# Why four components? not 8 components or more?

1. Sep 1, 2010

### wdlang

i am now studying dirac equation

why there are only four components? not more?

is it possible to test the number of components experimentally?

2. Sep 1, 2010

### humanino

This is related to the structure of the Lorentz group. Weyl spinors are the fundamental entities with two complex components. Dirac spinors comprise two Weyl spinors transforming in conjugate representations, in terms of Lie algrebras
so(3,1) = sl_2(C) x sl_2(C)
So we have the following representations
(0,0) scalar
(1/2,0) left and (0,1/2) right Weyl spinors
(1/2,0)+(0,1/2) Dirac bispinor
(1/2,1/2) 4-vector
...

3. Sep 1, 2010

### wdlang

thanks. any reference?

4. Sep 2, 2010

### tom.stoer

I guess Ryder's book on QFT, but I am not sure.

5. Sep 9, 2010

### michael879

The original Dirac hamiltonian (taken from wikipedia, the exact form isn't really important for this) is just:
$$\beta{m} -i\hbar\alpha\nabla$$
where alpha and beta need to be chosen according to their commutation relations. It turns out that the minimal solution to the commutation relations are 4x4 matrices (or quaternions). However, there are higher order solutions which would lead to larger component spinors.

6. Sep 10, 2010

### chrispb

I'm not positive about this, but I suspect that having bigger representations would mean they're either reducible or describe particles with greater spins, which is in principle possible. However, when you get up to spin 1 (fundamental) particles, you end up needing to introduce gauge degrees of freedom, and when you add spin 3/2 (fundamental) particles you end up needing to introduce both supersymmetry AND gravity into the mix. Oops.