Random Matrices in Three Short Stories – III

This is the third and final part of the series. For those who haven’t already, going through the first two posts, which can be found here and here, will be a good idea.

Three

Why do we want a quantum theory of gravity? We just want it, okay?

Hold on, this isn’t me. This was the American theoretical physicist John Preskill at a conference earlier this year [1]. And though, in almost all probability, he was trying to be funny, this does give an idea about how difficult (and often, frustrating) it is for theoretical physicists to answer why-do-we-want-this or what-use-doing-that questions. (By the way, apart from his seminal contributions to the fields of quantum information and quantum gravity, Preskill is also famous for winning a black hole bet against Stephen Hawking which Hawking conceded by offering him a baseball encyclopedia.)

But something about the notion of universality in random matrix theory before I go on to talk about the theory of quantum gravity, cosmological inflation, black holes, wormholes and time travel. (Okay, I was kidding about the last two – no wormholes and time travel here.)

Large matrices with random entries have some very intricate (and interesting) statistical properties. And the reason they come in so handy in our attempts to answer questions of different sorts is the applicability of the statistical laws of random matrix ensembles to all those systems which have the same symmetries as those of the ensemble.

These statistical laws often involve eigenvalues of the matrices. Eigenvalues are certain quantities associated with matrices. In fact, all square matrices (arrays of numbers with the same number rows and columns) can be characterised by their eigenvalues. And in most cases, computing the eigenvalues is not a very difficult task.

det(H-\lambda{I}) = 0
For a matrix H, all the possible values of λ satisfying the above equation are its eigenvalues. I is a square matrix the size of H with zeros along the diagonal going from left to right. And det() represents computing the determinant which again is a number associated with a matrix.

For a set of random numbers, it is very natural and useful to talk about the probability distribution – how probable the occurrence of each of the numbers is. Now, if your matrices have random elements, its eigenvalues will be random as well, which makes it useful to talk about the eigenvalue distributions of the ensembles.

These distributions in random matrix theory are universal in the sense that they don’t depend on the underlying structures as long as they have a common overall symmetry. In the context of a physical system, these eigenvalues correspond to the energy levels of the system and, in essence, contain the information about its dynamical properties. Superficially, this is what a lot of random matrix analyses is about. The energy spectra of systems which are difficult to interpret otherwise find meaning in the language of random matrix ensembles.

But understanding the underlying dynamics of many systems is not easy. And as we have come to realise in the case of black holes – it is certainly not.

Black holes are interesting; though calling them interesting might be an understatement. They are formed in a number of different ways all over the universe. Within the framework of the theory of general relativity, one can, in quite a straightforward manner, show that a black hole has a gravitational pull so huge that nothing can escape it, not even light.

But everything is not that straightforward. Though general relativity explains the force of gravitation, we believe our universe to be inherently quantum mechanical, that is, we expect every object in the universe to follow the laws of quantum theory. And taking quantum mechanical laws into consideration, one can show (as Hawking did for the first time in the early 1970s) that black holes radiate stuff [2]. Now, this may seem puzzling; in fact, it is puzzling. But what this radiation also indicates is that black holes are thermal objects – they have thermodynamic properties (say, temperature, for example) in a manner similar to how your daily cup of coffee does.

It is evident that black holes demand a better understanding than the one at present. And this can possibly be achieved by constructing a unified framework incorporating both gravity and quantum mechanical principles. This is somewhere random matrix analysis turns out to be useful – understanding energy spectra of black holes.

Depending on the theoretical framework in which the calculations are being done, black holes can be studied using different models. In principle, the details of the energy spectra can be worked out for each of these models. But as it turns out, not all black hole models are soluble. The trick is to then use random matrix models which can possibly mimic the expected properties.

Black holes have posed some of the most interesting challenges to those working in physics for the last hundred years. They have also motivated a quest for quantum gravity. However, a theory of quantum gravity will possibly also explain many other mysteries. One of them is cosmological inflation. Based on a large amount of astronomical data, it has been conjectured that our universe underwent a phase of extremely rapid expansion for a small fraction of second just after the big bang. This conjecture also explains the origin of the large scale structures in the universe pretty well. However, what is missing is a concrete theoretical underpinning of inflation itself. Among various proposals to explain inflation, there are a few which employ random matrix techniques as well [3]. But again, a complete understanding of inflation, like that of black holes, still belongs to the large set of open problems waiting to be solved!

References

[1] John Preskill, Quantum Information and Spacetime (I)https://youtu.be/td1fz5NLjQs, Tutorial at the 20th Annual Conference on Quantum Information Processing, 2017.
[2] Leonard Susskind, Black Holes and the Information Paradox, Scientific American, April 1997.
[3] M.C. David Marsh, Liam McAllister, Enrico Pajer and Timm Wrase, Charting an Inflationary Landscape with Random Matrix Theory, JCAP 11​, 2013, ​ [arXiv:hep-th/1307.3559]​.

Thanks are due to Divya Singh and Ramesh Chandra for providing essential feedback for this series of posts on Random Matrices.

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