‘Alien calculus’ could save particle physics from infinities
quantamagazine.org
quantamagazine.org
(I agree with your point and share your appreciation.)
Wonder how long before that’s solved. Hard to believe training on such large swathes of the web doesn’t result a compressed generic representation of language within its weights.
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.
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.
I used to work in the field of creative text generation for fiction. I’m genuinely very curious and I go out of my way to find compelling examples. There is definitely “good” output that’s on the right track. GPT-4 also does way better, but it still falls short.
This is also difficult to just evaluate objectively. If you find that GPT models have generated the best prose you’ve seen. That’s wonderful! I understand that my standards are quite high (high does not equate to “better” either)
In 6 different tabs, ask chatGPT GPT-4 version "Write a 200 word prose about X in the style of Y which perfectly mimics their style and perspective and label it Prose A,B,C,D,E,F"
Then open a new chatGPT GPT-4 chat and ask "Rank the pieces of prose below on how much they sound like something X would write, then detail your reasoning."
Then read the winning prose and be surprised at how much better "best of N" is.
....
And if that isn't enough, open two chatGPT GPT-4 windows side-by-side and prompt each with "You are Editor A/B. You will work with Editor B/A to make a piece of prose more accurately resemble authentic prose written by X"
Then give A the prose and copy its response to B. Copy their responses back and forth as they edit the prose.
After 5 or so back/forth it will be even more indistinguishable from the a genuine article.
....
And if that's not enough, you can have all of this done automatically programmatically with the API so you can just sit back and get the final result with no more work than putting in the topic and author's name.
Decoding methods also matter, and it’s a shame we aren’t given token probabilities (or any insight into model output) so we have more creative control over how to decode the output. Some of the better literature I’ve seen involving creative writing did have novel decoding methods
A tangent. Bard also confuses Yoda with Jar Jar Binks quite often. Try it out!
Regarding Bing Creative. It is delightful! I do like it a bit and whatever they’ve done to the system does make for some of the better output I’ve seen from LLMs
creative prompting gets most of the way there, already.
'Tell me the current weather as if you were Marcel Proust'
That’s not to say it isn’t impressive. It is and it accomplishes the job very well. We’re looking for progress not perfection, but I personally have very high standards from creative writing and GPT doesn’t meet my personal bar. However not everyone shares that bar and personal evals of GPT’s output are equally valid. Plenty of people find it to be great and at the end of the day that’s all that matters
And of the authors who think "my journey will be worth the destination, because I can really write!", some large percentage of them overestimate their ability and/or our interest level.
There was a time when people had more time to read than they had things to read. At that time, writing in a way that expanded the text made sense. But now, there's an essentially infinite amount to read, and still only 24 hours in a day. That favors getting to the point in a hurry. On the internet, you have from 5 to 30 seconds to persuade me that your thing is worth reading. If in that time you don't persuade me that you're going to deliver enough value to pay back my investment of reading your thing, I'm on to the next tab.
So if I'm going to read your thing, you have to pay me back more than those others are going to. You have to hold my attention more than the promise of all those other things.
But you're right that people different tastes. Some people like long-form writing. The same dynamic is still true in their choices, though - they're just using a different filter to decide what's going to pay them back the most.
To overcome the issues with traditional calculus, alien calculus involves replacing the standard real numbers used in calculus with "fuzzy numbers" that represent probabilities instead of definite values. These fuzzy numbers allow the equations to describe the behavior of particles more accurately because they capture the uncertainty inherent in subatomic systems.
Alien calculus calculates the probability that particles will be in certain positions or states, rather than trying to determine their exact positions or states. This is because particles at the subatomic level are constantly in motion and their position or state cannot be precisely determined at any given moment. By using probabilities, alien calculus can provide a more accurate description of the behavior of particles at this level.
Also check out https://ed.ted.com/search?qs=quantum+mechanics
Could be combined with the information bubble filter described in Stephenson’s Burn or Dodge in Hell.
