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I'm studying the Weyl's Equations from Section 1.5 of Perkins' Introduction to High Energy Physics.
This is the Dirac Equation: \begin{equation} E\psi=(\alpha.p+\beta m)\psi \end{equation} where $\alpha=\begin{bmatrix} 0 & \sigma \\ \sigma & 0 \end{bmatrix},\beta=\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}, \sigma $ being a Pauli Matrix and the 1 and -1 in $\beta$ are $2 \times 2$ matrices.

I just tried to play with this equation, assuming $p_y=p_z=0,\psi= \begin{bmatrix} \psi_1 \\ \psi_2 \\ \psi_3 \\ \psi_4 \end{bmatrix}$, the RHS of the equation gave me a spinor $\begin{bmatrix} p_x\psi_4+ m\psi_1 \\ p_x\psi_3+ m\psi_2 \\ p_x\psi_2- m\psi_3 \\ p_x\psi_1- m\psi_4 \\ \end{bmatrix}$. Now as I understand, $\psi_1,\psi_2$ are the positive energy eigenstates corresponding to two spins and $\psi_3,\psi_4$ corresponding to negative energy. Somehow it seems that these states have coupled in the process, and I do not understand this intuitively or physically. The following subsection talks about helicity conservation; as to how scalar bosons don't preserve helicity but vector/axial bosons do, and I'm guessing it has to do with the equations but can you please explain this in detail?

Ambica Govind
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1 Answers1

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I'm not sure what physical intuition you are after about the coupled components of Dirac spinors, which describe a particle and an antiparticle of both spin directions each. The Dirac Equation with your frame assumption amounts to finding the null vectors in the kernel of$$ \begin{bmatrix} E-m & 0 &0& -p_x \\ 0& E-m&-p_x & 0 \\ 0&-p_x& E+m& 0 \\ -p_x&0&0& E+m\end{bmatrix}, $$ namely , unnormalized, the transposes of $(1,0,0,p_x/(E+m))$; $~(0,1,p_x/(E+m),0)$; $~(0,p_x/(E-m),1,0)$; and $(p_x/(E-m), 0, 0, 1)$.

The matrix is only diagonal for vanishing momentum, for which the null vectors collapse to what you observe, positive and negative energy solutions, respectively. But note for m=0, $|p_x|=E$, the solutions merge to just two, as you'd expect, on acct of the double degeneracy.

Physically, you might think of this as an association of fermions to their antifermions with the opposite spin.

Cosmas Zachos
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