= Solution
Take a spatial <Fourier transform> of the <mode expansion of a Dirac field>:
$$
\widetilde\psi(t,\mathbf p)=\int d^3x\,e^{-i\mathbf p\cdot\mathbf x}\psi(t,\mathbf x)
=\frac1{\sqrt{2E}}\sum_r
\left[b_{-\mathbf p}^{r\dagger}v^r(-\mathbf p)e^{iEt}
+a_{\mathbf p}^ru^r(\mathbf p)e^{-iEt}\right].
$$
The <orthogonality> relations $u^{s\dagger}(\mathbf p)u^r(\mathbf p)=2E\delta^{sr}$ and $u^{s\dagger}(\mathbf p)v^r(-\mathbf p)=0$ isolate the <fermionic annihilation operator>. Projecting onto $v^s(-\mathbf p)$ instead isolates the <fermionic creation operator>:
$$
\boxed{\begin{aligned}
a_{\mathbf p}^s&=\frac{e^{iEt}}{\sqrt{2E}}u^{s\dagger}(\mathbf p)
\int d^3x\,e^{-i\mathbf p\cdot\mathbf x}\psi(t,\mathbf x),\\
b_{-\mathbf p}^{s\dagger}&=\frac{e^{-iEt}}{\sqrt{2E}}v^{s\dagger}(-\mathbf p)
\int d^3x\,e^{-i\mathbf p\cdot\mathbf x}\psi(t,\mathbf x).
\end{aligned}}
$$
The opposite time phases and reversed antiparticle <momentum> are essential. Both denominators are $\sqrt{2E}$, because the <Dirac spinors> have norm squared $2E$.
Back to article page
