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Dirac fermions is only the direct sum of left- and right-handed Weyl representations (which leads to time inversion, charge inversion and spatial inversion invariance of the theory). Two Weyl representations are mixed by the mass term in the Dirac equation. If we set mass to zero, we will get two uncoupled equations, each of which describes Weyl fermion. But ...

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Yes, the parity symmetry can be implemented only for the four-component Dirac fields and not for two-component Weyl fields. This fact, mathematically speaking, does not depend on the value of the mass. Physically speaking, however, I am not sure that a massless four component spinor makes much sense in standard theories (Sorry, I do not know anything about ...

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The real reason is in following. Let's assume Majorana field: $$\Psi_{M} = \Psi_{L} + \hat{C}\bar{\Psi}^{T}_{L}, \quad \hat{C} = i\gamma_{2}\gamma_{0}, \quad \Psi_{L} = \begin{pmatrix} \psi_{L} \\ 0 \end{pmatrix}.$$ By using this notation it's not hard to see that kinetic term is equal to $$\bar{\Psi}_{M}\gamma^{\mu}\partial_{\mu}\Psi_{M} = ... 0 The short asnwer to your question is that the overall factor \frac{1}{2} from the Lagrangian of a Majorana field (in the 4-component notation)$$\mathcal{L}=\frac{1}{2}(\bar{\psi}i\gamma^{\mu}\partial_{\mu}\psi -m\bar{\psi}\psi)$$compared to the general Dirac Lagrangian is usual for self-conjugate fields and it is introduced to ensure a consistent ... 1 To see that this is the simplest possible non-relativistic quantum field theory for fermions, it's useful to derive the dynamics. The canonical momentum for \psi(x,y,z) is the Lagrangian's derivative with respect to \partial_\tau \psi(x,y,z) – and it is \psi(x,y,z)^\dagger (up to signs and \pm i which depend on conventions). At any rate, the ... 2 The formula for the pressure$$ P=\frac{8\pi}{3h^3}\int^{p_0}_0 p^3\frac{dE}{dp}dp $$is valid for a simple reason: dE/dp is nothing else than the expression for the speed v. Check it for E=p^2/2m; the p-derivative is p/m=v. So the integrand only differs from the integrand for the electron number density n in the first formula by the extra factor ... 1 By starting from the Fermi theory and requirement of the tree-unitarity of this theory (it is similar to renormalizability, but only on a tree level) you may build theory of electroweak interactions (even with Higgs boson). I'm only show you how does it work. Fermi theory predicts growth the matrix element of neutrino-lepton scattering as E^{2} (or ... 3 Let's assume a typical fermionic mass-term (interacting leptons and quarks are spin 1/2-particles):$$ \tag 1 \bar{\Psi}\Psi = \bar{\Psi}\left(\frac{1 + \gamma_{5}}{2} + \frac{1 - \gamma_{5}}{2}\right)\Psi = \left| \bar{\Psi}\left( 1 \pm \gamma_{5} \right) = \left( (1 \mp \gamma_{5})\Psi\right)^{\dagger}\gamma_{0} \right| =  =\bar{\Psi}_{L}\Psi_{R} + ...

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