‘Alien calculus’ could save particle physics from infinities(quantamagazine.org)
quantamagazine.org
‘Alien calculus’ could save particle physics from infinities
https://www.quantamagazine.org/alien-calculus-could-save-particle-physics-from-infinities-20230406/
131 comments
I really appreciate how writers at Quanta turn extremely complex and dry topics into a pleasurable read by mixing simple analogies with history. I really admire the skill it takes to break down these topics and make them fascinating for someone with no understanding of them, such as me.
I like those articles too but I wish they include a short version with the major takeaways. In another industry that would be an executive summary. I don't always have the time to read all the story, so I end up fast reading it trying to find the important points and I'm never sure I really found them. In this case they seem to be the paragraphs after "Here’s an extremely rough cartoon version of the approach:"
Traditional calculus is not well-suited to describing the behavior of particles at extremely small scales because it leads to infinite values in calculations:
Traditional calculus is based on real numbers, which are continuous and can take on any value. However, at the subatomic level, particles behave more like waves than like particles with a definite position, and the value of some physical quantities, like energy, can become infinite at certain points, which makes the equations used in traditional calculus break down.
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.
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.
Wait, is this new? This is exactly how I'd assumed quantum calculations worked already
Yeah the above comment and it's reply is exactly what I'd assume GPT-4 would reply if asked what "alien calculus" is ;) (the linked article isn't about this at all, it's about technical details of perturbation series in mathematics as they relate to non-perturbative treatments)
You are correct that the probabilistic nature of quantum mechanics has been known for many years. However, what is new in the approach of "alien calculus" is the use of a mathematical framework that replaces traditional calculus with a more probabilistic approach to avoid infinities and better describe the behavior of particles at the subatomic level. Alien calculus is a novel approach that aims to improve upon traditional methods and has shown promise in resolving some of the issues that arise in particle physics calculations
Thanks! Would you be aware of any kid-friendly sources that could be used to inspire students? Of the “We love calculus because it explains how a ball moves. Now let me show you something really cool!”
There is a nice discussion of quantum mechanics here - note you have to click to "reveal all steps" at the very bottom - https://mathigon.org/course/probability/randomness
Also check out https://ed.ted.com/search?qs=quantum+mechanics
Also check out https://ed.ted.com/search?qs=quantum+mechanics
Awesome - thanks!
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Product opportunity: an LLM based tool to summarize works ti a personally customizable level. An executive will want the societal implications without the math. A scientist will want the equations without the analogies or number of elephants in size. A lay reader mag want to know the number of elephants without those equations getting in the way.
Could be combined with the information bubble filter described in Stephenson’s Burn or Dodge in Hell.
Could be combined with the information bubble filter described in Stephenson’s Burn or Dodge in Hell.
Sometimes the fun is in the journey, not (only) the destination
And sometimes it's not. Sometimes the journey is miserable, and we only do it to get to the destination.
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.
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.
That’s alright, not everything has to be catered to those with minimal time to read.
It's not exactly minimal time to read. It's too many other options to read. If you look at the number of things that people (or at least websites) suggest I read, I've got an average of less than a minute to give each one (and maybe less than a second).
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.
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.
All soon to be obsoleted by ChatGPT. /s
(I agree with your point and share your appreciation.)
(I agree with your point and share your appreciation.)
Sarcasm aside. Even the output from GPT-4 is quite bland and generic. Very good for performing tasks (I.e convert a blob of text into a knowledge graph or generate X code), but quite awful at the elegant prose we see at play
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.
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.
>Wonder how long before that’s solved.
creative prompting gets most of the way there, already.
'Tell me the current weather as if you were Marcel Proust'
creative prompting gets most of the way there, already.
'Tell me the current weather as if you were Marcel Proust'
I replied to another comment with my thoughts on prompting. I will add that I don’t consider mimicking another writers style to matter much. Seems like an easy cop out (just my opinion though).
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
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
Nobody outside OpenAI/Microsoft has access to the GPT-4 base model yet. Use Bing Creative for a differently finetuned GPT-4 than ChatGPT or "gpt-4".
GPT-4 comment isn’t true
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
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
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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.
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.
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.
Prompting works very well, but I have still not seen any output from ChatGPT that captivated me in a way my favorite writing has.
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)
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)
Try this:
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.
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.
that's exactly why it doesnt generate the most likely sequence of tokens! They are chosen at random based on the probabilities assigned by the model, so there is a chance of unusual output. In the API you can tweak the "temprature" which weights this towards more novel output
I’m very familiar with temperature and other parameters you can use to tweak output. They can take you decently far! GPT-2 can produce very coherent convincing output even today if you know what to tweak
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
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
Counterargument: "explain linked lists in the style of edgar allan poe"
My personal favorite is in the style of Yoda ;)
A tangent. Bard also confuses Yoda with Jar Jar Binks quite often. Try it out!