But I don't think there's any underlying notion of small changes to something continuous. It is not the slope of some smooth function.
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?
Good point, with the expense of even larger colliders, or where experimentation remains impossible resorting to mathematics is likely the only option.
It's possible that the mathematical techniques being developed here only work for renormalizable theories. In which case they wouldn't work at all for Quantum Gravity. This would likely signal that there is something else missing that is needed to build a quantum theory of gravity.
Nevertheless, I'm still in awe of Schwinger, Feynman et al who actually made renormalization work despite—according to Feynman—it being a 'dippy process'. These guys were truly physics and mathematics 'magicians'.
It has definitely been said that "String theory is part of 21st-century physics that fell by chance into the 20th century.". I can recall seeing variations of that quote in a few different books over the years. Interestingly though, a quick Google search (maybe I should have asked ChatGPT?) shows attributions to both Witten and Daniele Amati. So I'm not sure who said that particular bit first.
This is beautiful. Any examples of using Écalle's method to solve three-body problem?
In fact that’s pretty much the story of the development of physics. It turns out Newtonian mechanics is emergent from Relativity. Maxwells equations are emergent from quantum mechanics. The behaviour of bosons is emergent from the behaviour of quarks.
As I understand it there are theoretical reasons to suspect that quarks do not have any decomposition though. We’ll see.
However relativity is far more complex than Newtonian physics.
Which is more complex depends on what you mean. There are fewer laws in the possibility space of generally-relativistic physics than Newtonian physics. So, which metric is more important? Less pleasant calculations or a larger search space?
Your comment suggests you weigh the calculational simplicity more heavily, but most physicists would come down on the other side of the issue.
What's your reason for saying this? I'm more on the mathematical side of things so I don't know that many physicists. But it's been my understanding that equations that are extremely difficult to solve, impossible to solve except numerically, or involve infinities that can't be explained, are a major pain point for physicists. I mean, that's the entire premise of this article.
- 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 you mean
> The behaviour of _hadrons_ is emergent from the behaviour of quarks.
--
Anyway, back to your main idea, the problem is that most of the times the new underlaying theory is even worse than the original one
> Newtonian mechanics is emergent from Relativity
For Newtonian Mechanics you need only basic calculus. For General Relativity you need curvature, tensors, two definitions of derivatives and other nasty stuff. [I never saw the details, but it's in my todo list.]
> Maxwells equations are emergent from quantum mechanics
Well, electromagnetism is just the local gauge invariance of the U(1) group. The idea is very simple but each word in that sentence needs like one semester to be decoded. [I saw the calculations a long time ago. I don't remember the details, but I remember the general idea. I wrote a comment with a oversimplified version https://news.ycombinator.com/item?id=8189346 )
> there are theoretical reasons to suspect that quarks do not have any decomposition though.
My favorite reason is that for a classic small ball, the ratio of the magnetic moment to the inertia moment is 1 (once you fix some nasty details about units), but for an elementary quantum particle it is 2. For composite particles, there is no theoretical value. https://en.wikipedia.org/wiki/G-factor_(physics)
The experimental value for protons is 5.5 and for Neutrons is 3.8, that is not surprising because we are sure they are composed particles.
For Electrons and Muons it's slightly more than 2, but we understand that difference quite well (but not perfectly, and that is related to the main point of the article here).
I don't think it has been measured directly for Quarks, but my guess is that it's used in some parts of the calculation of some Feynman diagrams, and if it were very different from 2 someone would have noticed.
>most of the times the new underlaying theory is even worse than the original one
Oh quite. Macroscopic emergent behaviour is often apparently simpler than the underlying causes and is in some way a generalisation of them.
Well, I hope so. If reality was inconsistent and things happened arbitrarily with no rhyme or reason I think we’d be in big trouble.
I’m not totally unsympathetic to the view that maths is fundamental though. It’s an interesting way to think about it.
