Take the 2-minute tour ×
Physics Stack Exchange is a question and answer site for active researchers, academics and students of physics. It's 100% free, no registration required.

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?

share|improve this question
    
4 components means two spinors $\phi$ and $\chi$. You can obtain one from the other. See en.wikipedia.org/wiki/Dirac_spinor –  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.

share|improve this answer

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 $\xi^{A}$.

share|improve this answer
    
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

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.