(March 14, 2016 at 1:28 pm)Alex K Wrote:(March 14, 2016 at 12:26 pm)little_monkey Wrote: Me, misunderstanding?? No way,
Your Hamiltonian is important because we need it to do perturbation theory. It doesn't play much of a role in non-perturbative theories ( for instance, String Theory). The Lagrangian is far more important - it's with the Lagrangian you get your symmetries checked out, and most importantly, your Lorentz invariance is absolutely crucial to go from QM to QFT, and you get your theory Lorentz invariant through the Lagrangian. Moreover, the people in the 50's and 60' couldn't figure out the nuclear forces, both the weak and the strong. You don't know the force you're pretty much handicapped in developing any dynamical theory. So the whole plan was: try guessing the Lagrangian - you know if you have it right, you also know you have the right equation of motion. It was a nice way to circumvent not knowing the nuclear forces, and with Yukawa's idea, we could ignore "force" and replace it with "interaction".
Working with the Lagrangian in QFT is so much more fun for the reasons you mention, no argument about that. Especially if you're willing to employ path integrals from the get-go. Still, defining the S-Matrix and showing its unitarity is a bit of a shlep based purely on Lagrangian path integrals, compared to using the Hamiltonian time evolution.
Yeah, well, the problem is tit for tat: with the Lagrangian, it's manifestly Lorentz invariance, but then you need to check if it's unitarity. The Halmitonian gives you the reverse headache, it's manifestly unitarity but you need to check if it's Lorentz invariance. The advantage for the Lagrangian is what you need to calculate your time ordered product of fields in the LSZ formula, from there you get rid of your infinities if you have any by adding your counter terms. So you work more with the Lagrangian than the Hamiltonian.
Quote:However, your comment is not relevant for my point that Energy is important conceptually because - among other things - it is often directly related to the Hamiltonian which specifies the dynamics of the system. Whether going to the Lagrangian is more advantageous for some calculations is beside the point.
You've got that backward. It's your Lagrangian that will guarantee if you have the right equation of motion. Your Hamiltonian is just needed to get the free theory in there, of which you know the eigenstates and the eigenvalue. But once that's fixed, you work with the Lagrangian all the way to propagators to deal with various different interactions. By that point, you can forget that you have ever used the Hamiltonian.