But from what I’ve read, the deeper you go, the more it’s difficult to find something other than mathematics. When you ask what is a particle you find out it’s probably an excitation of a quantum field. So what is a quantum field if not a mathematical structure? Is there a physical reality to wave function collapse?
Maybe it doesn’t matter. The shut up and calculate crowd doesn’t seem to care.
It is an interesting speculation, but it’s also possible it’s just confusing the map for the terrain.
It also seems like a comfortable answer to the question about about what's happening every time I cause wave function collapse and split the universe. I'm just creating a new mathematical structure.
Aren't you describing a classical field?
As I understand it, a quantum field is essentially varying probabilities (the wavefunction).
> they produce consequences in the world
They don't produce consequences in the world, they are the world. You and I, we're excitations of a quantum field. We're of the wave function. The quantum field encodes all the information that is us.
Nobody calls the Standard Model a law, for example. The modern view is that the Standard Model is a low-energy effective field theory.
But, whatever supplants the SM, we still expect the principle of translation invariance to hold.
Until, that is, we have evidence for a paradigm shift. If we discover physics that really can't be described, for example, by dynamics happening in a geometric space, then we'll have to give up that principle. Strongly-coupled stringy dynamics seems to have non-geometric phases, for example.
So our statement of laws is more a description of the current best paradigm (say, the operating system), rather than our best model (the program).
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.
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.
What does that even mean? The physical world is not a formal system in any obvious sense.
Nor does "consistent" apply to mathematics, by the way, only to formal systems used/studied by mathematics. You cannot mathematically prove or disprove that mathematics is consistent or define precisely what that would mean because mathematics itself isn't rigorously defined. If you define it as "whatever mathematicians are doing", I guess it's in some sense inconsistent, since mathematicians often disagree.
On the other hand, if our universe is one of many arbitrary possible universes, I’d say it’s not a mathematical structure (although it could still be the only universe that exists, hypothetically).
The emergence of complex traits from simple rules still requires exotic mathematics to describe macroscopically.
Until you hit all the quantum weirdness and then it’s all wave functions and probabilities. That maybe comes out of something simple as well.
"Going down" simply means identifying laws that are more universal in that they can underly models of different systems, ideally "any known system". Quantum weirdness isn't significantly harder mathematically than what came before (we don't have an objective measure of how "hard" some piece of math is), it's just harder to relate to everyday experience. It's similar to how we got used to "masses attract each other" or "things just keep moving in a straight line", which seemed ridiculous to most of Newton's contemporaries.
We absolutely have a way to measure how hard a piece of math is: computational complexity. And quantum mecanichs is more computationally complex than newtonian mechanics (while general relativity is significantly harder still than both of them).
Is there a formal version of this claim somewhere? (Beyond theorems about quantum computers)
So, we can represent a physical theory as some algorithm A(init_condition, time, precision) = final_condition. Let's call the Newtonian model N and the quantum mechanical model QM.
My claim (which I think is completely uncontroversial) is that for the same init_condition, time, precision, you will need more computational steps to arrive at the same final_condition using QM than using N. This is the very definition of computational complexity.
If you want instead to compute the asymptotic complexity, we can take either time, precision, or some combination of the two, to be the parameter in respect to witch we compute complexity as a function.
Overall, this fact is quite uncontroversial since it is widely assumed that a classical computer requires an exponential amount of time to simulate a quantum computer running the same algorithm, for some algorithms such as Shor's. This is of course not yet proven, but it is widely considered very likely to be true. In fact, if it turns out to be false, it is quite likely that it will also turn out quantum mechanics is in fact reducible to classical mechanics (since the difference in computational speed is due to the special features of quantum mechanics as compared to classical, particularly the complex probability values that it arrives at).
To talk about "number of computational steps" (which I have never seen referred to as "definition of computational complexity") you need to specify a computational model. The number and the comparison between two programs will depend upon it and you can get whatever result you want by choice of the model.
