I am writting a text and I try to be consistent with my definitions. I have expanded my fields in modes in the following way $$\psi(x)=\int\frac{d^3\vec{p}}{(2\pi)^3\sqrt{2p^0}}\sum_s\bigg(u_s(\vec{p})b_s(\vec{p})e^{ip\cdot x}+\upsilon_s(\vec{p})d^{\dagger}_s(\vec{p})e^{-ip\cdot x}\bigg)$$ and I am using a mostly plus metric. Correct me if I am wrong, but I think in this notation the following hold $$\{b_s(\vec{p}),b^{\dagger}_r(\vec{q})\}=(2\pi)^3\delta(\vec{p}-\vec{q})\delta_{rs}$$ $$(\gamma\cdot p+m)u(\vec{p})=0, \hspace{1em} (\gamma\cdot p-m)\upsilon(\vec{p})=0$$ $$\bar{u}(\vec{p})(\gamma\cdot p+m)=0, \hspace{1em} \bar{\upsilon}(\vec{p})(\gamma\cdot p-m)=0$$ How can I use all the above (or some of the above) to determine the spinor orthogonality relations in the conventions I am using??

Also, if anyone is familiar with a textbook that uses those conventions, please refer that book to me:)



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