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‘Alien calculus’ could save particle physics from infinities

quantamagazine.org

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Re: ‘Alien calculus’ could save particle physics from infinities

#81

Post by mindcrime »

What I find fascinating about this, is the question of "What other applications (besides saving particle physics from infinities) could this branch of mathematics have"?

Every since I read The Universe Speaks in Numbers[1] I've been fascinated by those scenarios where there's some hard problem in science, and it turns out that the math needed to solve it was invented years before, to solve some other problem. And the eventual solution to the current problem is merely the serendipitous discovery that this other math exists and applies.

[1]: https://www.amazon.com/Universe-Speaks-Numbers-Reveals-Natur...

Re: ‘Alien calculus’ could save particle physics from infinities

#82

Post by r3trohack3r »

mindcrime wrote: post #74

Looks like the three Jean Écalle books mentioned in the article are available (legitimately) online: https://www.imo.universite-paris-saclay.fr/~jean.ecalle/publ... Unfortunately, as mentioned in TFA, they are in French, which makes them less than ideal for anyone who doesn't read/speak French. I wonder if these will ever see an English translation? Short of that, I wonder how well it would work to try to read throug…

ChatGPT isn't the only AI in town and there are services optimized for translation that might work better for this use case. Some accept HTML, PDF, etc. and send back a translation of the content.

Google's is available here: https://cloud.google.com/translate

That being said, taking something in French that is intensely academic and converting it to English might not be a straight-up translation task - having GPT-4 clean up and correct the translation with the context from the original French document might yield a better final product.

Re: ‘Alien calculus’ could save particle physics from infinities

#83

Post by m3kw9 »

mindcrime wrote: post #81

What I find fascinating about this, is the question of "What other applications (besides saving particle physics from infinities) could this branch of mathematics have"? Every since I read The Universe Speaks in Numbers [1] I've been fascinated by those scenarios where there's some hard problem in science, and it turns out that the math needed to solve it was invented years before, to solve some other problem. And th…

Would be cool to train GPT with all math books and have it a go at a few unsolved math problems

Re: ‘Alien calculus’ could save particle physics from infinities

#84

Post by MayeulC »

WindyLakeReturn wrote: post #66

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 0 Based on my limited knowledge of particle physics, physicists are currently able to calculate using Feynman diagrams be…

I only read the article halfway through, and I am no physicist, but my understanding was that this was a new mathematical method that allowed to calculate past the divergent part.

Possibly because some terms cancel out later? a bit like some limit calculations.

If that's the case, I was picturing it a bit like how imaginary numbers were initially introduced, to find real-valued solutions to 3rd+ degree polynomial equations. Step into another realm, perform your transformations, and find back a real solution. Laplace transforms come to mind as well, there are a phletora of such tools (fourrier, taylor series, etc) that allow to express the problem in a different space.

Re: ‘Alien calculus’ could save particle physics from infinities

#85

Post by [deleted] »

WindyLakeReturn wrote: post #66

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 0 Based on my limited knowledge of particle physics, physicists are currently able to calculate using Feynman diagrams be…

[deleted]

Re: ‘Alien calculus’ could save particle physics from infinities

#86

Post by MacsHeadroom »

maroonblazer wrote: post #56
simonh wrote: post #17

Earlier quoted context omitted.

It seems like there’s a tension. On the one hand generating the most likely sequence of tokens maximises the chance the response will make sense and be relevant. On the other hand it also guarantees you will get the most bland and unimaginative response.

I've only explored ChatGPT in the most cursory way, but wouldn't this be a function of the prompt it's given? E.g. "Summarize this idea in a way that uses novel comparisons to everyday phenomena. The target reader is someone who doesn't have domain knowledge of the field...etc." Or something like that.

Yes, you're correct. People who complain that chatGPT is bland don't realize that they have to specify a style to not get the average of all content.

It's just like if you ask an image AI for a "woman" you will get the average of all artists women and it will look very generic and bland. But with the right stylistic qualifiers in your prompt you will get something so captivating that it wins awards for its creativity.

Re: ‘Alien calculus’ could save particle physics from infinities

#87

Post by photonthug »

I did an undergraduate research project many years ago on Conway's surreal numbers (some people might be aware of Knuth's excellent book on the subject). This alien calculus of resurgence reminded me of that, so I went looking for a connection, and found one: https://arxiv.org/pdf/2208.14331.pdf

I was only barely a math nerd and very far from a physics nerd, but even as a pretty naive bystander the surreal numbers seemed to offer some hope for common problem of wrangling divergent sums. Anyone out there that can compare/contrast the approaches and challenges of integration/differentiation on the surreals vs this topic of resurgence in perturbation theory? Are these things even that fundamentally different or is it more like a difference in point-of-view/branding for the same techniques?

Re: ‘Alien calculus’ could save particle physics from infinities

#88

Post by WindyLakeReturn »

MayeulC wrote: post #84
WindyLakeReturn wrote: post #66

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 0 Based on my limited knowledge of particle physics, physicists are currently able to calculate using Feynman diagrams be…

I only read the article halfway through, and I am no physicist, but my understanding was that this was a new mathematical method that allowed to calculate past the divergent part. Possibly because some terms cancel out later? a bit like some limit calculations. If that's the case, I was picturing it a bit like how imaginary numbers were initially introduced, to find real-valued solutions to 3rd+ degree polynomial equ…

>allowed to calculate past the divergent part

My question was more in doubting if the divergent part actually existed.

n!x^n will eventually diverge for 0<x<1, regardless of how small x is, but was that really the issue? I thought the problem was more a question if g(n)x^n would diverge when g(n) involves computing n! Feynman diagrams. It can't be automatically assumed that computing n! Feynman diagrams leads to an answer that grows comparable to n!. Or maybe it can, but I didn't see that argument being made anywhere, though I may have missed it.

Re: ‘Alien calculus’ could save particle physics from infinities

#89

Post by scythe »

JPLeRouzic wrote: post #4
pvitz wrote: post #3

Écalle's own summary can be found here: https://www.imo.universite-paris-saclay.fr/~jean.ecalle/fich...

I am not a scientist, is this Alien calculus akin (somehow) to a derivative?

It involves defining a set of operators — like functors in CS, operators take one function and return another — which obey a modified form of the product rule for derivatives D[f*g] = g*Df + f*Dg. These operators are used to make the analytic continuation of divergent series consistent; because they are defined in terms of functions that cannot be calculated directly from their definition (hence analytically continued), they are "alien".

Re: ‘Alien calculus’ could save particle physics from infinities

#90

Post by simonh »

PartiallyTyped wrote: post #63
simonh wrote: post #22

Earlier quoted context omitted.

I see mathematics as a rigorous highly consistent descriptive language. Physics theories expressed mathematically are very precise descriptions of observed behaviour, but calling them laws is deceptive. The fact that they align precisely to observed behaviour just indicates that the behaviour of physical systems is highly consistent. Well, I hope so. If reality was inconsistent and things happened arbitrarily with no…

You shouldn’t see mathematics as consistent because they are not. A system can be complete, meaning all true statements can be built based on axioms, but it cannot be sound leading to contradictions. Alternatively a system may be sound, but not all true statements could be derived. And finally there are statements impossible to prove because the proof is undecidable.

You’re quite right, which is why I described it as highly rather than completely or perfectly consistent.

It’s an interesting question whether the physical world is perfectly or merely highly consistent. If it’s made of mathematics, it may be that it cannot be perfectly consistent. So if we ever find that it is perfectly consistent, that might be evidence that it isn’t made of mathematics.

Most likely we’ll never be able to tell, but we’ll see. Or at least maybe our descendants will.

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