# Importance of constrained Hamilton dynamics and BRST transformations

1. Jun 15, 2012

### sphyrch

(First of all apologies for the long wall of text)

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?

(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know how beneficial it is to study constrained Hamiltonian approach, because I'm not sure whether it's obsolete, having been replaced by better approaches (like path integrals?) which I should rather pay attention to. I'm not even sure what is the extent of applicability of constrained Hamiltonian dynamics..)

Thanks

EDIT: 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.

2. Jun 15, 2012

### vanhees71

I don't think it's obsolete. You gain a lot of insight about non-abelian gauge theories studying them also in the covariant operator formalism. It's of course good to begin with the path-integral formalism to understand where the more complicated operator formalism comes from. Then you should first study the Abelian case (QED) to get familiar with the concepts (Gupta Bleuler quantization is also an intermediate step).

Another point is that you should be aware that also in the path-integral formalism you should always start in the hamiltonian form, because this is the only correct one. Only in special cases, by integrating out the canonical field momenta, you come to the "naive" Lagrangian path integral. That's often the case but not always!

Already for simple qft models you obtain a wrong result if studying many-body QFT with finite chemical potentials when using the naive Lagrangian path integral as compared to the Hamiltonian path integral and then integrating out the canonical field momenta. I recommend to do this exercise for a simple free charged-boson QFT with chemical potential!

3. Jun 15, 2012

### dextercioby

Even a formal, algebraic approach to the subject of constraints is useful, not to mention the advanced one in terms of fiber bundles. There's no way to <guess> the path integral for QCD (as an example) without knowing where the ghosts come from.