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it is well known that the intrisic spin is closely related to the rotation in space. However, in 1d , it is impossible to define rotation, therefore it is meaningless to talk about spin in 1d.However, one can see many papers talking about electrons in 1d with spin Is this meaningful?

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Spin is an intrinsic property, it's is not to be confused with the rotation in a space in which it is embedded. –  renormalizedQuanta Sep 6 '12 at 3:18
If you write down the dirac equation in 1d, there is just 2 gamma matrix with dimension 2, and the spinor contains two components, which correspond to the positive and negative energy solution. My question is where is the space foe spin, as in the 3d dirac equation. –  zyzeng Sep 6 '12 at 12:50

2 Answers 2

While Cristi's answer is correct, for spinors in one dimension, I think most papers discussing electrons in one dimension actually refer to one spatial dimension + time. Hence the spin group would be $Spin(1,1)$, which has a two-dimensional Majorana representation.

However, I don't think this is what is usually meant by the spin of the one-dimensiona electron. Instead, it seems to be more common to consider three-dimensional electrons constrained to move in one dimension. Considered as a one-dimensional field theory, the (three-dimenional) spin of those electrons has no direct geometric interpretation, but effectively functions as a flavor structure.

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I am not familiar with the electrons in 1d with spin, but I can tell you about the math theory of spinors for this case.

The spin group $Spin(r,s)$ is defined as the double cover of the special orthogonal group $SO(r,s)$. This definition applies also to the case when $n=r+s=1$: for one dimension, $SO(1)$ has only one element, which is $1$, and the spin group is $Spin(1)=O(1)$, which has as elements only the transformations $\pm 1$. For the spinor representations in this case, I quote Wikipedia:

"In 1 dimension (a trivial example), the single spinor representation is formally Majorana, a real 1-dimensional representation that does not transform."

The Majorana spinors are described for example here, and page 13 contains the one dimensional case.

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