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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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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