papalex on Nostr: To add some details on the “complicated” side of things: In standard QM, we work ...

To add some details on the “complicated” side of things:

In standard QM, we work with a complex Hilbert space where every vector has a clear physical meaning. Moving to QFT, we shift to Fock space, which aggregates these states. We replace individual particle states with field operators (ladder operators) that create the vacuum excitations. But we still work with a conventional complex Hilbert space.

However, there is no single "correct" Fock space here. Thanks to Haag’s Theorem, there is an infinite choice of unitarily inequivalent representations, meaning the way we define our particles is fundamentally linked to the state of the vacuum.

Then, things get messy when we move to gauge theory (which apparently we need to describe nature). We start by quantizing the fields, but this introduces massive redundancy because we quantized redundant degrees of freedom. The raw Fock space gets flooded with "unphysical" states that don't satisfy gauge symmetry. In many cases, these raw spaces contain negative-norm states, making a direct probabilistic interpretation impossible. To recover a physical Hilbert space, we have to "mod out" the gauge orbits and ghosts (aka undo the mistake of quantizing too much).

Finally, what I did not yet mention: transitioning to relativistic QM (aka Dirac equation) forces us to switch from complex scalar fields to slices of spinor bundles, which accounts to space time geometry.