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You have discovered the fact that the Dirac spinors form a reducible representation of Spin(3,1) $\simeq$ SL(2,C), the covering group of SO(3,1)$^+$. The left and right Weyl spinors, which have two components, are irreducible representations.

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Let the undotted index $a$ correspond to the irrep $(\frac12,0)$ of the Lorentz group. Then, $F_{ab}$ corresponds to the tensor product $(\frac12,0)\otimes (\frac12,0)$ which decomposes into the two irreps $(1,0)$ and $(0,0)$ of the Lorentz group. $(1,0)$ is given by the symmetric part of $F_{ab}$ while the antisymmetric part gives the scalar ...

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I believe I'm ready to answer my own question. The pin group can alternately be defined as the set of all invertible elements $S_{\Lambda} \in \mathrm{Cl}(p,q)$ satisfying $S_{\Lambda} S_{\Lambda}^{\tau} = \pm 1$ and $$\alpha(S_{\Lambda}) \gamma^a S_{\Lambda}^{-1} = {\Lambda^a}_b \gamma^b$$ for some element ${\Lambda^a}_b \in \mathrm{O}(p,q)$. The map ...

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I think its really important to differentiate between helicity and chirality. Helicity is the spin angular momentum of a particle projected onto its direction of motion. For a massive particle this quantity is frame dependent. Furthermore, since angular momentum is conserved, as a particle propagates helicity is conserved. On the other hand, chirality is an ...

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Simply think of a Weyl Spinor as a Dirac spinor where the other two components are set to zero. Equivalently, a Weyl spinor (of chirality +1) belongs to the two-dimensional subspace with eigenvalue 1 under the action of the projection operator $P_+=\frac{(1-\gamma^5)}2$. For negative chirality, use the other projection operator, i.e., ...

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In many contexts, we would like to determine how Lorentz transformations act on the mathematical objects that characterize a particular theory. In the case of classical, Lorentz-invariant field theories on Minkowski space for example, we need to specify how Lorentz transformations act on the fields of the theory. This leads naturally to determining how ...

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The answer does come down to the representation theory of the Lorentz group. A good discussion can be found in the first volume of QFT by Weinberg (and in other places as well). One thing to note is that you postulate a 4-dimensional representation of the Lorentz group. This postulate comes in when you assume that your objects have 4 components. Now a ...

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