If the mass of neutrino is not zero, why we cannot find right-handed neutrinos and left-handed anti-neutrinos? I am learning P&S's Introduction of quantum field theory. My teacher said that if the mass of neutrino is exactly 0, then we should not observe any right-handed neutrinos and left-handed anti-neutrinos according to Weyl's theory. It is because
$$
\left( i\gamma^\mu \partial_\mu - m \right) \psi
=
\left(
\begin{array}{cc}
-m & 
i\left(\partial_0 + \vec\sigma \cdot \vec\nabla\right)
\\
i\left(\partial_0 - \vec\sigma \cdot \vec\nabla\right)
& -m
\end{array}
\right)
\left(
\begin{array}{c}
\psi_L \\ \psi_R
\end{array}
\right)
=0.
$$
If we set m=0, then we will have:
\begin{align}
i\left(\partial_0 - \vec\sigma \cdot \vec\nabla\right) \psi_L &= 0;
\\
i\left(\partial_0 + \vec\sigma \cdot \vec\nabla\right) \psi_R &= 0.
\end{align}
And if these formulas can describe neutrinos we must conclude that there are two types of neutrinos.
My confusions are:


*

*It is known that the mass of neutrino is not 0, therefore there are chances (depending on the mass) to find right-handed neutrinos and left-handed anti-neutrinos. But we don't.

*If I accept that there are only these two types of neutrinos (right-handed anti-neutrinos and left-handed neutrinos), this property of neutrino causes the parity violation, so parity violation must relate to angular momentum and therefore spinor space. Am I write?

*Are there any reasons why neutrino is so weird that is has mass but still has helicity + Or - 1. This goes against with what I have learned that only when a particle  moves at the speed of light can its spin aligned with its momentum.
 A: We don't observe the right handed neutrinos directly because, to good approximation, only left handed neutrinos interact with the weak force, and the weak force is the only mechanism we have observed neutrinos to interact with at all. As far as I know, directly observed right handed neutrinos would require observing it interacting via gravity. We can barely detect really massive black holes colliding via gravity waves, so there's basically no hope there, and I don't see us being able to produce a lump of stationary neutrinos we could put on a scale or feed into a Cavendish type experiment.
That leaves us with avenues that we're already pursuing: experiments with neutrino beams (like MiniBooNE), neutrino detectors (like Super-Kamiokande, SNO, and IceCube), and 'missing energy' experiments like were used to infer the existence of neutrinos in the first place. All of these experiments interact directly only with left handed neutrinos, and infer the existence of right handed neutrinos from neutrino oscillations, and (perhaps someday) a measurement of neutrino mass beyond the current upper limits around 2 eV.
A: About your questions:

"It is known that the mass of the neutrino is not 0, therefore there are
  chances (depending on the mass) to find right-handed neutrinos and
  left-handed anti-neutrinos. But we don't."

Actually no. The way you wrote it down assumes that the mass is a so-called Dirac mass. In that case the left-handed and right-handed neutrinos would be connected via the mass term. Therefore one would need both left- and right-handed neutrinos. However, one can also have a different kind of mass term called a Majorana mass, which does not require the right-handed neutrino.

"If I accept that there are only these two types of neutrinos
  (right-handed anti-neutrinos and left-handed neutrinos), this property
  of neutrinos causes the parity violation, so parity violation must
  relate to angular momentum and therefore spinor space. Am I [right]?"

Parity, as you probably know, is the transformation where something is replaced by its mirror image. So, under parity, a left-handed particle is replaced by a right-handed particle and visa verse. Parity violation implies that nature does not behave the same as its mirror image. However, this does not have anything to do with angular momentum. In other words, the violation of parity does not mean that angular momentum is not conserved.
For the weak interaction, it was found that parity is maximally violated. This is incorporated into the Standard Model in that only the left-handed neutrino couples to the weak force. So the left-handed neutrino, together with the left-handed electron, form a weak doublet, while the right-handed electron is a singlet under the weak force. So if there is a right-handed neutrino, then it is believed that it should be a singlet, otherwise we should have seen it being produced via the weak force in high energy experiments.
Because of this difference in the transformations of left- and right-handed neutrinos under the weak interaction, one cannot represent the mass of the neutrinos as a Dirac mass, because such a Dirac mass term would explicitly break the gauge symmetry associated with the weak interaction. Therefore one needs to use the Majorana mass term to represent the mass of the neutrino in the Standard Model.

"Are there any reasons why neutrinos are so weird that they have mass,
  but still has helicity + Or - 1. This goes against with what I have
  learned that only when a particle moves at the speed of light can its
  spin aligned with its momentum."

Firstly, what is helicity? It is the component of the spin along the direction of the momentum vector, regardless of the mass of the particle. It can either be parallel or anti-parallel to the momentum, which are represented by $+1$ and $-1$, respectively. So a particle with mass can have a helicity of either $+1$ and $-1$, provided it is not a scalar.
But now what about neutrinos? If they are always left-handed, but at the same time have mass, shouldn't one be able to boost to a frame in which the neutrinos are at rest? Well, strictly speak the left-handedness refers to chirality and not to the helicity. These two are not exactly the same thing, because they don't share eigenstates. As a result, although neutrinos have left-handed chirality, they can still be a mixture of both helicities. In general, neutrinos are always observed as fast-moving (relativistic) particles, so that the difference between chirality and helicity is vanishingly small.
Hope these answers address your questions.
