If we define spacetime as a Lorentzian manifold $(E^4,g)$, how can we define the property that $$\iiint_{E^3}\Psi^*\Psi\ d^3\mathbf{x}=1$$ for a wavefunction $\Psi$? Because there's no global sense of time.
With my approach, I started with a Lorentzian manifold $(M,\,g)$, the frame bundle $LM\xrightarrow{\pi_{LM}} M$, and the associated bundle $E\equiv (LM\times\mathbb{C})/GL(\mathrm{dim}\,M,\mathbb{R})\xrightarrow{\pi_E}M$. Then, a wavefunction is a section $\psi:M\to E$ of this $\mathbb{C}$-vector bundle $E\xrightarrow{\pi_E}M$. This is so $\psi(p)=[\mathbf{e},z]=\{(\mathbf{e}\lhd g,g^{-1}\rhd z)\in L_pM\times \mathbb{C}\ |\ g\in GL(\mathrm{dim}\,M,\mathbb{R})\}$ is independent of frame.
We can also look at a wavefunction as a function $\Psi:LM\to\mathbb{C}$ defined by $\psi(p)=[\mathbf{e},\Psi(\mathbf{e})]$. We can additionally pull $\Psi$ back using a local section $\sigma:U\to LU$ to view it as a local complex-valued function $\sigma^*\Psi:U\to\mathbb{C}$. Whichever global method we take, how do we parameterize $\Psi$ over time so that we can normalize? I tried using the volume form with respect to the metric but this integrates over spacetime instead of just space. $$\int_M \Psi^*\Psi\ vol_g\neq 1$$
How do we define this property of a wavefunction? Do we need quantum field theory for this task?