= Equal-time canonical anticommutator of a Dirac field
{c}
{title2=$\{\psi_\alpha(t,\mathbf x),\psi_\beta^\dagger(t,\mathbf y)\}=\delta_{\alpha\beta}\delta^3(\mathbf x-\mathbf y)$}
For mode normalization $1/\sqrt{2E_{\mathbf p}}$, particle and antiparticle operators satisfy $\{a_{\mathbf p}^s,a_{\mathbf q}^{r\dagger}\}=\{b_{\mathbf p}^s,b_{\mathbf q}^{r\dagger}\}=(2\pi)^3\delta^{sr}\delta^3(\mathbf p-\mathbf q)$. The <Dirac spinor> completeness relations imply $\sum_s[u_s(\mathbf p)u_s^\dagger(\mathbf p)+v_s(-\mathbf p)v_s^\dagger(-\mathbf p)]=2E_{\mathbf p}I$. Inserting this into the <mode expansion of a Dirac field> gives the displayed local anticommutator. The reversed momentum in the antiparticle completeness term is essential.
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