# How important are constrained Hamiltonian dynamics and BRST transformation as a formalism?

I am to study BRST transformations, for which I'm currently trying to understand constrained Hamiltonian dynamics to treat systems with singular Lagrangians. The crude recipe followed is Lagrangian -> Hamiltonian -> Dirac brackets and their quantization. I have been told that these techniques are used in QFT, string theory/high energy , etc.

My question is are these formalisms indeed good and useful to learn? I'm confused because there are other formalisms and recipes (for example directly working with the Lagrangian). From a modern perspective, are these relevant? And does there exist any other formalism that I should not ignore? I will be starting my graduation soon; is it late to be studying constrained H formalism/BRST?

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What is "directly working with the Lagrangian"? The BRST transformation is just the fermionic symmetry that mixes up the gauge fixing determinant with the gauge fixed gauge field. It is not something seperate from the ghost fields that you get by directly inserting gauge fixing terms and determinants by hand. Polchinsky discusses it essentially perfectly in his string theory book--- I don't think one can ever improve on his presentation. –  Ron Maimon Jun 15 '12 at 2:07
Sorry but I'm not very clear about it myself. Directly working with Lagrangian as in one doesn't bother with the Hamiltonian in that approach, direct quantization from the Lagrangian stage. I have also heard that there are other approaches like Feynman path integrals, etc. But I don't know about their relative merits, and that's what I want to be clear about. –  1989189198 Jun 15 '12 at 7:10
I want to know how beneficial it is to study constrained Hamiltonian approach, because I'm not sure whether it's obsolete, having been replaced by better approaches which I should rather pay attention to. I'm not even sure what is the extent of applicability of constrained Hamiltonian dynamics... –  1989189198 Jun 15 '12 at 7:15
What you want is a translation from Dirac to path-integral. I agree that this will be useful, but I don't know if it exists in the literature. The formalism of "first class constraints" and "second class constraints" etc, should be equivalent to (and generalized by) ghosts and BRST, but I don't know the exact correspondence. One should work it out. Perhaps you can ask the question "What is the relation between path integral methods for dealing with constraints and Dirac's methods, involving the Dirac bracket?" This is a very specific question and it would be useful. –  Ron Maimon Jun 15 '12 at 18:08