Tarek (OP) e-mailed me to contribute to this thread. Here's the response that I gave him (slightly edited for clarity).
I see why this was confusing, my apologies! I was perhaps too glib in the post. Iwas implicitly talking about a chiral rotation but wanted to present it somewhat more intuitively.
Let me try to spell it out more carefully, and hopefully I can also explain why this is the same as what is being explained in QFT textbooks.
The point is the difference between helicity and chirality. Helicity has to do with angular momentum, i.e. spin up or spin down relative to the momentum of the particle. In the massless limit, helicity and chirality are identical. For our purposes, let's treat "chirality" as a label: I have a left-chiral fermion and a right-chiral fermion, just treat these as two different fermions. (This is probably not a great use of the term 'chirality', but hopefully it elucidates the point.)
A left-chiral fermion comes pre-packaged with a left-helicity and a right-helicity state. The anti-particle of a left-helicity, left-chiral fermion is a right-helicity, left-chiral fermion. (Again, you could argue what we mean by chirality here---but for our purposes, it's just a label for this fermion field and is distinct from helicity.)
Suppose I have a left-chiral fermion and a right-chiral fermion. These are two completely different types of fields. With some foresight, let's call the left-chiral fermion an electron and the right-chiral fermion a positron. We will clarify later the connection to the physical electron and positron (its antiparticle), but for now these are just names.
We have a total of four kinds of particles:
- left-chiral (electron) left-helicity
- left-chiral (electron) right-helicity
- right-chiral (positron) left-helicity
- right-chiral (positron) right-helicity
To emphasize once again, "left/right-chiral" is just part of the fermion's name. We could now ask how do #1 and #3 differ, since "chirality" is just some name. Note that this is the relevant comparison, if we want to compare apples-to-apples, then we should compare objects of the same helicity. A left-helicity state is obviously different from a right-helicity state, but how do the two left-helicity states above differ?
In the Ryder formula referenced in the comments to your question, you're comparing the left-helicity and right-helicity transformations. Indeed, both left-helicity and right-helicity transform with the same phase under a z-axis rotation. (Where z is assumed to be collinear with the momentum.)
But as I said, that's not the apples-to-apples comparison we're interested in. If we want to compare left-helicity states, then we should compare the left-chiral left-helicity state (left-helicity electron) with the right-chiral left-helicity state (left-helicity positron). But the "right-chiral left-helicity" state (the left-helicity positron) is the antiparticle of the "right-chiral right-helicity" (right-handed positron) state, so this thing picks up a complex conjugation.
This complex conjugation is precisely what flips the phase.
Thus: the left-helicity positron transforms with an opposite phase as the left-helicity electron.
If you want, this really all boils down to the following (which may be more confusing):
You have electrons (left-chiral) and positrons (right-chiral). Under a rotation about the z-axis, the left-helicity electrons pick up a phase that is opposite to the phase the left-helicity positrons pick up.
This may be confusing because it makes you pause and say "wait, i thought electrons and positrons were antiparticles of one another?" Here we need to clarify the words that we used. The `left-chiral' fermion (electron) and the right-chiral fermion (positron) mix when you introduce mass term. Just look at which two-component spinors are connected when you write out a Dirac mass. Indeed, you can think of a mass term as a 2-point vertex where a left-helicity electron goes in and a left-helicity "anti-positron" goes out. In the Dirac picture there's a total of four different states--just look at the four components of a Dirac spinor compared to the two components of a Weyl spinor. The four components of a Dirac spinor are the marriage of two Weyl spinors. Each Weyl spinor has a left- and right-helicity piece. Alternately, this is why you have plane wave spinors called u, ubar, v, and vbar.
Does this clarify things? (probably not, I know this is a bit of a thorny issue.)
It might help if you looked at the chiral particle content of the Standard Model. (Or, e.g. the particle content of the MSSM, which is most naturally written with respect to `chiral superfields' which contain Weyl spinors). Usually people write the particles as: Q, uR*, dR*, L, eR*, where the star indicates a conjugate so that all five fields are written as left-chiral fields. Note that the doublet L contains both a left-chiral electron (eL) and a left-chiral neutrino. The eR* has the quantum numbers of a positron. The fields eL and eR each contain left- and right-helicities. But the left-helicity component of eR picks up an opposite phase under rotations as the left-helicity component of eL.
You can also see the two-component bible for a more thorough discussion.