A tangent. Bard also confuses Yoda with Jar Jar Binks quite often. Try it out!
Seems mostly solved with prompting. I.e. "Rewrite this essay in the style of John Steinbeck." Works quite well.
Écalle's own summary can be found here: https://www.imo.universite-paris-saclay.fr/~jean.ecalle/fich...
Man, I know a good bit of graduate-level math well and that is incomprehensible to me. Either it's very poorly written or it's targeted at an audience who are already experts in, specifically, Borel transforms and I guess functional analysis?
I am not a scientist, is this Alien calculus akin (somehow) to a derivative?
The capital delta acts like a "normal" derivative if you want to say so. In eq. 4, you can see the product rule which is one of the most defining features for any sort of calculus. However, I must admit that this summary will take me a lot of time to digest...
Yes, it obeys enough algebraic laws that calling it a derivative is useful.
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.
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.
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".
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?
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?
Finding a way around the problem of infinities in perturbative quantum theory/QFT would not only put the science on a more sound mathematical footing but would also greatly benefit it in other ways. It seems to me that with a rigorous mathematical foundation it's likely phenomena will emerge that would never have been obvious without it. We can only hope the 'discovery' of Écalle's work leads us along that path.
Is t the problem of quantum gravity related? Does this being us closer to a grand unified theory?
It's related in the sense that the vanilla attempts to quantize gravity lead to infinities as described in the article. But the nature of these infinities is more troubling in the case of quantum gravity than those found in the rest of the theories that compose the Standard Model. Infinities appear very early in the series expansion of quantized gravity, at second order of the series expansion for the free theory. And the techniques developed by physicists to deal with them ( https://en.wikipedia.org/wiki/Renormalization ) do not work at all.
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.
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.
I recall reading a book on String Theory in the late 1990s edited by Ed Witten where he mentions the problems with physics and the limits of mathematical techniques. It's long ago so I can't paraphrase his words accurately but he did so with such eloquence that I've long remembered the fact. He pitched the problem in terms of both existing physics and of new '21st Century' ideas inklings of which had somehow trickled down into the 20th Century and that physicists needed new mathematics to deal with them (I hope I've not mangled his words too much).
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'.
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'.
He pitched the problem in terms of both existing physics and of new '21st Century' ideas inklings of which had somehow trickled down into the 20th Century and that physicists needed new mathematics to deal with them (I hope I've not mangled his words too much).
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.
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.
Not surprised, it's a memorable quote. Some months ago, I mentioned the book to the colleague who originally loaned it to me with the view of borrowing it again but he no longer has it. I then did a quick search and couldn't place it. This has spurred me to look again, perhaps starting with GPT.
A better understanding of Quantum Physics, or Physics in general will always bring us closer to either finding a grand unified theory, or being able to mathematically exclude the possibility of it.
"…or being able to mathematically exclude the possibility of it."
Good point, with the expense of even larger colliders, or where experimentation remains impossible resorting to mathematics is likely the only option.
Good point, with the expense of even larger colliders, or where experimentation remains impossible resorting to mathematics is likely the only option.
> Tunneling is one of many nonperturbative phenomena in quantum physics, but nonperturbative effects are everywhere: The branching growth of snowflakes, the flow of a liquid through a pipe with holes, the orbits of planets in a solar system, the rippling of waves trapped between round islands, and countless other physical phenomena are nonperturbative.
This is beautiful. Any examples of using Écalle's method to solve three-body problem?
This is beautiful. Any examples of using Écalle's method to solve three-body problem?
The three-body problem is solved nowadays (the existence and structure of possible solutions is known). It can also be simulated very precisely using numerics. There is no need for perturbation theory.
Sincere question: could it be that the underlying structure of the universe is really simple but since we have no idea what it is we have to use exotic mathematics for it?
That’s quite possible. A lot of what we think of as fundamental physics might turn out to be emergent behaviour.
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.
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.
> It turns out Newtonian mechanics is emergent from Relativity.
However relativity is far more complex than Newtonian physics.
However relativity is far more complex than Newtonian physics.
The kinds of physical influences that are allowed in Newtonian physics are more general than those allowed in relativity. Relativity requires physics to satisfy constraints.
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.
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.
> 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.
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.
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'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.
Theoretical physicists are like enterprise architects, there seems to be a preference for generality of prescription over practical experimentation/implementation.
This could the universe's way of telling us that knowledge requires work, mathematically speaking.
That’s an interesting point, but I don’t think the complexity gradient from underlying causes to macroscopic effects always goes the same way. The macroscopic dynamic effects of an earthquake on a building can be modelled fairly easily. Trying to model the earthquake’s dynamics through the earth at the molecular level is a lost cause though.
> The behaviour of _bosons_ is emergent from the behaviour of quarks.
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.
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.
