‘Alien calculus’ could save particle physics from infinities

Everything is simple, given the right notation (and the concepts underlying it).

The original Maxwell theory of electromagnetism is about 10 rather involved equations. Maxwell-Heaviside form is 4 simpler equations. A formulation using differential 3-forms is 2 simple equations. A formulation using geometric algebra / Clifford algebra is one utterly simple equation.

Theoretical physicists are like enterprise architects, there seems to be a preference for generality of prescription over practical experimentation/implementation.

So one question I have in the introduction section is that it seems the article misses the difference between the number of Feynman diagrams to calculate a_n and the value of a_n. They point out that the the number of Feyman diagrams grows ~n!, which is much larger than the rate x^n shrinks (given 0Based on my limited knowledge of particle physics, physicists are currently able to calculate using Feynman diagrams because a_n does grow less than x^n shrinks. There are some equations (I think dealing with specific forces/fields) where the constant it larger than others which makes calculates much harder. x ~=.7 shrinks much slower than x~=.007. Yet even then the general trend does hold and it does allow for making calculations which can then be tested against experimental data.

What we find is that our calculations do match the experimental data. It isn't a perfect match, there is room for error and confidence intervals and such. The important point is that what this article suggest doesn't seem to happen. If at some point a_n grew much faster than x^n shrunk, then the real world solution would diverge and our answer from calculating n out to 5 wouldn't closely match the data. It almost sounds like the article is suggesting things will diverge only once we calculate out for n>100 or so, but reality doesn't await for those calculations. If this problem really existed, it would happen because reality is calculating out n all the way to infinity even while the physicists can not.

So I'm left with two possible conclusions.

1. The article is misunderstanding the relationship between the number of Feynman diagrams needed to calculate a_n and a_n itself.

2. The real critique is that the current model is wrong because the model diverges, not that reality itself diverges. Thus while this model is approximate for what we currently calculate, it is inherently wrong.

The second issue is an interesting idea. A model that looks correct and is correct for all calculations done so far, but which may no be correct for more detailed calculations but which we do not and will not have the computation power to test at that level.

My reasons are

- I'm a professional physicist and most people I know professionally think this way. People like symmetry, it helps clarify things, simplify things, provides powerful principles. If the cost is practical difficulties, well, that's just the cost of doing business; the physical understanding offered by simpler rules is beneficial. I know at least 2 people mentioned in the article would agree with that.

- I have no conceptual problem saying that the only solutions are numerical in nature if the principles are clear. Nobody promised physics should be easy. In fact some of the people mentioned in the article also have shown how their formal understanding might unlock better numerical methods!

- Some infinities are worse than others, and a modern effective field theory perspective makes me not worry about most examples.* With a Wilsonian understanding, renormalization is perfectly simple to understand. For theories which have perturbative UV fixed points you can formulate a lattice discretization which flows to that fixed point and you never encounter any infinity along the way.

Theories without a perturbative UV fixed point, well, that's where the trouble lies. Either there is no UV fixed point, in which case the theory is not valid for all energy scales and the troubling divergences point to an energy scale beyond which your theory is invalid. Or there IS a UV fixed point but it can only be found nonperturbatively.

Handed a QFT with no perturbative UV fixed point, how should you decide?

Well, one step back: should you, as a physicist, care?

For instance, why should we worry whether QED as a standalone theory is UV-complete? We know that in the real world electrodynamics mixes with the weak force at high energy. So whether QED as a standalone theory is UV-complete is a question that I'm not worried about. It is an interesting mathematical question, and that can only be answered with new techniques, such as resurgence. For standalone QED it's a question of pure mathematics, as far as anyone can tell. That's what these tools are good for, at the moment.

HOWEVER. I do admire the program of trying to show that more quantum field theories even exist mathematically, beyond the handful that we already know (which tend to have exotic properties). That seems important to me. But if it's false that's ALSO extremely interesting, it suggests that there are additional principles that we ought to understand.

* except for gravity, where the divergences are SO bad that even the EFT approach has problems.

I think this ends up being a question that is the cousin of Bertrand's paradox. In that case, the English words in the original question, despite feeling concrete in what they ask for, leave enough vagueness to give different ways to solve the problem that all seem to satisfy the query but give incompatible answers. I say this because I see two similar phrases in your query that seem to carry equal levels of assumptions.

First is the idea of simple. If something has a few very well defined rules that are understood in isolation, but whose emergent behavior is beyond our ability to define, is it simple? Conway's Game of Life is somewhat the default example. 2 very simple rules (or perhaps more, depending upon specifically how you count them), but it gives rise to a Turing complete system. Math itself is another example, as mathematicians seek to find simple rules from which math arises, yet even for the subsets of math that are limited to such rules, is it really fair to call it simple?

The second idea is that of an underlying structure. Does the universe have an underlying structure, and even if it does, does that exist in side of some more foreign concept? What happens before the big bang? Why did the big bang happen when it did? Are there other universes, both from the many worlds interpretation of quantum mechanics, and universes that entirely separate from our own. These seem questions that feel almost entirely in the realm of science fiction, not physics, but there are plenty of theoretical physicists who dive into this field even though it currently doesn't produce testable hypothesis and is thus outside the scope of proper science.