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 \Lambda a cosmological constant. This equation is the same as the Liouville equation

\partial_t^2\phi-\partial_x^2\phi+\Lambda e^{b\phi}=0

and all the problem is to find the scalar function \phi. 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

\partial_t^2G+\Lambda e^{bG}=\delta(t)

and this equation can be solved exactly:


being \epsilon and \phi two arbitrary constants that may depend on the spatial coordinate. This Green function solves for the propagator after we have rescaled time by\sqrt{\epsilon}\tanh\phi and the \Lambda constant as \Lambda/\epsilon\tanh^2\phi. 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.

3 Responses to Liouville theory in the infrared limit

  1. hermann says:

    Has a large cosmological constant physical meaning?

  2. mfrasca says:

    It is the only way to have non-linearity so high in 2d to be comparable with the 4d case of strong gravitational fields. QFT in 2d gravity taking \Lambda small has been developed since ’80 and gives the same identical conclusions about spectrum.


  3. […] Curiously enough, I was able to see such solutions only in the Smilga’s book. I think this was Smilga’s idea and was also my source of inspiration.  I was in need of these solutions to treat classical Yang-Mills equations with a gradient expansion against a lot of unmanageable chaotic solutions. I would like to remember here that this approach is quite common in physics. For interested readers, I invite them to look at this beautiful Wikipedia entry about BKL solution. This is the way this approach is used in general relativity with a widespread example as the Kasner solution. This is an exact solution of Einstein equations that depends solely on time. Exactly as happens to the solutions obtained by a Smilga’s choice from Yang-Mills equations. Indeed, I suspect that Kasner solution may be helpful to quantize Einstein equations in the infrared limit. Currently I have no time to exploit this but I have given a hint about here. […]

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