5 ms·

I don't understand this mysticism. Legendre transformation is just one example of canonical transformations that you can do switch from one set of variables to

by tagrun 7y ago

I don't understand this mysticism. Legendre transformation is just one example of canonical transformations that you can do switch from one set of variables to another, typically out of convenience (you can rest assured physicists are aware of this, and I would be more conservative before saying it lies "at the heart" of anything --it just links Hamiltonian mechanics to Lagrangian mechanics, but they're basically two different ways we use to describe the same physical thing, each parametrization with their own intuitive features--, and it doesn't really show up in quantum mechanics). Yes, you can go beyond the actual purpose of "I just want to get a function of x,y starting from a function of x,z" and do illustrative geometric interpretations for every single canonical transformation out there, but physics books tend be practical and not to dwell on such mathematical "curiosities". Still, Legendre transform is arguably the most important one, and some books do go further. On top of my mind, you can see Analytical Mechanics by Hand & Finch, preceding that blog post by 2 decades. (Interestingly, he claims to have read this book --not carefully, apparently, because the equation he finds surprising is there; it's just written in words relating two equations, rather than as a new symbolic equation).

jessriedel 7y ago
You mistakenly say the Legendre Transform doesn't really show up in quantum mechanics, which is likely because you didn't notice when it appeared in the construction of the path integral. (Or maybe you just haven't used quantum mechanics outside of the non-relativistic domain, in which case you've probably only used the Hamiltonian formulation and don't know why? I don't know your background.) That's an easy mistake to make if you think about it as just some coordinate transform, since then you won't notice it when it's used in a different guise than when it was taught to you. Many field theorists would say the path integral is the most beautiful idea in theoretical physics -- Noether's theorem probably being the only idea that is more popular -- and if you don't understand how it bakes in the Legendre transform you don't understand it well. Other than that technical mistake, which I think is revealing, the rest of your comment is hard to respond to because it takes the form "this thing you claim is important for given reasons just isn't; it's not a big deal". So all I can do is try to flesh out my reasons, and hope you'll engage with them. The Lagrangian and Hamiltonian formulations are the two primary ways to write down fundamental physical laws. There is no third formulation of remotely comparable importance or clarity, so the Legendre transform has a special role not enjoyed by other coordinate transforms. If you want to actually understand why laws are formulated how they are, and especially if you want to be prepared to move beyond them in case new physics requires it, you need to have a deep understanding of how the Lagrangian and Hamiltonian formulations are linked. Otherwise it's impossible to answer questions like: "If the Lagrangian formulation put space and time on equal footing, and if the Hamiltonian formulation gives a preferred role to time (generating time evolution), could we give a similarly preferred role to space?" "More generally, why isn't there a third or forth major formulation of mechanics?" "Where does the incredible richness of symplectic geometry come from, and why isn't there anything similar associated with the Lagrangian formulation?" "How would any of this change if there was more than one time dimension?" Regardless, Hand & Finch is awful. I encourage you to quote the piece of Hand & Finch on the Legendre Transform that you think is clear, as I have quoted the books I think are confused. And then we can see whether students find my explanation or their explanation clearer. If you say "well, the ideas are encoded in Hand & Finch even if the students don't understand them", you've missed the point. I make zero claims of novelty, and neither, I'm sure, would Hand & Finch. (These ideas are nearly two centuries old, so pointing out that their textbook is two decades old is silly.) My complaint, rather, is that these old ideas crucial to the invention of mechanics are not being faithfully transmitted to generations of students who take mechanics for granted.
- newacctjhro 7y ago
  > Otherwise it's impossible to answer questions like: "If the Lagrangian formulation put space and time on equal footing, and if the Hamiltonian formulation gives a preferred role to time (generating time evolution), could we give a similarly preferred role to space?" "More generally, why isn't there a third or forth major formulation of mechanics?" Hey, can you answer those questions? Or point to the right answers. Thanks!
  - jessriedel 7y ago
    I'm mostly unable. It was musing about these questions that got me interested in really understanding the Legendre transform. I can say that dynamical equations that try to generate spatial translation from initial data on a time-like slice, rather than time translation from a space-like slice like Hamiltonian dynamics, are doomed because there is no well-posed initial-value problem (except in certain special cases involving massless particles), e.g., you generically cannot infer what's far from a spatial plane even if you know everything that happens on that plane for all time. Related topics: http://www.scholarpedia.org/article/Hyperbolic_dynamics http://www.scholarpedia.org/article/Hyperbolic_dynamics https://en.wikipedia.org/wiki/Well-posed_problem https://en.wikipedia.org/wiki/Well-posed_problem Also, I would say that the fact that the Legendre transform is a manifest involution gives (quite) weak evidence that there are no other major formulations to find. Of course, it's possible to use a hybrid strategy, Routhian mechanics, with Lagrangian and Hamiltonian formulations on different degrees of freedom: https://en.wikipedia.org/wiki/Routhian_mechanics https://en.wikipedia.org/wiki/Routhian_mechanics
- tagrun 7y ago
  Ugh. So much noise. I was going to avoid this but I'll bite once. I'm a professor in condensed matter physics, doing research actively, quite well aware of Lagrangian mechanics and relativistic QFT, path integrals, and I also taught QFT. That's my background. You're mistakenly equating Legendre transform to Lagrangian, and basically saying "there is Legendre transform in quantum mechanics because there's Lagrangian". No. And no, saying no isn't a "technical mistake". First, when building theories from symmetries, you typically start from a Lagrangian as the fundamental quantity, as opposed to taking the Legendre transform of something. Second, when strictly doing quantum mechanics, Legendre transform itself doesn't make any sense because everything is an operator. It's useful before you do a canonical quantization of a classical theory, because you can only do Legendre transform between classical Hamiltonian or Lagrangian, but that's still something you do when doing classical mechanics, not quantum mechanics. Even then you need to be careful with noncommuting operators --Legendre transform won't magically give you the correct Lagrangian and you need to do some trial and error to find the correct form (which happens even with the QED Lagrangian with derivatives and field tensor terms). Thinking that ordinary Legendre transform will just work in quantum mechanics is naive and plain wrong. Path integrals don't have anything to do with relativity, they work perfectly fine for non-relativistic QM as well. And no, you don't need path integrals for doing relativistic QM/QFT either. There are third and fourth and other formulations of mechanics (Louville theorem, Hamilton-Jacobi equation, ...). You're just dismissive about them because they don't serve your narrative. Hand & Finch reads perfectly fine to me, everything is there, it's not "encoded", you just need to read the equation and the text. There will always be confused students, even if you spell everything out. And this whole attitude reminds me of Feynman's comment on mathematical physicists or mathematicians doing physics. I personally have better things to do than dwelling on mysteries of Legendre transform, or deep meanings and magical connections of the number pi.

