3 ms·

For a more current example of a mathematical technique that preceded formalization by a considerable amount, consider renormalization. Particularly renormalizat

by btilly 2y ago

For a more current example of a mathematical technique that preceded formalization by a considerable amount, consider renormalization. Particularly renormalization over a calculation that takes place over Feynman diagrams. For decades physicists were happily using this to predict experiment, while mathematicians were tearing their hair out trying to make some formal sense of this, even if only in a limited context. I'd have to do some poking around to find out whether mathematicians are happy about it yet, even though the idea is older than I am.

hansvm 2y ago
For Feynman diagrams in particular, isn't the status quo that they're known to not converge (i.e., the motivation generating the idea is definitely wrong), but the first few terms are still predictive for some reason?
- eigenspace 2y ago
  It's not that the motifivation / generating idea is definitely wrong, the problem you're alluding to here is just that they form an asymptotic series[1]. An asymptotic series is one that diverges if you sum the whole series, but will accurately approximate some function if you take a finite partial sum. The reason that so many perturbative series are only asymptotic in quantum mechanics actually makes a lot of sense if you think about what these perturbative series are saying. Typically, each order of the perturbative series is meant to account for the physics at smaller and smaller length scales, but we know that our physical theories aren't applicable at infinitely small length scales because we know that at the very least, we need a theory of quantum gravity. So in some sense, the problem is just that we shouldn't even want to sum these series to infinite order because there is no reason to think that the ultraviolet behaviour in that series has any physical meaning. [1] https://en.wikipedia.org/wiki/Asymptotic_expansion https://en.wikipedia.org/wiki/Asymptotic_expansion
  - hansvm 2y ago
    That's a bit of a strange perspective: 1. We have some ideas. 2. Those ideas can't possibly be correct, because quantum gravity can't quite fit in. 3. Suppose anyway that the ideas are good enough; look at this cool equation. 4. Yeah, it doesn't... technically...converge, but the first few terms are pretty good. Your comment about not knowing things in the limit yet makes perfect sense, but the motivating argument for that math in the first place is limiting behavior. It makes a ton of sense that none of these ideas converge, but it's peculiar that some of them are basically right for small terms when they're created from a foundation of convergence.
    - eigenspace 2y ago
      > but the motivating argument for that math in the first place is limiting behavior. No it isn't. The entire reason we truncate the series and dont sum the entire series off to infinity is exactly that the limiting behaviour isn't well defined.
eigenspace 2y ago
I wouldn't say physicists were 'happily' using renormalization back in the day. Rather, some of them were happy and many others were very concerned. Then, the ideas behind renormalization were formalized in terms of (poorly named) renormalization group flows, and by the early 90s it was on quite solid mathematical footing (though there remains to this day quite a bit of confusing folklore and fear around the topic, complete with plenty of bad pedagogy that has given many an undergrad as well as many the interested mathematician a lot of anxiety) However, simultaneously with us coming to understand renormalization better, we've also come to realize it's really not such a big deal and it was supremely overused back in the day. Nowadays, most modern field theorists think in terms of 'effective field theories' and are not nearly so interested in trying to sum infinite perturbative series so we have a lot less use for renormalization (though it does still have its place)
- btilly 2y ago
  That makes sense. I ran across the idea, informally, back in the 1990s in grad school. And what I heard then was concerning. I heard physicists saying that it was fine. But I also saw that their notion of fine and mine did not agree... I had not really updated my understanding much since then.

by btilly 2y ago

hansvm 2y ago

For Feynman diagrams in particular, isn't the status quo that they're known to not converge (i.e., the motivation generating the idea is definitely wrong), but the first few terms are still predictive for some reason?

eigenspace 2y ago
It's not that the motifivation / generating idea is definitely wrong, the problem you're alluding to here is just that they form an asymptotic series[1]. An asymptotic series is one that diverges if you sum the whole series, but will accurately approximate some function if you take a finite partial sum. The reason that so many perturbative series are only asymptotic in quantum mechanics actually makes a lot of sense if you think about what these perturbative series are saying. Typically, each order of the perturbative series is meant to account for the physics at smaller and smaller length scales, but we know that our physical theories aren't applicable at infinitely small length scales because we know that at the very least, we need a theory of quantum gravity. So in some sense, the problem is just that we shouldn't even want to sum these series to infinite order because there is no reason to think that the ultraviolet behaviour in that series has any physical meaning. [1] https://en.wikipedia.org/wiki/Asymptotic_expansion https://en.wikipedia.org/wiki/Asymptotic_expansion
- hansvm 2y ago
  That's a bit of a strange perspective: 1. We have some ideas. 2. Those ideas can't possibly be correct, because quantum gravity can't quite fit in. 3. Suppose anyway that the ideas are good enough; look at this cool equation. 4. Yeah, it doesn't... technically...converge, but the first few terms are pretty good. Your comment about not knowing things in the limit yet makes perfect sense, but the motivating argument for that math in the first place is limiting behavior. It makes a ton of sense that none of these ideas converge, but it's peculiar that some of them are basically right for small terms when they're created from a foundation of convergence.
  - eigenspace 2y ago
    > but the motivating argument for that math in the first place is limiting behavior. No it isn't. The entire reason we truncate the series and dont sum the entire series off to infinity is exactly that the limiting behaviour isn't well defined.

eigenspace 2y ago

I wouldn't say physicists were 'happily' using renormalization back in the day. Rather, some of them were happy and many others were very concerned. Then, the ideas behind renormalization were formalized in terms of (poorly named) renormalization group flows, and by the early 90s it was on quite solid mathematical footing (though there remains to this day quite a bit of confusing folklore and fear around the topic, complete with plenty of bad pedagogy that has given many an undergrad as well as many the interested mathematician a lot of anxiety) However, simultaneously with us coming to understand renormalization better, we've also come to realize it's really not such a big deal and it was supremely overused back in the day. Nowadays, most modern field theorists think in terms of 'effective field theories' and are not nearly so interested in trying to sum infinite perturbative series so we have a lot less use for renormalization (though it does still have its place)

btilly 2y ago
That makes sense. I ran across the idea, informally, back in the 1990s in grad school. And what I heard then was concerning. I heard physicists saying that it was fine. But I also saw that their notion of fine and mine did not agree... I had not really updated my understanding much since then.