As promised in my preceding post (see here), I have posted yesterday a preprint on this matter on arxiv (see here). I have presented all the exact solutions I was able to obtain at a classical level and I have given a formulation of the quantum field theory for a massless quartic theory. The key point in this case is the solution of the equation for the propagator
being the given exact classical solution. As usual, I have used a gradient approximation and the solution of the equation
that I know when the phase in is quantized as , being an integer and an elliptic integral. This gives back a consistent result in the strong coupling limit, , with my preceding paper on Physical Review D (see here).
The conclusion is rather interesting as quantum field theory, given from such subset of classical solutions, is trivial when the coupling becomes increasingly large as one has a Gaussian generating functional and the spectrum of a harmonic oscillator. This is in perfect agreement with common wisdom about this scalar theory. So, in some way, Jacobi elliptical functions that describe nonlinear waves behave as plane waves for a quantum field theory in a regime of a strong coupling.
3 Comments | Applied Mathematics, Physics | Tagged: Exact solutions of PDEs, Quantum Field Theory, Quartic massless scalar field theory, Scalar Field Theory, Strong Coupled Theory | Permalink
Posted by mfrasca
Following my exchange with Lubos Motl (see here) I try to explain what an unconventional view is for people working in QCD. Of course, I agree with Lubos sight that it does not matter how unconventional is your view and so more attracting. What really counts is that this view agrees finally with experiments. But for QCD we have an important goal to reach, a goal that can hit all QFT and whatever else will follow: Our ability to manage a strongly coupled theory and this is a thing that nobody is able to do today in its full generality. This would be a large impact technology as it could possibly apply to any field of physics.
Currently, people tries to approach Yang-Mills theory with lattice computations that grant a non-perturbative solution to such a theory. From a strict theoretical point of view we know from QFT that a tower of non-perturbative equations exists to obtain n-point functions of the theory and these are Dyson-Schwinger equations and can always be obtained for any QFT. What you need here is a proper truncation of the tower and you are done. But this is the most serious difficulty with this approach as it is generally hard to evaluate how good is the chosen truncation and one can also incur in a dramatic error.
So, unconventional views here are those that evade both approaches given above and are able to recover lattice results. I would like to cite the work of Sorella et al. ( see their latest preprint) where they consider an extended Yang-Mills Lagrangian to recover lattice results. Other works have been put forward e.g. by Cornwall as in a pioneering paper to be found here and this produced several interesting works by Aguilar, Natale and Papavassiliou that, numerically solving Dyson-Schwinger equations, are trying to support Cornwall’s view. Finally, I have modified Bender et al. approach that did not work changing it into a gradient expansion recovering straightforwardly lattice results. I apologize for any missing contribution but anyhow I would appreciate whoever would help me to enlarge this list and I will be happy to do it.
What is the most valued of these approaches? Surely one would appreciate the most general ones, those that are not just fitted for the specific aims but that can expand to a large extent to all fields of physics and this possibility exists. So, the stake is high and lattice computations defined the aims. Next years we will see what physicists creativity deserves to us.
8 Comments | Physics | Tagged: QCD, QFT, Strong Coupled Theory, Unconventional Views | Permalink
Posted by mfrasca