# General definition of vector, spinor, and spin

I am looking for basic and exact definitions of fundamental physical concepts in graduate level.

• Spin: Intrinsic degree of freedom for quantum particles and obey angular momentum algebra.

• Spinor: Eigenstates of unitary groups

• Vector: Eigenstates of orthogonal groups

• Dirac Spinor: (operator? or eigenstate?) describes spin 1/2 particles

• Weyl Spinor: (operator? or eigenstate?) describes spin 1/2 massless particles (like neutrino?)

• Majorana Spinor: (operator? or eigenstate?) describes spin 1/2 particles which have equal to own antiparticles.

• I recently saw a lecture by Atiyah on spinors. He started by saying something along the lines of: "Nobody but God really knows what spinors are. And perhaps Dirac, but that doesn't change much." – Danu Aug 30 '14 at 9:49
• @Danu Here it is: youtu.be/SBdW978Ii_E Watching it right now :) – Nephente Aug 30 '14 at 13:07
• Possible duplicates: physics.stackexchange.com/q/41211/2451 and links therein. – Qmechanic May 18 '15 at 4:03

Everything depends on how your fields (vectors and spinors are fields in the classical theory, and when you quantize in QFT, they become operator-valued fields) transform when you make a Lorentz transform:

• An scalar is a field that doesn't change at all: $\phi'(x') = \phi(x)$. Examples are the Higgs and pions.

• A vector field is a field that transform like a relativistic four-vector $A'_\mu (x') = {\Lambda^\nu}_\mu A_\nu(x)$, where $\Lambda$ is a Lorentz transformation. Examples are the electromagnetic field (photons) and gluons.

• An spinor field transform using a different set of matrices $$\psi'(x') = \exp\left[(-i \vec{\theta} \pm\vec{\eta}) \cdot \frac{\vec{\sigma}}{2}\right] \psi(x)$$ where $\vec{\theta}$ are the angles of rotation along the axes, $\vec{\eta}$ the rapidity and $\vec{\sigma}$ the Pauli matrices. As you can see, Pauli matrices are 2x2 matrices, so this transforms acts on objects with two components, the Weyl spinors. I have written two signs, $\pm$, because there are two types of transformations that act on two types of spinors: $-$ for left-handed spinors $\psi_L$ and $+$ for right-handed spinors $\psi_R$. But Weyl spinors have two problems: when you make a parity transformation ($\vec{r} \to -\vec{r}$), spinors change their handeness, and we know that QED and QCD are invariant under parity. And the other, as you say, is that Weyl fields must be massless.

• The Dirac spinor solves both problems. It is just (in the chiral representation) a left-handed and a right-handed Weyl spinors side-by-side $$\psi = \begin{pmatrix} \psi_L\\ \psi_R \end{pmatrix}$$ The Dirac spinor can have mass (although massless Dirac spinors are fine). Electrons, muons, taus, neutrinos and quarks are described Dirac fields.

• The Majorana spinor is a special Dirac spinor, where the left-handed and the right-handed parts are not independent. This relationship means that a Majorana particle is equal to its antiparticle. Therefore, the Majorana field has no electric charge. Although you only need one Weyl spinor to determine a Majorana spinor, Majorana fields still can have mass. It is conjectured that neutrinos might be Majorana particles (there are several experiments researching this).

So, where is spin? Angular momentum is a conserved quantity related to rotations. Whe you apply Nöther's theorem to a field, you get two terms: one depends on the movement of the particles (the orbital angular moment) and the other not (the spin). The spin part is related to the type of Lorentz transform that the field uses: in scalar fields there is no spin term (they have spin 0), in spinor fields it is a representation of rotations of dimension 2 (spin 1/2), and in vector fields a representation of rotations of dimension 3 (spin 1).