Let's have Dirac spinor $\Psi (x)$, which formally corresponds to $$ \left( 0, \frac{1}{2} \right) \oplus \left( \frac{1}{2}, 0 \right) $$ representation of the Lorentz group.
What representation is true for $\Psi (x) \Psi^{+}(x')$? I expect something like $$ \left[\left(\frac{1}{2}, 0 \right) \oplus \left( 0, \frac{1}{2} \right)\right]\otimes \left[\left( 0, \frac{1}{2} \right) \oplus \left( \frac{1}{2}, 0\right)\right] = $$ $$ =\left(\frac{1}{2}, 0 \right) \otimes \left(\frac{1}{2}, 0 \right) \oplus \left(\frac{1}{2}, 0 \right) \otimes \left( 0 , \frac{1}{2}\right) \oplus \left( 0, \frac{1}{2} \right) \otimes \left(0, \frac{1}{2} \right) = $$ $$ \tag 1 =\left[\left( 0, 0\right)\oplus (1, 0) \right]\oplus \left( \frac{1}{2} , \frac{1}{2}\right) \oplus \left[\left( 0, 0\right)\oplus (0, 1) \right], $$ but I'm not sure.
Also I know that $$ \tag 2 [\Psi (x), \Psi^{+}(y)]_{+} = i\left( i\gamma^{\mu}\partial_{\mu} + m\right)\gamma_{0}D_{m}(x - y), $$ where $D_{m}(x - y)$ is a lorentz scalar function, so formally $(2)$ doesn't coinside with $(1)$. How to compare it with $(1)$?
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