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Everything depends on how your fields (vectors and spinors are fields in the classical theory, and when you quantize in QFT, they become operator-valued fields) transform when you make a Lorentz transform: An scalar is a field that doesn't change at all: $\phi'(x') = \phi(x)$. Examples are the Higgs and pions. A vector field is a field that transform like ...


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Be careful: The spin constribution to the number current is proprotional to $\nabla \times S$ where $S$ is the spin density. The spin contribution to the momentum density is $(\nabla\times S)/2$ because the $g=2$ gyromagnetic ratio makes spin twice as effective at contributing to the number (electric) current as to the momentum density. See my contribution ...


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Dirac equation $\left( i \gamma^\mu \partial_\mu - m\right) \psi=0$ gives in momentum space $$ \left(\gamma^\mu p_\mu - m\right)\psi(p)=0 $$ which becomes in the rest frame $p^\mu= (m,\vec{0})$ $$ \left(\gamma^0 - 1\right)\psi(p)=0 $$ Now you can show that $\gamma^0 - 1$ is a projection operator of rank $D/2$, or more explicitly work in the Dirac ...


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Can these two pictures be connected in some way? Yes, that's why the Wikipedia spinor article features a picture of a Möbius strip: GNUFDL image by Slawekb, see Wikipedia The Mobius strip also features in the Mathspages Dirac's belt article where you can read that it's "reminiscent of spin-1/2 particles in quantum mechanics, since such particles must be ...



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