Ted Jacobson’s deep understanding

A few weeks ago I published a post about Ted Jacobson and his deep understanding of general relativity (see here).  Jacobson proved in 1995 that Einstein equations can be derived from thermodynamic arguments as an equation of state. To get the proof, Jacobson used Raychaudhuri equation and the proportionality relation between area and entropy holding for all local acceleration horizons. This result implies that exist some fundamental quantum degrees of freedom from which Einstein equations are obtained by properly managing the corresponding partition function. To estabilish such a connection is presently not at all a trivial matter and there are a lot of people around the World trying to achieve this goal even if we lack any experimental result that could lead the way.

Today in arxiv appeared a nice paper by Ram Brustein and Merav Hadad that generalize Jacobson’s result to a wider class of gravitational theories having Einstein equations as a particular case (see here). This result appears relevant in view of the fact that a theory exploiting quantum gravity could have as a low-energy limit some kind of modified Einstein equations, containing at least coupling with matter. Anyhow, we see how vacuum of quantum field theory seems to become even more important in our understanding of behavior of space-time.

Liouville theory in the infrared limit

Today I want to report a quite interesting result that I have discussed in the comments of a preceding post of mine (see here):  2d general relativity has no confinement as a quantum field theory. 2d general relativity can be written down as

being a cosmological constant. This equation is the same as the Liouville equation

and all the problem is to find the scalar function . As you know this equation can be solved exactly. About quantum field theory for 2d gravity there is really a large body of literature due the importance of this equation. I just point out to you this paper but there is much more about.

So, if you want to study this equation in the infrared limit, you have just to take the cosmological constant going to infinity. Then, to solve this problem we have to use strong perturbation theory (or a gradient expansion) giving at the leading order the equation for the propagator

and this equation can be solved exactly:

being and two arbitrary constants that may depend on the spatial coordinate. This Green function solves for the propagator after we have rescaled time by and the constant as . What can we learn from it? We see that this is not a periodic function and so it cannot be expressed through a Fourier series. This implies that the quantum spectrum is not discrete and so the theory has no bound states in the infrared limit of an increasingly large cosmological constant. This is a substantial difference with respect to a quartic scalar field theory that has a discrete spectrum in the same limit producing confinement.

As shocking as this result may seem, it can be straightforwardly extended to general relativity. We know that the solution, in the gradient expansion of the Einstein equations, is the Kasner solution that is not periodic at all. The situation is made more complicate by BKL scenario. In this case we have a sequence of oscillatory epochs making an overall chaotic scenario. So, we cannot find a class of periodic solutions to build an infrared quantum field theory that in this way seems to have no bound state again in a regime of strong nonlinearities (strong gravitational fields). I should say that a more detailed analysis would be helpful here opening the possibility to have an infrared formulation of QFT for Einstein equations.

Physics laws and strong coupling

It is a well acquired fact that all the laws of physics are expressed through differential equations and our ability as physicists is to unveil their solutions in a way or another. Indeed, almost all these equations are really difficult to solve in a straightforward way and are very far from the exercises at undergraduate courses. During the centuries people invented several techniques to manage such equations and the most generally known is surely perturbation theory. Perturbation theory applies when a small parameter enters into the equations and a series solution is so allowed. I remember I have seen this method the first time at the third year of my “laurea” course and was Giovanni Jona-Lasinio that showed it to me and other fellow students.

Presently, we see as small perturbation theory has become so pervasive that conclusions derived just at a perturbation level are sometime believed always true. An example of this is the Landau pole or, generally, what implies the renormalization program. It is not generally stated but it is quite common the prejudice that when a large parameter enters into a differential equation we are stuck and nothing can be done than using our physical intuition or numerical computation. This is true despite the fact that the inverse of such a large parameter is indeed a small parameter and most known functions have both a small parameter and a large parameter series as well.

As I said elsewhere this is just a prejudice and I have proved it wrong in a series of papers on Physical Review A (see here, here and here). I have given an overview in a recent paper. With such a great innovation to solve differential equations at hand is really tempting to try to apply it to all fields of physics. Indeed, I have worked for a lot of years in quantum optics testing the approach in a lot of successful ways and I have also found applications to condensed matter physics appeared on Physical Review B and Physica E.

The point is quite clear. How to apply all this to partial differential equations? What is the effect of a large perturbation on such equations? Indeed, I have had this understanding under my nose since the start but I have not been so able to catch it immediately. The reason is that the result is really counterintuitive. When a physical system is strongly perturbed all the terms that imply spatial variation can be neglected. So, a strong perturbation series is a gradient expansion and the converse is true as well. I have proved it numerically in a quite easy way using two or three lines of Maple. These results can be found in my very recent paper on quantum field theory (see here and here). Other results can be found by yourself with similar simple means and are very easy to verify.

As strange as may seem this conclusion, it has obtained a striking confirmation through numerical computations in general relativity. Indeed, I have applied this method also to general relativity (see here and here). Indeed this paper gives a sensible proof of the Belinski-Khalatnikov-Lifshitz or BKL conjecture on the behavior of space-time approaching a singularity. Indeed, BKL conjecture has been analyzed numerically by David Garfinkle with a very beatiful paper published on Physical Review Letters (see here and here). It is seen in a striking way how all the gradient contributions from Einstein equations become increasingly irrelevant as the singularity is approached. This is a clear proof of BKL conjecture and our approach of strong perturbations at work. Since then Prof. Garfinkle has done a lot of other very good work on general relativity (see here).

We hope to show in future posts how this machinery works for pdes. In case of odes we have already posted about (see here).