> 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.