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A projective unitary representation of ${\rm SO(3)}$ satisfies $$U(R_1)U(R_2)=e^{i\phi(R_1,R_2)}U(R_1R_2)\tag{1}$$ where $R_1,R_2\in {\rm SO(3)}$. How to show that the $j=1/2$ representation, $U(R(\th …

The title of the question is not quite correct. I offer a a partial answer and I hope it helps to some extent! I might expand it a bit later. The way the angular momentum is first defined in classica …

It is to be emphasized that $\psi_L$ and $\psi_R$ are not 2-component spinors; $\psi_L$($\psi_R$) are still 4-component spinors with lower(upper) two entries being zero and upper(lower) two entries being … Let $$\psi_L=\begin{pmatrix}\chi\\0\end{pmatrix},~~\psi_R=\begin{pmatrix}0\\\zeta\end{pmatrix},\tag{3}$$ where $\chi$ and $\zeta$ are two-component spinors, called Weyl spinors. …

A massless spin-1/2 particle can be represented by 2-component Weyl spinors. This can be seen by expressing the Dirac equation with $m=0$ in the Weyl basis. … The solutions are now 4-component Dirac spinors. …

Weyl spinors are massless. Is the converse also true? Does any massless spin-1/2 fermion have to be a two-component Weyl spinor? … Is there a reason for not using two-component Weyl spinors for the electron when it is massless? …

Each spin-1/2 particle is associated with a $(2\times\frac{1}{2}+1)$=2-dimensional vector space $\mathbb{V}$ as far as its spin degree of freedom is concerned. A composite system of two spin-1/2 parti …

Theoretically, after a rotation of $2\pi$, a fermion wavefunction picks up a minus sign, and it is after a rotation through $4\pi$ that it returns to its initial quantum state. Now, the wave-functions …

Neutinos are either Dirac particles or Majorana particles but can’t be both at the same time. Then how can we write a general mass term as the sum of a Dirac mass term and a Majorana mass term? When w …

In Chiral representation, a Majorana spinor looks like: $$\psi=\begin{pmatrix} \psi_L\\ -i\sigma^2\psi_L^*\end{pmatrix}$$ In this representation the Right handed field is the charge-conjugate of th …

In chiral basis, $\psi=\begin{pmatrix} \psi_L\\ \psi_R \end{pmatrix}$ and therefore, $\overline\psi=\psi^\dagger\gamma^0=\begin{pmatrix} \psi^\dagger_L & \psi^\dagger_R \end{pmatrix}\gamma^0=\begin{pm …

In quantizing a scalar field, we impose commutation relation between the field operators by hand. On the other hand, anti-commutation relation is imposed between Dirac field operators by hand. As a co …

Does the concept of both helicity and chirality make sense for a massive Dirac spinor? A massive electron in the chiral basis is written as a column made up of $\psi_L$ and $\psi_R$. What is the signi …

The massless neutrinos can be represented by two component Weyl spinors. …

EDIT: - Let $\psi_L$ and $\psi_R$ be 2 component left-handed and right-handed Weyl spinors. Their transformation properties are known. … This is possible because we started from the definition of left-handed and right-handed Weyl spinors and their transformation properties are known. Right? …