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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