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I'm recently learning the Dirac equation and we're following the more historical approaching working in the Dirac basis. At first it seems OK that the upper two components are interpreted as positive energy and the lower two negative. However, when I learnt that after a boost the spinor (say initially at (1000)) becomes something like this [ tex ] \begin{array}{cc}1\\0\\P_z\\P_+\end{array} [ /tex ] with some constant factors.

So I'm now really curious that what is the spin direction (the direction in which this spinor is the eigenvector of the spin operator) now. I can see that it's definitely not in the z direction anymore, but I don't quite see how I can get the spin direction. And actually how do people interpret the lower indices in this case.

Thanks!