Solving Dyson-Schwinger equations


Sunday I posted a paper of mine on arxiv (see here). I was interested on managing a simple interacting theory with the technique of Dyson-Schwinger equations. These are a set of exact equations that permit to compute all the n-point functions of a given theory. The critical point is that a lower order equation depends on higher order n-point functions making the solution of all set quite difficult. The most common approach is to try a truncation at some order relying on some physical insight. Of course, to have a control on such a truncation could be a difficult task and the results of a given computation should be carefully checked. The beauty of these equations relies on their non-perturbative nature to be contrasted with the severe difficulty in solving them.

In my paper I consider a massless \phi^4 theory and I solve exactly all the set of Dyson-Schwinger equations. I am able to do this as I know a set of exact solutions of the classical equation of the theory and I am able to solve an apparently difficult equation for the two point function. At the end of the day,  one gets the exact propagator, the spectrum and the beta function. It is seen that this theory has only trivial fixed points. I was able to get these results on another paper of mine. So, it is surely comforting to get identical results with different approaches.

Finally,  I can apply  the mapping theorem with Yang-Mills theories, recently proved thanks also to Terry Tao intervention, to draw conclusions on them in the limit of a very large coupling. In the paper you can find a formulation of this theorem as agreed with Terry, a direct consequence of my latest accepted paper on this matter (see here).

I think this paper adds an important contribution to our understanding of Dyson-Schwinger equations presenting an exact non-trivial solution of them.


On the arrow of time again


Lorenzo Maccone’s argument (see here)  is on the hot list yet. Today, a paper by David Jennings and Terry Rudolph (Imperial College, London) appeared (see here) claiming Maccone’s argument being incorrect. Indeed, they write down Maccone’s argument as follows

Any decrease in entropy of a system that is correlated with an observer entails a memory erasure of said observer

but this erasure is provided by quantum correlations. The key point is the link between quantum correlations and local decrease of entropy as seen by classical correlations. Jennings and Rudolph interpret Maccone’s view as the reduction of information at a quantum level entails a reduction of information at a classical level and we do not observe such events. These authors show counterexamples where this does not happen arguing that Maccone’s argument does not explain rather worsens the problem as quantum correlations can decrease while classical ones can increase.

I guess that this comment will undergo the standard procedure of Physical Review Letters for it and Lorenzo Maccone will produce a counterargument facing in this way a review process. As it stand, it appears a substantial open problem to the original Maccone’s proposal but relies in an essential way on the interpretation Jennings and Rudolph attach to it.

Being this a really exciting matter, it will be really interesting to following the way events will take place.

KAM tori reforming to be published


Today, the Editor of Journal of Mathematical Physics communicated to me that my paper on KAM tori reforming has been accepted for publication. This is one of the most prestigious journals in mathematical physics and the result is really important and I am very happy for this very good news. Thank you very much, folks!

This paper was the subject for a demonstration with Mathematica (see here) for the Wolfram Demonstration Project. You can find a post of mine about this here.

This question is indeed crucial for ergodicity and the arrow of time so much discussed in these days. See here for a post about this matter.

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