by tagrun 7y ago

jessriedel 7y ago

You mistakenly say the Legendre Transform doesn't really show up in quantum mechanics, which is likely because you didn't notice when it appeared in the construction of the path integral. (Or maybe you just haven't used quantum mechanics outside of the non-relativistic domain, in which case you've probably only used the Hamiltonian formulation and don't know why? I don't know your background.) That's an easy mistake to make if you think about it as just some coordinate transform, since then you won't notice it when it's used in a different guise than when it was taught to you. Many field theorists would say the path integral is the most beautiful idea in theoretical physics -- Noether's theorem probably being the only idea that is more popular -- and if you don't understand how it bakes in the Legendre transform you don't understand it well. Other than that technical mistake, which I think is revealing, the rest of your comment is hard to respond to because it takes the form "this thing you claim is important for given reasons just isn't; it's not a big deal". So all I can do is try to flesh out my reasons, and hope you'll engage with them. The Lagrangian and Hamiltonian formulations are the two primary ways to write down fundamental physical laws. There is no third formulation of remotely comparable importance or clarity, so the Legendre transform has a special role not enjoyed by other coordinate transforms. If you want to actually understand why laws are formulated how they are, and especially if you want to be prepared to move beyond them in case new physics requires it, you need to have a deep understanding of how the Lagrangian and Hamiltonian formulations are linked. Otherwise it's impossible to answer questions like: "If the Lagrangian formulation put space and time on equal footing, and if the Hamiltonian formulation gives a preferred role to time (generating time evolution), could we give a similarly preferred role to space?" "More generally, why isn't there a third or forth major formulation of mechanics?" "Where does the incredible richness of symplectic geometry come from, and why isn't there anything similar associated with the Lagrangian formulation?" "How would any of this change if there was more than one time dimension?" Regardless, Hand & Finch is awful. I encourage you to quote the piece of Hand & Finch on the Legendre Transform that you think is clear, as I have quoted the books I think are confused. And then we can see whether students find my explanation or their explanation clearer. If you say "well, the ideas are encoded in Hand & Finch even if the students don't understand them", you've missed the point. I make zero claims of novelty, and neither, I'm sure, would Hand & Finch. (These ideas are nearly two centuries old, so pointing out that their textbook is two decades old is silly.) My complaint, rather, is that these old ideas crucial to the invention of mechanics are not being faithfully transmitted to generations of students who take mechanics for granted.

