Exploring Spin in Different Dimensions: 2+1 and Beyond

[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

 

[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.