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Apparently, there are two different routes to get to quantum field theory from single-particle quantum mechanics: (I'm going to use nonrelativistic quantum mechanics for this discussion. I think the same issues apply in relativistic quantum mechanics.)
Route 1: Many-particle quantum mechanics
Start with single-particle QM, with the Schrodinger equation: [itex]- \frac{1}{2m} \nabla^2 \psi = i \frac{\partial}{\partial t} \psi[/itex]
Once again, start with the single-particle wave function.
Maybe it's inevitable because there is only one QFT possible for free nonrelativistic fields satisfying Galilean symmetry. But it still seems strange, because the first route doesn't start with a general principle about commutation relations; those rules are just consequences of the way creation and annihilation operators work on Fock space, together with the symmetry/anti-symmetry rules for identical particles. It doesn't presuppose any general commutation rule relating a field to its canonical momentum.
Route 1: Many-particle quantum mechanics
Start with single-particle QM, with the Schrodinger equation: [itex]- \frac{1}{2m} \nabla^2 \psi = i \frac{\partial}{\partial t} \psi[/itex]
- Now, extend it to many (initially, noninteracting) particles: [itex]\Psi(\vec{r_1}, \vec{r_2}, ..., \vec{r_n}, t)[/itex]
- Introduce creation and annihilation operators to get you from (a properly symmetrized) [itex]n[/itex]-particle state to an [itex]n+1[/itex]-particle state, and vice-verse.
Once again, start with the single-particle wave function.
- Instead of viewing [itex]\psi[/itex] as a wave function, you view it as a classical field.
- Describe that field using a Lagrangian density [itex]\mathcal{L} = i \psi^* \dot{psi} - \frac{1}{2m}|\nabla \psi|^2[/itex]
- Derive the canonical momentum using [itex]\pi = \dfrac{\partial}{\partial \dot{\psi}}[/itex]
- Impose the commutation rule: [itex][\pi(\vec{r}), \psi(\vec{r'})] = -i \delta^3(\vec{r'} - \vec{r})[/itex]
Maybe it's inevitable because there is only one QFT possible for free nonrelativistic fields satisfying Galilean symmetry. But it still seems strange, because the first route doesn't start with a general principle about commutation relations; those rules are just consequences of the way creation and annihilation operators work on Fock space, together with the symmetry/anti-symmetry rules for identical particles. It doesn't presuppose any general commutation rule relating a field to its canonical momentum.