The conversion of theory -> model -> program is also not at all straightforward. The complexity strongly depends on the conversion. If you specify "best possible conversion" for model to program (which is probably the easiest part of the huge amount of things you need to define), you cannot prove anything about it anymore.
The complexity difference between asymptotic classical and quantum calculations is usually determined by its scaling in the "size of state space". Why do you focus on time and precision? I don't know for sure but it seems reasonable to me that the complexity with respect to those two might actually be the same for common system models. Regardless, you have to choose some parameters and you provided no arguments that this choice is in any way objective.
> In fact, if it turns out to be false, it is quite likely that it will also turn out quantum mechanics is in fact reducible to classical mechanics
Strongly disagree for any reasonable notion of "reducible". Quantum theory is different from classical in many qualitative and experimentally accessible ways (e.g. Bell inequalities). Computational complexity isn't everything.
To end, I want to further contest your defining complexity of a theory according to "length of computation to get model prediction". An interesting alternative would be "length of theory specification under optimal compression " (i.e., "Kolmogorov complexity of a theory"). This notion obviously also isn't rigorously defined yet but I think it's about as hard to definine as your notion. Neither of them seems obviously the "right one" to me in this context.
Apart from that, I hope upon some reflection you realize that your sentence is an illustration of my point.
Of course, you don't help anyone make progress in physics just by suggesting they should "use some philosophy for a change".
You aren't really pointing it out, you're just claiming that without any evidence at all.
You should be able to back up your claim by showing some example (not fully worked out, just some idea) of a philosophy-inspired physical theory that is simpler than current theories but more or less as precise in its measurable predictions of what will happen next in sine physical system.
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.
This way, the nabla symbol is very helpful in turning a group of (usually) 3 related PDEs into one pretty understandable one. Same for vector and matrix forms of common 2D and 3D transforms. Once you understand how these symbols work (they have a very regular structure), you can think at a bit higher level, and juggle with transforms that won't fit in your head in the scalar form, requiring toilsome operations on paper (or equivalent).
Say, general relativity is hard as it is; without various notational tricks which abstract away some complexity, it would likely be completely unwieldy.
A change of notation is like a refactoring in programming: a good set of powerful and well-defined functions makes the code to solve a problem significantly easier to produce and to understand, and harder to make an error.
Every system of equations can be written as "A = 0" for a suitable definition of A. That doesn't make it simple.
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.
Based 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.
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.
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.
There are then two natural questions: 1. if the perturbation series diverges, why doesn't the universe explode? and 2. If the series diverges, why can we use it at all?
Let's first talk about the first part: why doesn't the universe explode? Well, it's because the perturbation series is not actually what is going on, the real answer is the solution to the full set of equations. It's just that we're using a perturbation expansion as a crutch. It's sort of like if the universe's function is 1/(1-x) but we constantly insist on using 1+x+x^2+... Clearly the first function is completely well behaved at x=2 but the second one is not. If we notice that our series explodes for x=2 we should not immediately assume that the universe also must explode, it's just that our representation of the true physics is not faithful. This is perhaps a bad example because the series in question is convergent for some x, just not for x=2. The perturbation expansions in question are more subtle since the never converge.
This then leads into the second question: if the series diverges, how can we even use it? Well the idea here is that it's not just any divergent series (like my silly example with 1/(1-x) above) but rather an asymptomatic series. This means that as long as you truncate the series at some point it is in fact reasonably close to the target function for a sufficiently small value of the parameter. It's just that the more terms you want to include, the sooner the approximation breaks in terms of the parameter. So, if you want to include 10 terms it might be a decent approximation until x ~0.1 but if you include 100 terms it might only be a good approximation until x~0.01. Now, within the overlapping range (x<0.01) it's better to have 100 terms than 10 terms, so it's not like including more terms is bad in all ways. But you see the issue: if you include 1000 terms you get a better approximation for your function for values x<0.001 than you had with 100 terms but now your approximation breaks much sooner. If you want to include all the terms your approximation breaks the moment you leave the point x=0.
