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When (falsely) quantizing the Dirac-field, Peskin/Schroeder (Introduction to Quantum Field Theory) get with $$\psi(\vec{x})=\int\dfrac{d^3p}{(2\pi)^3}\dfrac{1}{\sqrt{2E_{\vec{p}}}}e^{i\vec{p}\cdot\vec{x}}\sum\limits_{s=1,2}\left(a_{\vec{p}}^su^s(\vec{p})+b_{-\vec{p}}^sv^s(-\vec{p})\right)$$ and $$ \left[a_{\vec{p}}^r,~a_{\vec{q}}^{s\dagger}\right] = \left[b_{\vec{p}}^r,~b_{\vec{q}}^{s\dagger}\right] = (2\pi)^3\delta^{(3)}(\vec{p}-\vec{q})\delta^{rs}$$ the relationship $$\left[\psi(\vec{x}),~\psi^{\dagger}(\vec{y})\right] = \int\dfrac{d^3p~d^3q}{(2\pi)^6}\dfrac{1}{\sqrt{2E_{\vec{p}}2E_{\vec{q}}}}e^{i(\vec{p}\cdot\vec{x}-\vec{q}\cdot\vec{y})}\times\sum\limits_{r,s}\left(\left[a_{\vec{p}}^r,~a_{\vec{q}}^{s\dagger}\right]u^r(\vec{p})\bar{u}^s(\vec{q})+\left[b_{-\vec{p}}^r,~b_{-\vec{q}}^{s\dagger}\right]v^r(-\vec{p})\bar{v}^s(-\vec{q})\right)\gamma^0$$

This is all easy to see but I am wondering where the $\gamma^0$ at the end comes from?

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$$\psi(\vec{x})=\int\dfrac{d^3p}{(2\pi)^3}\dfrac{1}{\sqrt{2E_{\vec{p}}}}e^{i\vec{p}\cdot\vec{x}}\sum\limits_{s=1,2}\left(a_{\vec{p}}^su^s(\vec{p})+b_{-\vec{p}}^sv^s(-\vec{p})\right)$$ so $$\psi^{\dagger}(\vec{y})=\int\dfrac{d^3q}{(2\pi)^3}\dfrac{1}{\sqrt{2E_{\vec{q}}}}e^{-i\vec{q}\cdot\vec{y}}\sum\limits_{s=1,2}\left(a_{\vec{q}}^{\dagger s}u^{\dagger s}(\vec{q})+b_{-\vec{q}}^{\dagger s}v^{\dagger s}(-\vec{q})\right)$$

This gives the relationship $$\left[\psi(\vec{x}),~\psi^{\dagger}(\vec{y})\right] = \int\dfrac{d^3p~d^3q}{(2\pi)^6}\dfrac{1}{\sqrt{2E_{\vec{p}}2E_{\vec{q}}}}e^{i(\vec{p}\cdot\vec{x}-\vec{q}\cdot\vec{y})}\times\sum\limits_{r,s}\left(\left[a_{\vec{p}}^r,~a_{\vec{q}}^{s\dagger}\right]u^r(\vec{p})u^{\dagger s}(\vec{q})+\left[b_{-\vec{p}}^r,~b_{-\vec{q}}^{s\dagger}\right]v^r(-\vec{p})v^{\dagger s}(-\vec{q})\right)$$

Now, $$\gamma^0 = \begin{pmatrix} I & 0 \\ 0 & -I \end{pmatrix}$$ and so we can define the Dirac adjoint of $u$, $\bar{u} = u^{\dagger}\gamma^0$ which also gives $u^{\dagger} = \bar{u}\gamma^0$ as $\left(\gamma^0\right)^{-1} = \gamma^0$ and we obtain the required result

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  • $\begingroup$ Ah, that simple. Stupid me... $\endgroup$ – Simon Fromme Jul 29 '16 at 21:41
  • $\begingroup$ It should be $e^{-i\vec{p}\vec{x}}$ though in the second line. $\endgroup$ – Simon Fromme Jul 29 '16 at 22:10
  • $\begingroup$ Yes, of course... Stupid me :P I'll fix that now $\endgroup$ – Mithrandir24601 Jul 30 '16 at 7:40
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Because $\bar u \gamma^0 = u^\dagger$

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  • $\begingroup$ Could you expand on this? Otherwise, it could very easily get deleted as low-quality (it was already flagged as such). $\endgroup$ – heather Jul 29 '16 at 17:09

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