Timeline for Dirac equation as Hamiltonian system
Current License: CC BY-SA 3.0
8 events
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| Aug 20, 2014 at 11:31 | comment | added | firtree | @kηives I've checked my notes and immediately found a mistake: cancelled $\gamma^0\partial_0$ and $\gamma^\mu\partial_\mu$ terms :-) Fixed that, I came to the same $\mathcal{H}=-\bar{\psi}i\gamma^i\partial_i\psi+m\bar{\psi}\psi$ ($i=1,2,3$) as you have. Thanks! Ah, if only I had seen it yesterday! | |
| Aug 19, 2014 at 18:04 | comment | added | kηives | @firtree I think you should end up with more terms then something proportional to mass only. Check my answer up above and see if that is correct and you get the same thing. | |
| Aug 18, 2014 at 23:03 | comment | added | firtree | @kηives I've tried both Lagrangians and I found that Hamiltonian is either $m\bar{\psi}\psi$ or $\tfrac{1}{2}m\bar{\psi}\psi+\mathrm{h.c.}$, being independent of any spatial derivatives of $\psi$. Is it okay for a fermion field? If I set $m=0$, I would have no Hamiltonian at all, and no evolution of the system, which seems wrong (waves should propagate properly, because Dirac equation works okay in the $m=0$ case). | |
| Nov 8, 2012 at 0:36 | comment | added | kηives | The true Lagrangian to use is $\mathcal{L}' = \frac{1}{2}(\mathcal{L}+\mathcal{L}^{\dagger})$ this way you won't run into a problem with the canonical momenta for either of the independent fields. | |
| Nov 7, 2012 at 21:14 | comment | added | kηives | @juanrga I think the canonical variables are independent. If you don't want to treat the variables independently, that's your choice, but I think they are independent since to write down the full Lagrangian you need to add the hermitian conjugate. Usually no one does, but that's their choice. | |
| Nov 7, 2012 at 11:20 | comment | added | Sasha | Yes, this is constrained system. I'm trying to extend phase space to super space and get super Poisson bracket that would give a meaningful answer. By the way, everything you wrote is classical field theory, because your spinors are fields on space-time not operators (I look at the situation from QFT point of view, not QM). | |
| Nov 6, 2012 at 20:51 | history | edited | juanrga | CC BY-SA 3.0 |
added 224 characters in body; added 8 characters in body
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| Nov 6, 2012 at 20:43 | history | answered | juanrga | CC BY-SA 3.0 |