quote
While such open questions indicate that we are still far from a complete description of nature, it
is impossible to deny our successes, and in particular the success of mathematics in science. The
reason for this deep connection is far from obvious and is a recurring subject among scientists
and philosophers. The theoretical physicist Eugene Wigner explores this connection in his
widely-discussed essay, The Unreasonable Effectiveness of Mathematics in the Natural Science
(Wigner, 1960). He readily acknowledges that the correspondence is not one-to-one, since only a
small fraction of mathematical concepts are employed in the context of physical theories. Despite
this, Wigner sees a strong indication that the connection is more than coincidence based on the
incredible ability, particularly obvious in physics, to extrapolate a mathematical description far
beyond its original domain: “… the mathematical formulation of the physicist’s often crude
experience leads in an uncanny number of cases to an amazingly accurate description of a large
5
class of phenomena. This shows that the mathematical language has more to commend it than
being the only language which we can speak; it shows that it is, in a very real sense, the correct
language.
end of quote
While I like this essay I wish to point out that we have a long way to go in terms of optimization of mathematical analysis
quote
Differential Dyson–Schwinger equations for quantum chromodynamics
Marco Frasca
The European Physical Journal C volume 80, Article number: 707 (2020) Cite this article
761 Accesses
7 Citations
1 Altmetric
Metricsdetails
A preprint version of the article is available at arXiv.
Abstract
Using a technique devised by Bender, Milton and Savage, we derive the Dyson–Schwinger equations for quantum chromodynamics in differential form. We stop our analysis to the two-point functions. The ’t Hooft limit of color number going to infinity is derived showing how these equations can be cast into a treatable even if approximate form. It is seen how this limit gives a sound description of the low-energy behavior of quantum chromodynamics by discussing the dynamical breaking of chiral symmetry and confinement, providing a condition for the latter. This approach exploits a background field technique in quantum field theory.
1 Introduction
The main difficulty of quantun chromodynamics (QCD) is that, at low energies, the theory is not amenable to treatment using perturbation techniques. This implies that some non-perturbative methods should be devised to solve them. The most widespread approach is solving the equations of the theory on a large lattice using computer facilities. This permitted to obtain, with a precision of a few percent [1, 2], some relevant observables of the theory. This method improves as the computer resources improve making even more precise the comparison with experiment. Use of numerical techniques is a signal that we miss some sound theoretical approach to compute observables.
A similar situation is seen for the correlation functions of the theory. Studies on the lattice of the gluon and ghost propagators, mostly in the Landau gauge, [3,4,5] and the spectrum [6, 7] proved that a mass gap appears in a non-Abelian gauge theory without fermions. Theoretical support for these results was presented in [8,9,10,11,12,13] providing closed form formulas for the gluon propagator. Quite recently, the set of Dyson–Schwinger equations for this case was solved, for the 1- and 2-point functions, and the spectrum very-well accurately computed both in 3 and 4 dimensions [13,14,15]. Confinement was also proved to be a property of the theory [14, 16].
Indeed, the Dyson–Schwinger equations were considered, since the start, the most sensible approach to treat a non-perturbative theory like QCD at low-energies [17,18,19] and, more recently, [20]. In any case, the standard technique is to reduce the set of equations, that normally are partial differential equations, to their integral form in momentum space. Some years ago, Bender, Milton and Savage [21] proposed to derive the Dyson–Schwinger equations and treat them into differential form. This way to manage these equation was the one used to find the exact solution [13]. This technique appears more general as it permits to work out a solution to a quantum field theory also when a background field is present. This is a rather general situation when a non-trivial solution of the 1-point equation is considered. Such a possibility opens up the opportunity of a complete solution to theories that normally are considered treatable only through perturbation methods. The idea is that, knowing all the correlation functions, a quantum field theory is completely solved.
The aim of this paper is to derive the Dyson-Schwinger equations for QCD in differential form.
end of quote
Let me run this by you, so you can see it. QCD is, according to this abstract NOT particularly well solved by PERTURBATION THEORY
Why is this so important ? There have been cases of series solutions of the QCD equations having very large contribution from terms way down in the line of series expansion solutions
It means that there is more work to do.
In about 100 years from now if we have not killed ourselves off, our solutions to many of these equations may look very different
