-glue duality that should be studied by many more people than one Vače Sahakian.

Incidentally, after a 7-month-long hiatus, Matt Strassler wrote a blog post about their somewhat innovative search for dimuon resonances.

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