Think of a modified Mott experiment: You place a single particle in the center of an empty perfect detector. The particle is described by a wave function, which will spread outwards and interact at exactly one point P. But if physics is local, how does nature around a different point Q know the particle already interacted at P? What keeps nature from messing up, and "double-interacting" the particle, or forgetting to let the particle interact?
Historically, people came up with various non-satisfying solutions:
- Hidden variables, the particle takes a classical path after all
- Non-locality, P and Q communicate (in a way that doesn't allow signaling)
- Particle number is just conserved. Nature does magical bookkeeping. ("shut up and calculate")
Of course, looking at the math, this is not so mysterious after all! One-particle states belong to a different Hilbert space than multi-particle states (which live in a Fock space). There is no way one state could evolve into the other, so no way for nature to mix up. The way QM attaches a "global variable" named "particle number" to the wavefunction is by putting different $n$-particle wavefunctions in different buckets. But this is still not very satisfying. It is merely a construction that ensures the right outcome. It hard codes particle number conservation, rather than explaining it.
This makes me wonder if particle number conservation is really so exact. Small violations might point to an underlying mechanism. Has it ever been tested? Are there experimental limits on particle number non-conservation? Like where you put one photon in a box, and in rare cases five, or none, come out.
Clarification: I'm of course well aware that particle number is not conserved absolutely. You can have one photon in the initial state, and multiple of lower energy in the final state. You can have particles decay, and you can have them convert. And then there are intermediate, virtual particles. Momentum and energy between initial and final state are always 100% conserved. I'm not talking about any of this (although some of it might be relevant for the answer).
What I'm asking about is this: In the Born interpretation of the wave function, the amplitude squared is the probability to detect a given particle somewhere. E.g. it'll be with 40% probability in a certain region, and with 60% outside of the region. But the probability is zero to detect particles in both disjunct regions. Has someone tested if that is really the case?
Prepare a 1 GeV proton, put it somewhere at rest ($x$ and $p$ small, macroscopically). Look a bit later again. On average, there will still be one 1 GeV proton there. But does it happen sometimes that you'll find two, or none? Do we know for sure that the wave function collapse (nature's bookkeeping, particle conservation, or whatever you want to call it) always works as thought? Can one give a bound on how large deviations can maximally be?