Yes Hadrons sorry. Posting in a hurry.
>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.
>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.
It's strange because from galaxies to protons/neutrons/electrons it looks like the underlying theory is simpler. But if you try to look inside the protons it gets nastier each time you go down in the ladder of theories.
This is Stephen Wolfram’s bet. He thinks the universe is based on simple computation rules.
The emergence of complex traits from simple rules still requires exotic mathematics to describe macroscopically.
The emergence of complex traits from simple rules still requires exotic mathematics to describe macroscopically.
I’m just a computer guy, but I also see that as likely. When it comes down to it there’s less than 100 elements, made of a handful of forces and subatomic particles that make up the incredible universe that we live in. It seems somehow to get simpler as you go down.
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.
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.
The first fundamental step in any physics model (or theory) is to separate the easily describable "laws" from the almost impossible to describe "state". The perhaps surprising question is why anything at all can be separated but if that wasn't the case, we wouldn't be having this conversation.
"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.
"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.
> 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.
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).
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).
Computational complexity is defined for programs with respect to a parameter in the limit where that parameter is large. What's the objectively correct parameter for this comparison that every theory of physics has, after you convert it to a program in the objectively correct way, which you supposedly have?
The purpose of a physical theory is ultimately to derive a prediction of how a physical system will evolve after some time, within a given precision of measurement. The prediction can then be compared to a real measurement of the real system (or several, in the case of statistical predictions).
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).
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).
There is usually no obvious way to convert the initial condition of an arbitrary quantum system to that for a classical system. The other way around, there is some consensus what the quantization of standard classical systems should be, but to define this transition in general is, again, not straightforward. Also, you obviously don't arrive at the same final condition for two different theories, otherwise they would be equivalent.
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.
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.
Is there any formal proof of this computational complexity ladder you mention? Saying quantum is more complex than classical seems to imply P != NP. I don’t know nearly enough about general relativity to know how complex that is, though I’d have assumed it’d be less than quantum.
> 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)
Is there a formal version of this claim somewhere? (Beyond theorems about quantum computers)
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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.
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.
Number of equations != How simple something is
Humans have a limited capacity of holding something before their "mind's eye", like 5-7 items. The simpler the equation is, the easier it is to understand and operate. Of course, it only works provided that the parts of the equation are well-understood, too.
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.
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.
Yes, good notation helps dealing with complexity. My point was that "number of equations" involved in a model or theory is not an objective measure of, really, anything.
Every system of equations can be written as "A = 0" for a suitable definition of A. That doesn't make it simple.
Every system of equations can be written as "A = 0" for a suitable definition of A. That doesn't make it simple.
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.
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.
Some think the underlying structure of the universe is mathematics. That is, the universe isn’t merely describe by mathematics, but it is a mathematical structure.
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 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.
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.
I wish I understood it better, but you don’t have to go much below classical mechanics before I’m lost.
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.
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.
The deeper we go the more objects we find that we can only describe using mathematics, for sure. That’s not the same as them being mathematics though. We don’t know what the essential nature of these objects is.
It is an interesting speculation, but it’s also possible it’s just confusing the map for the terrain.
It is an interesting speculation, but it’s also possible it’s just confusing the map for the terrain.
If mathematics is the only way we know to describe them, then they might be mathematics. That seems like the simplest possible explanation, so that's probably why I suspect it's probably the correct.
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.
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.
What do you mean something other than mathematics? It's just a language problem nothing else. Just because the English language lacks the descriptive power it does not mean those objects are "mathematical", just as a round ball is not "English" in it's nature just because you can describe it's properties and behavior using that language. A quantum field is just that, a field, it exists just as much as a magnetic field exists and that you can observe directly. It is an area of space where a specific "force" you might say has impact on objects that enter that area of space. It's properties are measurable and they produce consequences in the world and it has a precise description using the language of mathematics
> It is an area of space where a specific "force" you might say has impact on objects that enter that area of space.
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.
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.
Typically in physics we derive laws from principles. For example, the law of conservation of momentum is derived from the principle of translation invariance.
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).
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).
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.
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.
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.
> whether the physical world is perfectly or merely highly consistent
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.
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.
I’ve wondered that before. The question would be how to differentiate the two possibilities. I suppose if you could somehow prove that our universe is the only one capable of existing while still meeting certain consistency requirements, then it might be fair to call that a “mathematical structure”.
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).
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 universe could be simpler than the current models suggest, but that would require taking a step back too far for the comfort of today’s STEM-oriented mind. For as long as natural sciences consider philosophy a load of hand-wavy abstract inapplicable hogwash they will be stuck iterating on existing physical models towards local maximum.
Can you give an example of your philosophy physics being applied to a real world problem?
What do you mean by “philosophy physics”?
Apart from that, I hope upon some reflection you realize that your sentence is an illustration of my point.
