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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 $h$?

ann marie cœur
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    I see three questions here. (1) What's the relation, if any, between helicity and chirality in $d\neq 4$ dimensional spacetime? (2) When is chirality defined? (3) How is helicity defined in $d\neq 4$ dimensional spacetime? Questions (2) and (3) are prerequisites for (1). Question (3) is almost a duplicate of Analogue of helicity in higher dimensions and concrete formula, but not quite. For (2), are you familiar with Clifford algebra and spinors? For (3), are you familiar with the concept of the "little group" in the context of massless particles? – Chiral Anomaly Sep 19 '21 at 15:18
  • thanks - what you wrote is very helpful. maybe you can update to an answer, I know chirality can be defined when you have a gamma5 in even spacetime dimensions. – ann marie cœur Sep 19 '21 at 15:31
  • Chirality is defined only in even-dimensional spacetime, and it's always binary (the only options are left-handed and right-handed). In contrast, helicity is not always binary. In 2-dimensional spacetime (1-dimensional space), angular momentum does not exist because the rotation group is trivial. In $2n$-dimensional spacetime for $2n\geq 4$, a massless particle has a $2^{n-2}$ linearly independent helicity states for each chirality. So, for $2n\geq 6$, a massless particle has a continuum of possible helicity states (for each chirality) instead of only one. Is this what you're looking for? – Chiral Anomaly Sep 20 '21 at 01:12

1 Answers1

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In a general space-time dimension, helicity is the eigenvalue of a rotation that leaves the direction of motion unchanged. The group of such rotations is called the “little group”. For a massless particle, the little group plays an especially important role, since it is not possible to bring this particle to a rest frame where all rotations have an equivalent role.

In 4 dimensions, the little group is U(1), the group of rotations about the direction of motion. This has one generator, which is exactly the projection of $\vec S$ along the direction of motion, that is, the usual helicity. This is a very simple situation. In 2 dimensions, there is no little group. In 6 dimensions, the little group is SO(3) or SU(2).

I restrict myself to even dimensions here because that is where $\Gamma$, the chirality, is defined. In 2 dimensions, $\Gamma = \gamma^0\gamma^1$, in 4 dimensions in is the usual $\gamma^5$, etc. In general dimensions, the minimal size of the representation for a Dirac spinor is $2^{[d/2]}$, that is, 2 in 2 or 3 dimensions, 4 in 4 or 5 dimensions, and 8 in 6 or 7 dimensions. In even dimensions, it is possible to split the massless spinor representation into two parts with chirality ($\Gamma$ eigenvalues) +1 and -1. These have dimensions 1 in 2-d, 2 in 4-d, and 4 in 6-d. In 2-d (1 space dimension), the $\Gamma = +1$ eigenstate is fermion moving only to the right along the line at the speed of light, and the $\Gamma = -1$ state is a fermion moving the left along the line. In 4-d, the chirality eigenstates are the usual right- and left-handed spinors. In 6-d, the 4-dimensional representation is composed of two representations which are both SU(2) spinors. In reduction to 4 dimensions, this representation decomposes as $$ 4 \to (+1, 2, 1) + (-1, 1, 2) $$ where the first quantum number is the $\Gamma$ eigenvalue and the next two are the $(j_R,j_L)$ quantum numbers of the $SO(3,1)$ representation. For those who are attuned to this, the 4 in 6-dimensions is “vectorlike” when reduced to 4 dimensions.

This sort of analysis can be done in any even dimension. It is necessary to take into account also the possibility of Majorana fermions. In dimensions 2, 10, …, spinors can be both chiral and Majorana, further reducing the degrees of freedom. This is all worked out in books on supersymmetry, for which the size of spinor representations in higher dimensions is important. I recommend the treatment in “Supergravity”, by Freedman and van Proeyen.

Michael Peskin
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