Gamma matrices and spin

1. Oct 31, 2013

Troy124

Dear PhysicsForum,

We have just treated the Dirac equation and its lagrangian during our QFT course, but we have only gone in depth in 3+1 dimensions.

My question is about what happens to spin in 2+1 dimensions. In 3+1 dimensions we have to use 4 by 4 gamma matrices, but in 2+1 dimensions we could use 2 by 2 gamma matrices, so would this imply two 'degrees of freedom' instead of four? I tried to calculate this by myself and I found out that you still have particles and anti-particles and you also have spin up and spin down. However, particles are always associated with spin up and anti-particles with spin down. Is this correct?

And what happens to spin in different dimensions? In 1+1 dimensions, for example, is there spin at all? Or what happens for higher dimensions than 3+1?

Regards,
Troy

2. Nov 12, 2013

PhilDSP

Hi Troy,

I'm not aware of anyone having studied what you seem to mean by 2+1 dimensions, that is, 2 spatial dimensions plus time. Initially spin was modeled in only 3 spatial dimensions because time was a variable outside of the spin mechanism, so to speak. Pauli refined that technique and developed his own variation of the Schrödinger equation with spin before Dirac. Remember that a two-component spinor effects spin in 3 spatial dimensions. It encodes 2 orthogonal vectors plus their orientation.

Clifford algebra is used for spin in higher dimensions. It can be regarded as a generalization of complex numbers and quaternions.

Last edited: Nov 12, 2013