Apart from that, I hope upon some reflection you realize that your sentence is an illustration of my point.
I reflected and I have no idea what you're talking about, since all you said was "physics is wrong" while giving no evidence.
I guess the intended meaning was that you chose your request ("apply what you say to a problem") based on certain philosophical criteria. I think that's actually a surprisingly valid point, given the original (downvoted) comment.
Applying what someone claims is science is not a "philosophical criteria", that's how evidence and proof works.
Which is a philosophical criterion. Just 1000 years ago people would have found such a call for "science" strange and deeply insulting to their authority, which they would have needed to dare speculate about things such as math or physics. Most theories were made from "first principles" while things that were established by evidence (tried it a few times, it always worked, let's keep doing it) were mostly not considered science.
I have no idea what you are talking about and I don't think you do either. The person above thought that physics would be simpler if people used 'philosophy' (whatever that means).
Just in case you care to know what I'm talking about: I'm pointing out that philosophy had a lot of influence in changing the way science is done (leading to faster progress). This process is ongoing.
Of course, you don't help anyone make progress in physics just by suggesting they should "use some philosophy for a change".
Of course, you don't help anyone make progress in physics just by suggesting they should "use some philosophy for a change".
> I'm pointing out that philosophy had a lot of influence in changing the way science is done (leading to faster progress)
You aren't really pointing it out, you're just claiming that without any evidence at all.
You aren't really pointing it out, you're just claiming that without any evidence at all.
Your claiming that there exists some simpler physical theory that could be derived from philosophy.
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.
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.
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<x<1). If the number a_n calculated by Feyman diagrams doesn't grow proportional to the number of Feyman diagrams, then it is still entirely possible for x^n to shrink faster than any change in a_n.
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.
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.
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.
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.
>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.
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.
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It is generally believed that the perturbation expansion that we see in realistic quantum field theories are what are known as asymptomatic expansions. These are series that have a radius of convergence of zero (i.e. they only converge when the expansion parameter is exactly zero and diverge for all non zero values).
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.
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.
That's such a great comment; you have answered many of the questions I had after reading the article! Thank you!
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...
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...
Would be cool to train GPT with all math books and have it a go at a few unsolved math problems
The issue is that GPT is very bad at logical reasoning. Unfortunately, already for humans it is very difficult sometimes to recognize tiny (but devastating) logical errors in mathematical proofs.
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 through it using ChatGPT or something to translate the prose bits.
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.
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.
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.
Fyi, this is from the french. "Alien" is a bad translation. The better word is "stranger". Alien just sounds more scifi, akin to how "alien hand syndrome" is the more scifi translation of le mans etrangers, which is best translated as "stranger's hand".
https://www.webmd.com/brain/what-is-alien-hand-syndrome
https://www.webmd.com/brain/what-is-alien-hand-syndrome
Or perhaps better "foreigner", legal alien, someone from over the border.
The alien derivative is an operation which moves you from one sector to another. A bit like crossing a branch cut.
The alien derivative is an operation which moves you from one sector to another. A bit like crossing a branch cut.
Using different math approach is super fruitful.
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.
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.
I learned how to use differential equations to solve circuits as an undergraduate in electrical engineering, and also how to derive the more efficient methods from the differential equations by assuming solutions are (possibly complex) exponentials. No PhD required.
> If we used differential equations to solve [RC networks], 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.
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.
Can those Feymann diagrams be calculated by a computer? I understand that it raises expotentially, but why the 891 and 320 thousand diagrams cannot be solved by some tool?
Of course! But the R&D for those tools are very peculiar, at least as I understand it from a friend who works directly on this domain.
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:
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:
Yes, but you very quickly run out of steam.
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.
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.
Here's Écalle himself lecturing on the topic in English:
https://youtu.be/pTcmkBocYZU
https://youtu.be/pTcmkBocYZU
That links to 20 more related lectures on resurgence here: https://www.youtube.com/watch?v=pTcmkBocYZU&list=PLx5f8IelFR...
The way I read this: this method of finding non-pertrubative terms requires calculating perturbative terms first, that’s why it can’t be applied to QED and other more-or-less realistic theories. But of course string theorists can use this method to write more papers.
I'm much more interested in whether it can provide a better way to deal with the strong force. QCD is almost impossible to calculate with in most cases because everything diverges.
People can calculate perturbative processes in QED. Some observables are known to 7 loops (!), last I paid attention.
One sentence from Wikipedia:
> "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!
> "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!
Yeah, sounds 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.
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.
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Is the implication that each nonperturbative term represents a specific kind of field interaction? Thus would there be a finite set of nonperturbative terms that, if exhaustively found, would render the trans-series convergent?
That's an interesting idea but I think if it was the case, the article would have said so. What I got from it is that they haven't even got this to work in anything except for simplified toy universes. So I doubt anyone can answer your question yet.