To find these points, consider some $p_1$ on $\gamma$ not equal to $p$ or $q$, then consider the set $I^-(p_1)$ containing points that live on timelike curves that have $p_1$ as it's *endpoint*. Next, since spacetime is open, consider any convex neighbourhood $U_p$ of $p$ that contains $p_1$, i.e. a set such that we can connect $p$ to $p_1$, and then restrict the points of this convex set to those which live on some timelike curve that has $p_1$ as their endpoint, i.e. $U_p \cap I^-(p_1)$.
Thus the convex set allowed us to find points 'near' $p$ in Minkowski space, intersecting it with $I^-…

Using $\sum u \overline{u} - v \overline{v} = 1$ in the Dirac Hamiltonian, We Have

$\mathrm{H} = \int d^3x(\pi \partial_0 \psi - \mathcal{L}) = \int d^3x(i\psi^+ \partial_0 \psi) = \int d^3x(\overline{\psi} i \gamma^0 \partial_0 \psi) $

$ \ \ \ = \int \dfrac{d^3p}{(2\pi)^3}(\hat{\overline{\psi}} \varepsilon \hat{\psi}) = \int \dfrac{d^3p}{(2\pi)^3} \varepsilon (\hat{\overline{\psi}} \gamma^0 (\sum u\overline{u} - v \overline{v}) \gamma^0 \hat{\psi}) $

$ \ \ \ = \sum \int \dfrac{d^3p}{(2\pi)^3} \varepsilon [(\hat{\overline{\psi}} \gamma^0 u) ( \overline{u} \gamma^0 \hat{\psi} ) - ( \h…

Can't get the same thing to work for

$Q = \int d^3 x \overline{\psi} \gamma^0 \psi = \int d^3 x \psi^+ \psi$

Trying

$Q = \sum \int \dfrac{d^3 p}{(2\pi)^3} ( b^+ b - d^+ d) + C = \sum \int \dfrac{d^3 p}{(2\pi)^3} (b^+ b + d d^+) $

$ \ \ \ = \sum \int \dfrac{d^3 p}{(2\pi)^3} [(\int d^3 x e^{ipx} \overline{\psi} \gamma^0 u)(\int d^3 x e^{-ipx} \overline{u} \gamma^0 \psi) + (\int d^3 x e^{-ipx} \overline{\psi} \gamma^0 v) (\int d^3 x e^{ipx} \overline{v} \gamma^0 \psi)] $

$ \ \ \ = \sum \int \dfrac{d^3 p}{(2\pi)^3} [ \hat{\overline{\psi}} \gamma^0 u \overline{u} \gamma^0 \hat{\psi} + \hat…