|MadSci Network: Physics|
thanks for the good question. First of all, Dirac and Weyl fermions are matter fields which satisfy or solve certain equations, namely the Dirac equation and the Weyl equation, respectively. The Dirac equation describes spin 1/2 particles with mass. In contrast to the Schroedinger equation, it includes effects from the theory of special relativity. The Weyl equation is "obtained" from the Dirac equation by putting the mass equal to zero. The name Dirac electron (or Weyl electron) does not automatically imply that the physical electron is meant, but a particle satisfying the Dirac resp. Weyl equation.
Putting the mass to zero has a lot effects on the solutions of the equation. It simplifies the situation a lot and therefore they have a different structure. Therefore, there is also a difference between a massless Dirac fermion and a Weyl fermion. But there is a mathematical connection: A Dirac fermion can be described as a pair of Weyl fermions. For a nice overview see e.g. Weyl spinors and Dirac's electron equation.
The property of Parity is an important concept in physics. In most interactions this quantity is conserved, except in weak interactions. Parity appears in two fashions, right and left handed resp. It is closely related to the spin. Weyl spinors have a definite parity, and therefore are not invariant under parity transformations, violate the parity symmetry. But neutrinos appear in nature de facto only left-handed. And are therefore described by Weyl spinors. On the other hand, Dirac fermions include left and right handed Weyl spinors. Therefore, a Dirac spinor implements unbroken parity symmetry.
The spin of any particle, also of a Weyl and a Dirac fermion, can point in any direction. If projected in a direction the value of the spin is quantized, for fermions ...-5/2,-3/2,-1/2,1/2, 3/2, 5/2, ... for bosons ...-3,-2,-1,0,1,2,3,... E.g. you can project the spin in the direction of movement, i.e. in the direction of momentum, so-called helicity. For a spin 1/2 particle the helicity can be positive or negative. The parity transformation transforms between the two possibilities.
I hope this answers a few of your questions ;)
Best regards, Michael
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