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).
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.
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.
> 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.
Sorry, that was my mistake. I didn't expect that someone knowledgeable would say the Legendre transform "doesn't really show up in quantum mechanics". Even if you want to dispute whether the switch between Lagrangian and Hamiltonian formulations of quantum mechanics is truly a Legendre transform, rather than a generalization that doesn't deserve that name, it seems incumbent on you to state clearly that this was your (semantical) argument rather than pretending that this was obvious.
I don't know how a better way to "objectively" establish this than just to Google "path integral legendre transform"; the first hit is arXiv:1612.00462 (in a respectable journal by respectable authors, etc.) whose first sentence is "Legendre transforms play important roles in quantum field theory" before going on to review their central role in the divergence issues that appear immediately when you introduce perturbative expansions in QFT. Neither of us care about this random paper, but it just doesn't seem reasonable to dismiss Legendre transforms as not really showing up without further qualification.
Anyways, that was just to explain my mistaken assumption about your background. Nonetheless, I do apologize as I should have been more careful. I didn't mean to cause offense.
> You're mistakenly equating Legendre transform to Lagrangian, and basically saying "there is Legendre transform in quantum mechanics because there's Lagrangian"
No, I'm saying that if you have the same quantum system (including field theories) that you can treat in the Lagrangian and Hamiltonian formulations, the connection between them makes use of the Legendre transform.
> And no, saying no isn't a "technical mistake".
Now that I understand better what you were trying to say, I withdraw that language with apologies. I still think you're mistaken, but it's about aesthetics, semantics, and pedagogy rather technical assertions.
> 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.
It's certainly true that we can build field-theoretic Lagrangians from symmetries and calculate important quantities without touching a Legendre transform. But the same can be said about classical mechanics, and it doesn't mean the Legendre transform doesn't have a central organizing role in our body of knowledge.
> Second, when strictly doing quantum mechanics, Legendre transform itself doesn't make any sense because everything is an operator. ... Thinking that ordinary Legendre transform will just work in quantum mechanics is naive and plain wrong.
I definitely did not suggest that one could do a Legendre transform of operators as if they were commuting variables. But saying that the Legendre transform doesn't play a role in quantum systems for that reason is like saying Maxwell's equations don't play a role in QED because E and B are commuting variables in Maxwell's equations.
> Path integrals don't have anything to do with relativity, they work perfectly fine for non-relativistic QM as well.
Yes, I'm well aware. The reason I parenthetically asked "Or maybe you just haven't used quantum mechanics outside of the non-relativistic domain...?" is because most students first use it in a relativistic QFT course (rather than just seeing it in Sakurai). I should have said "field-theoretic context" to be clearer, although of course you can use them even without fields.
> And no, you don't need path integrals for doing relativistic QM/QFT either.
Sure, you can do some QM/QFT without path integrals just like you can do classical mechanics solely in the Lagrangian framework or solely in the Hamiltonian framework. But, unlike classical mechanics and unlike quantum field theories where there is a larger diversity of techniques, non-relativistic QM with a finite number of degrees of freedom is usually done solely in the Hamiltonian framework, so people rarely think about the connection to Lagrangians.
> 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.
Louville's theorem is a result within Hamiltonian mechanics. I've never heard anyone suggest that it's a distinct formulation/framework, and I'm honestly confused as to what you could mean by this. Can you explain your viewpoint more, or point me toward somewhere where I could read more about it?
(Or do you just mean using Louville's equation to evolve a probability distributions over phase space rather than Hamilton's equation to evolve individual phase-space points? This is no less Hamiltonian mechanics than probabilistic Turing machines are Turing machines.)
Whether the Hamilton-Jacobi equation qualifies as a similarly fundamental formulation as the Lagrangian or Hamiltonian formulations is at least an arguable assertion, although one I certainly disagree with. (Recall my "remotely similar importance" criteria.) I'd be happy to debate you on this, but it sounds like you're mostly uninterested in aestetic/ontological questions, e.g., what makes a formulation qualify as "fundamental". That's fine, you don't have to care about those debates to do good science, but that's the nature of our disagreement here, and I think you would need to address it before saying I'm just trying to serve my narrative. (Honestly, what ulterior purpose do you think I have? Scoring fake internet point by writing blog post rants?)
> Hand & Finch reads perfectly fine to me, everything is there,
I previously suggested that you quote the parts you think are clear, as I have quoted the parts I think are confused in other books. (Since you're the one who thinks it's explained well in Hand & Finch, it makes a lot more sense for you to quote the relevant section where you think that happens than for me to quote all the sections where I think it doesn't.) You can certainly dismiss this exercise as not worth your time, but I can't really engage with you on this because you're just asserting the same thing as your first comment, not actually gathering evidence for it.
"Reads perfectly fine to me", a professor, is not a good criterion for deciding which books do a good job of explaining to students. Curse of knowledge and all that.
> 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.
I'm not saying treatments like Hand & Finch are bad because some confused students exist, I'm saying they're bad because most students remain confused and better, much-less-confusing treatments are possible. I challenged you to compare my explanation to Hand & Finch's by seeing which is actually understood more by students.
To be concrete, I predict that less than 20% of any class of undergraduate students (e.g., Princeton, where I took Hand & Finch with confused classmates) will be able to correctly report that two functions are each other's Legendre transforms when their derivatives are inverses. (If you think that sort of failure rate is just par for the course, then I think we have very different teaching standards and there's not much more to discuss.)
Before I wrote the blog post, I surveyed two active research faculty and three postdocs and none knew this fact. I similarly challenge you to ask your colleagues to explain the Legendre transform on the spot and see the fraction of them that can say something more rigorous than H = pv - L.
> I personally have better things to do than dwelling on mysteries of Legendre transform, or deep meanings and magical connections of the number pi.
If you haven't spent any time thinking about it, wouldn't it be better to just not have an opinion one way or the other? "The Legendre transform isn't deep" is a very different assertion to "it's pointless to worry which parts of physics are the deepest".
Regardless, I think you're making a mistake that ultimately will have a small but non-trivial negative impact on your students.
> 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!
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:
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:
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).