Why do we think that QFT perturbation theories generally have zero radius of convergence? Well, look at QED, the quantum theory of E&M. If the theory had any nonzero radius of convergence, that also means that the theory would need to make sense for negative coupling constants. However, what would E&M look like for negative coupling? Well, we'd still have electron/positron virtual pair creation from the vacuum since the interactions of the theory are still the same. However this time around they wouldn't attract each other anymore but instead repel each other causing an instability in the vacuum of the theory. We would just constantly be producing these particle/anti-particle pairs and they'd form two separate clusters where all the electrons attract each other and all the positions attract each other but they pairwise repel. In other words, the vacuum would break. This suggests that QED with a negative coupling constants doesn't make sense. But this contradicts the fact that the radius of convergence of the perturbative expansion is nonzero.
That's not to say that all QFTs must have zero radius of convergence, but similar arguments can (I think) be made for the type of QFTs that we actually see in nature.
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...
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 through it using ChatGPT or something to translate the prose bits.
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.
The alien derivative is an operation which moves you from one sector to another. A bit like crossing a branch cut.
For example, alternating current in presense of capaciators and induction coils is well handled by switching to imaginary (complex) resistance/current/voltage calculation, and transition processes in electric chains are handled by operational calculus.
If we used differential equations to solve these, which indeed look natural for these tasks, we will not be able to accomplish any calculations without a PhD.
That's overstating the argument[1]. In fact "differential equations" are naturally abstractable too. Decades and decades of workaday electrical engineers have been writing SPICE models successfully. The fact that one abstraction looks like "math" and the other "software" is just an aesthetics thing.
That's not to say that there's no point in teaching complex impedance. There absolutely is. But abstraction works in mysterious ways and some abstractions are more "beautiful dead ends" than others.
[1] Also needs to be mentioned that linear RC networks are a pretty small subset of the actual problem that needs solving. Transistors are kinda important too.
Moreover, we know the perturbative series does not converge for many theories.
For QED, for example, the radius of convergence must be 0, by an argument due to Dyson. Roughly:
1. The perturbative series is an analytic function of the fine structure constant α, which is proportional to the electron charge squared.
2. Like charges repel and therefore α>0. If like charges attracted the vacuum would be unstable against the continuous creation of electron-positron pairs from the vacuum and the electrons going to one part of the universe and the positrons to the other.
3. Because of this instability the series converges for no α<0, and since the perturbative expansion is analytic in α cannot converge for any α>0 either.
Essentially there are a whole variety of different interactions, some of them which interact with themselves. You wind up with a tiny zoo of bizarre diagrammatic creatures that you have to figure out herding patterns for - and only then can you get the tool to do the herding for you. I seem to remember involved a whole deal of abstract group theory.
It's way easier to draw the diagrams on paper to solve all cases of an interaction, anyhow; after all that nightmare you wind up with some result. And from what I recall, the number of cases also grows exponentially too D:
> "Resurgent functions" are divergent power series whose Borel transforms converge in a neighborhood of the origin and give rise, by means of analytic continuation, to (usually) multi-valued functions, but these multi-valued functions have merely isolated singularities without singularities that form cuts with dimension one or greater.
Cool!
Regardless, I think we can all agree super compact massively heavy objects do in fact exist. We have pictures of black holes, we can see infrared time-laps images spanning decades of stars whipping around an undefined point in space.... they certainly do exist. Does all that matter collapse to an asymptotic point beyond Planck space? Perhaps not, it could simply be really compact degenerate matter, inside the Schwarzschild radius, like a quark-gluon plasma, or whatever might go above such high energies. And whatever that stuff is, it could perhaps not collapse to single point, it just gets really hot, and really dense.
Recently Eric Weinstein has been making the rounds on internet podcasts, for example Joe Rogan and the likes... getting what we might call academically belligerent about singularities, and all the "(re)normalisation" that gets explained away to balance equations. His characterisation of the situation is charismatic, and to some extent persuasive. But I dunno, he seems kinda weird.