Is the helicity operator $h=\hat p \cdot S$ such as the one defined in eq (3.54) of Peskin QFT book well-defined in other dimensions? For example, there Peskin focused on 3d space and 1d time (4d). The helicity operator $h=\hat p \cdot S$ is related to the chirality of massless fermions in this 4d.
My question is that whether the helicity operator $$h=\hat p \cdot S$$ is also related to the chirality of massless fermions also in other dimensions? (other than 4d)
For example in 1d space and 1d time (2d), we still can have a left-moving fermion ($\hat p<0$) and right-moving fermion ($\hat p > 0$). How about their spins? It seems that their spins are locked with their momentum, for example, we can say that left-moving fermion has a spin up ($S>0$), while the right-moving fermion has a spin down ($S<0$).
Then we see that the left-moving fermion ($\hat p<0, S>0$) $$h=\hat p \cdot S <0,$$ the right-moving fermion ($\hat p>0, S<0$) $$h=\hat p \cdot S <0.$$
So they do not have the different helicity $h$ if $h=\hat p \cdot S$ is the way to define helicity. Because the chirality of left-moving can be be $-$ chiral, and right-moving fermion can be be $+$ chiral. They should have different signs.
How about other even space time dimensions, when we can define chirality? Can we find analogy of the helicity operators? How to define such $h$?