newacctjhro 7y ago
> Otherwise it's impossible to answer questions like: "If the Lagrangian formulation put space and time on equal footing, and if the Hamiltonian formulation gives a preferred role to time (generating time evolution), could we give a similarly preferred role to space?" "More generally, why isn't there a third or forth major formulation of mechanics?" Hey, can you answer those questions? Or point to the right answers. Thanks!
- jessriedel 7y ago
  I'm mostly unable. It was musing about these questions that got me interested in really understanding the Legendre transform. I can say that dynamical equations that try to generate spatial translation from initial data on a time-like slice, rather than time translation from a space-like slice like Hamiltonian dynamics, are doomed because there is no well-posed initial-value problem (except in certain special cases involving massless particles), e.g., you generically cannot infer what's far from a spatial plane even if you know everything that happens on that plane for all time. Related topics: http://www.scholarpedia.org/article/Hyperbolic_dynamics http://www.scholarpedia.org/article/Hyperbolic_dynamics https://en.wikipedia.org/wiki/Well-posed_problem https://en.wikipedia.org/wiki/Well-posed_problem Also, I would say that the fact that the Legendre transform is a manifest involution gives (quite) weak evidence that there are no other major formulations to find. Of course, it's possible to use a hybrid strategy, Routhian mechanics, with Lagrangian and Hamiltonian formulations on different degrees of freedom: https://en.wikipedia.org/wiki/Routhian_mechanics https://en.wikipedia.org/wiki/Routhian_mechanics
tagrun 7y ago
Ugh. So much noise. I was going to avoid this but I'll bite once. I'm a professor in condensed matter physics, doing research actively, quite well aware of Lagrangian mechanics and relativistic QFT, path integrals, and I also taught QFT. That's my background. You're mistakenly equating Legendre transform to Lagrangian, and basically saying "there is Legendre transform in quantum mechanics because there's Lagrangian". No. And no, saying no isn't a "technical mistake". First, when building theories from symmetries, you typically start from a Lagrangian as the fundamental quantity, as opposed to taking the Legendre transform of something. Second, when strictly doing quantum mechanics, Legendre transform itself doesn't make any sense because everything is an operator. It's useful before you do a canonical quantization of a classical theory, because you can only do Legendre transform between classical Hamiltonian or Lagrangian, but that's still something you do when doing classical mechanics, not quantum mechanics. Even then you need to be careful with noncommuting operators --Legendre transform won't magically give you the correct Lagrangian and you need to do some trial and error to find the correct form (which happens even with the QED Lagrangian with derivatives and field tensor terms). Thinking that ordinary Legendre transform will just work in quantum mechanics is naive and plain wrong. Path integrals don't have anything to do with relativity, they work perfectly fine for non-relativistic QM as well. And no, you don't need path integrals for doing relativistic QM/QFT either. There are third and fourth and other formulations of mechanics (Louville theorem, Hamilton-Jacobi equation, ...). You're just dismissive about them because they don't serve your narrative. Hand & Finch reads perfectly fine to me, everything is there, it's not "encoded", you just need to read the equation and the text. There will always be confused students, even if you spell everything out. And this whole attitude reminds me of Feynman's comment on mathematical physicists or mathematicians doing physics. I personally have better things to do than dwelling on mysteries of Legendre transform, or deep meanings and magical connections of the number pi.