Understanding the Role of Spinors in Quantum Mechanics

[Milsomonk]

Hey guys,
Hope all is well. I'm trying to get my head round some of the Quantum Mechanics of spin. I fully understand why the Pauli equation acts on a two component spinor wavefunction, where I'm a little confused is why the Dirac equation then acts on a 4 component spinor wavefunction. I get that it is in a sense four coupled equations but why is spin implied? I've done a fair bit of reading in the griffiths book but any extra insight would be very much appreciated.

 

[vanhees71]

There is a hand-waving answer and one based on group theory.

Let's start with the hand-waving answer: Relativistic QT is most easily formulated in terms of quantum field theory, because when particles collide at relativistic energies, it's possible that new particles get created, i.e., the particle number is not conserved. It turns out that you only get a local quantum field theory, i.e., a theory such that under Lorentz transformations the field operators transform as the corresponding classical fields, in a local way, i.e.,
$$\psi'(x')=S \psi(\Lambda^{-1} x'),$$
where $S$ is an appropriate matrix acting on the field components, it turns out that you always need a field operator that consists of both a annihilation and a creation operator.

At the same time this helps to solve the problem with the negative-frequency solutions for free particles. You don't like to interpret those as particles with negative energy, because this would mean that there's no stable ground state. In QFT that's no problem since you just write a creation operator in front of the negative-frequency solutions and a annihilation operator in front of the positive-frequency solutions. Then you get two sorts of particles with the same mass but opposite charges: particle and its corresponding antiparticle. If you work this out for particles with spin 1/2 you get to the Dirac equation.

As I said, it's the hand-waving answer, and you need also a lot of hand waving doing this with math. The more advanced idea is to think in terms of group theory. As it turns out you can extend the spin-1/2 representations of rotations (SU(2) matrices) to the full Lorentz group. This leads to a two-valued representation of the proper orthochronous Lorentz group in terms of SL(2,C) matrices. Now there's another representation also two-dimensional which is not equivalent, namely the conjugate complex representation (the matrices are still SL(2,C) matrices). Then it turns out that you need the direct product of these two representations to also define the parity (space reflection) transformation. So all together you get a four-dimensional spinor, and the corresponding representation of the orthochronous Lorentz group defines how these Dirac spinors transform under these transformations.

 

[Milsomonk]

Thanks! that helps a lot. Sometimes a hand wavy answer and a more formal one are just what you need.