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Comment by antognini

3 years ago

I have idly wondered whether or not there could be a completely different approach to QCD from the usual perturbative techniques. I remember reading in one of Zee's books that back 80s he pointed out to Feynman that the path integral formalism that QFT is based on has no natural way to treat something as simple as a particle in a box. And an object like a proton seems to be more like a particle in a box than a free particle undergoing an interaction.

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antognini

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morbia  3 years ago

Yeah as someone who spent 4 years of his life calculating a second order term (Next-to-Next-to-Leading-Order), I have often wondered the same thing! In my original post I grossly simplified how challenging it is to calculate terms in perturbative QCD, even when in the perturbative regime. To name a few:-

* Two loop calculations are extremely challenging on an algebraic level

* You get low energy (called 'infrared red') infinities appearing at low energies. These need to cancel between all your contributing terms, and getting them to cancel is really really challenging.

* The numerical Monte Carlo approaches become extremely computationally intensive because of high dimensional integrals and numerical instability caused by point 2

It was not uncommon for calculations of single terms to involve multiple PhD students over a decade or more.

Throughout my PhD I certainly felt like something was fundamentally 'wrong' with the approach. Alas, I wasn't smart enough to rewrite the field with a whole new way of thinking so bailed instead.

  • selectodude  3 years ago

    >It was not uncommon for calculations of single terms to involve multiple PhD students over a decade or more.

    Forgive my ignorance, but what does calculating this sort of look like? I am not a mathematician or even math-adjacent.

    • morbia  3 years ago

      There are many components to such a calculation that can be split up amongst different research groups. I won't go into detail on all of those but essentially it boils down into two main categories of components to the calculation:-

      * Analytical integrals - This is a big algebraic task where you're trying to compute an equation that can be written by hand. For example, if you have 1-loop diagram [1] then the particle in the loop effectively becomes an integral over all possible momentum configurations that particle can have. One-loop is a hard problem but reasonably 'solved', 2-loops is extremely challenging.

      * Numerical integrals - This is typically using Monte Carlo techniques to numerically integrate over all possible momentum configurations of the incoming and outgoing particles. Because you can have many particles, it becomes a high dimensional integral pretty quickly. Monte Carlo scales well with dimensionality, but not that well. Therefore you need serious computation power for non-trivial numerical integrals.

      Added to this fact is a fun feature of these calculations that infinities spring up all over the place. You have both a numerical and analytical game of getting these guys to cancel (they do, the calculation must be finite) but it is not a straightforward task at all.

      [1] https://en.wikipedia.org/wiki/One-loop_Feynman_diagram

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