Motivation for spinors

Question

1. After it was found that the gamma matrices couldn't be Pauli matrices and only had to be larger and even, why was their need to define a new algebraic object (i.e a Dirac spinor)?

2. Why couldn't a vector or a tensor do?

3. Plus what exactly is a spinor as compared to vectors?

Answer 1

As I had wrote here, the Lorentz group can be represented as the direct product of two $SU(2)$ or $SO(3)$ groups. The $SU(2)$-realization of the irrep, $\left( \frac{n}{2}, \frac{m}{2}\right)$, refers to the spinor tensor $\psi_{a_{1}...a_{n}\dot{b}_{1}...\dot{b}_{m}}$, where the sum $n + m$ represents the spin value of representation: $\frac{n + m}{2} = s$. If $s$ is half-integer, we must work with spinor indices. If $s$ is integer, we may convert all spinor indices to vector one and forget about spinors.

The one-particle representations of the Poincare group (which describes of particle with definite mass and spin/helicity) are also characterized by spinor tensors $\psi_{a_{1}...a_{n}\dot{b}_{1}...\dot{b}_{m}}$, which are the irrep of the Lorentz group. So we assume that at least for describing the each free spin-half particle we must work with spinor indices.

The Stern-Gerlach experiment had showed that electron has spin one half. So as the Poincare irrep with spin $\frac{1}{2}$ and mass $m$ it must be described by 2-spinor $\psi_{a}$ (or by the irrep $\left(\frac{1}{2}, 0\right)$ ) of by 2-spinor $\psi_{\dot {a}}$ (or by the irrep $\left(0, \frac{1}{2}\right)$) . But it is not hard to show, that this representation isn't invariant under $C, P, T$-transformations (while the real theory is invariant), so we must to take the direct sum of $\left(\frac{1}{2}, 0\right)$ and $\left(0, \frac{1}{2}\right)$. Corresponding object is called Dirac spinor.

Moreover, all half-spin free particles in $C, P, T$-invariant theory are described as object like Dirac spinor which satisfies the Dirac equation (look here).