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Electron is a spin $\frac{1}{2}$ particle, so needs 2-component wave function but the Dirac theory of electron is based on 4-component wave function, are all components of it independent or only two of them are independent?

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4 components means two spinors $\phi$ and $\chi$. You can obtain one from the other. See – jinawee Dec 29 '13 at 9:27

2 Answers 2

You have discovered the fact that the Dirac spinors form a reducible representation of Spin(3,1) $\simeq$ SL(2,C), the covering group of SO(3,1)$^+$. The left and right Weyl spinors, which have two components, are irreducible representations.

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The 4-component Dirac spinor $\psi$ is formed by stacking two two-component Lorentz spinors $\xi^{A},\eta^{\dot{B}}$. The Dirac equation is the pair of coupled equations, $$ \hat{p}^{A}_{\dot{B}}\eta^{\dot{B}}=m\xi^{A} $$ $$ \hat{p}^{\dot{A}}_{B}\xi^{B}=m\eta^{\dot{A}} $$ where $p^{A}_{\dot{B}}$ is equivalent to the relativistic momentum operator $\hat{p}_{\mu}=i\partial/\partial x^{\mu}$. The spinor form of the momentum operator can be written, $$ \hat{p}^{A}_{\dot{B}}=2i\frac{\partial}{\partial X^{\dot{B}}_{A}} $$ and the Hermitian tensor $X^{\dot{A}}_{B}$ is equivalent to the spacetime point $x^{\mu}$. Now try the plane wave ansatz, $$ \xi^{A}=u^{A}\exp(iK^{C}_{\dot{D}}X^{\dot{D}}_{C}/2) $$ $$ \eta^{\dot{A}}=v^{\dot{A}}\exp(iK^{C}_{\dot{D}}X^{\dot{D}}_{C}/2) $$ where $u$ and $v$ are constant spinors. Substituting in the Dirac equation, $$ mv^{\dot{A}}=-u^{B}K^{\dot{A}}_{B} $$ shows that (in this case) the $v$ spinor is fixed by the choice of the $u$ spinor: only two of the four components of the Dirac spinor are free. The wave number is found by putting the last equation into the Dirac equation, $$ m^{2}v^{\dot{A}}=v^{\dot{C}}K^{\dot{A}}_{B}K^{B}_{\dot{C}}=v^{\dot{C}}k_{\mu}k^{\mu}\delta^{\dot{A}}_{\dot{C}}=v^{\dot{A}}k_{\mu}k^{\mu} \ . $$ So, the two components $\eta^{\dot{A}}$ are fixed by the choice of the $\xi^{A}$.

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Is this also the case when particle is not free (or non-plane wave solutions)? – richard Dec 29 '13 at 13:00
@richard : if one boosts a dotted or undotted spinor along the z axis, one component grows and the other reduces so for a large boost a four component Dirac spinor would have only two components. I think this makes sense in a hand-waving way. – Stephen Blake Dec 29 '13 at 14:14

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