# Left-/righthanded fermions

## Main Question or Discussion Point

Hey
I have a basic question about the Standard Model. In this forum and on other places the expression left-/righthanded fermions. Can someone explain the difference between these two types of fermions.

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neu
nrqed
Homework Helper
Gold Member
Hey
I have a basic question about the Standard Model. In this forum and on other places the expression left-/righthanded fermions. Can someone explain the difference between these two types of fermions.

It's a bit confusing at first because there are two concepts involved and people use the term right and left handed in two different contexts.

There is something called "chirality" and something called "helicity". Unfortunately, left and right handed is used to refer to eigenstates of chirality sometimes and of helicity sometimes.

The chirality operator is $$\gamma_5$$. You can define define positive and negative chirality fermions as the ones that are eigenstates of the chirality operator (it turns out that the eigenvalues are plus or minus 1)

$$\gamma_5 \Psi= \pm \Psi$$.

Those eigenstates are called left and right handed fermions unfortunately (because it has nothing to do with another concept of left and right associated to helicity..see below).

In general, a Dirac fermion is not a chirality eigenstate.

So, why should we care? We have to cae because the weak interaction couples the weak gauge bosons to chiral states! So we can't couple the weak gauge bosons directly to Dirac fermions.

However, one can always construct chiral eigenstates out of an arbitrary Dirac fermion by using what we call the projectors

$$P_\pm = (1 \pm \gamma_5)/2$$ .

Then, for any Dirac fermion $$\Psi$$,
the projection $$P_+ \Psi$$ is automatically an eigenstate of the chirality operator with eigenvalue +1 ( check it! It's obvious if you use the fact that $$\gamma_5^2 = \gamma_5$$). And people call this a right-handed state so they write

$$\Psi_R = P_+ \Psi$$

Likewise, we can get a left-handed state using

$$\Psi_L = P_- \Psi$$

Helcity is a totally different concept. There is something called a helicity operator that measures the projection of the spin along the motion of the particle. The only states which are helicity eigenstates are massless states. It turns out that for a massless particle, the eigenstates of the helicity operator are the same as the eigenstates of the chirality operator which is probably why people use right-handed and left-handed in the two contexts.

Okay. Thank you for the explanation and the link!

Is there an operator that measures the spin component not along the momentum of the fermion but say perpendicular to it?

In the usual non-relativistic QM, you have Sz but also Sx, and Sy operators. Why something similar is not considered for the solutions of Dirac's equation?

Hans de Vries
Gold Member
Is there an operator that measures the spin component not along the momentum of the fermion but say perpendicular to it?

In the usual non-relativistic QM, you have Sz but also Sx, and Sy operators. Why something similar is not considered for the solutions of Dirac's equation?

Yes, this is simply done with the Pauli matrices.

The reason that people generally talk about the spin being along the momentum
direction is because in high energy physics the particles move very close to c.

The spin-component along the momentum increases the closer the particle gets to
the speed of light. The spin-components perpendicular to the momentum however
stay constant. This is true for any object, also for a rotating rock for example.

Regards, Hans