# Defining spacetime as a Lorentzian manifold, how to define the property $\iiint_{E^3}\Psi^*\Psi d^3{\bf x}=1$?

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?

• Integrating on the entire spacetime shouldn't work since the probability is only one at a given time. The thing usually done is to make it so that the wavefunction is of unit norm for any Cauchy surface, regardless of foliation. Jun 15, 2020 at 23:34
• I believe that @Slereah just did express it mathematically Jun 16, 2020 at 3:18
• volume integrals can't be invariant, because volumes transform under lorentz transformations. Any definition of a volume is going to have a trailing timelike lorentz index. Apr 24, 2022 at 15:33

If you quantize the manifestly covariant Polyakov action for the point particle, written in its 1st-order form as $$S[x^{\mu}(\tau), p_{\mu}(\tau), \lambda(\tau)] = \int d\tau \left( p_{\mu} \dot{x}^{\mu} - \lambda \left( g^{\mu \nu} p_{\mu} p_{\nu} - m^2 \right) \right),$$

you will end up with wavefunctions which are functions over the spacetime like you wanted, with $$x$$ and $$p$$ acting by $$\hat{x}^{\mu} \Psi(x) = x^{\mu} \Psi(x), \quad \hat{p}_{\mu} \Psi(x) = - i \hbar \partial_{\mu} \Psi(x).$$

However, these functions aren't elements of the physical Hilbert space. You have to implement the constraint that is fixed by the Lagrange multiplier $$\lambda$$ as a quantum operator: $$\left( \hbar^2 \Box + m^2 \right) \Psi(x) = 0.$$

This is of course just the Klein-Gordon equation.

Naively, to recover the physical Hilbert space you should look at the rigged Hilbert space of functions over space-time, and select the kernel of the constraint operator to be your physical Hilbert space. However, this operation is only well defined when $$0$$ is in the discrete spectrum of the contraint operator, and it is easy to see that $$0$$ is in the continuous spectrum of the Klein-Gordon operator. You therefore need additional input solve this theory.

The solution comes from noticing that the solutions of the Klein-Gordon equation are completely determined by $$\Psi(t_0, \vec{x})$$ and $$\dot{\Psi}(t_0, \vec{x})$$ for some $$t_0$$. Therefore your wavefunctions are two-component functions over the spatial slice parametrized by $$\vec{x}$$. The two components correspond to particles and anti-particles.

However, it isn't possible to define a positive definite inner product on the space of solutions.

That is not the case for Dirac's equation, where a positive definite inner product can be defined.

In either case, second quantization makes the problem of defining an inner product on the space of solutions to a relativistic field equation obsolete. The modern treatment is to start with the field Lagrangian instead of a particle Lagrangian and derive a positive Hilbert space of the QFT.