What is conclusive proof of the (in)existence of a magnetic monopole?
There is no conclusive proof either way.
Conclusive proof for existence might be something like:
Conclusive proof for nonexistence is trickier. How would we have known if there was no Higgs boson? The answer is that the Higgs boson plays a particular theoretical role, and decades ago it was already known that it wouldn't be able to play that role if it were heavier than 1000 GeV. Over the years, as more and more evidence was gathered through high-energy experiments, the range of allowable masses consistent with theory was narrowed down. A few months before the Higgs was positively identified at the LHC, enough evidence had been collected to show that the Higgs could not be heavier than 140 GeV. And if your collider is powerful enough so that it should be able to produce 140 GeV particles, and you still don't find the Higgs, then the theory must be wrong. So that's how we would've known.
But when it comes to magnetic monopoles things are trickier, because there are several different theories that predict their existence, so we don't know exactly what role they're supposed to play in particle physics and we can't accurately predict their properties. So, in the immediate future, we can keep building more and more powerful particle accelerators, and if we still don't find any monopoles, it might just be because they're still not powerful enough, and there could be theories that predict their existence at even higher energies. Wikipedia tells me that they probably can't be heavier than 1017 GeV, because if they were, then the universe's energy density would be too high, and it would have already collapsed. However, 1017 GeV is way beyond any energies we can plausibly achieve during this century, so it may be a long time before monopoles can be conclusively ruled out.
[1] Of course, those experiments haven't been working thus far, so they're not going to suddenly start working, but if, say, we discovered some hitherto unknown material in a meteorite or something, and placed it near one of these apparatuses and it suddenly started registering monopoles, we'd be forced to consider the possibility that the material was emitting magnetic monopoles. Several different types of experiments would have to be performed, so we'd know it really was due to monopoles and not some interfering effects, as it's unlikely the same interference would yield the same apparent signals in multiple different types of experiments.
Conclusive proof for existence might be something like:
- Production of a new particle at the LHC with five-sigma certainty, confirmed to have properties in precise agreement with a theory that predicts their existence.
- A reproducible method of preparing of a B-field with nonzero divergence.
- Precisely reproducible success of conventional detection experiments like that performed by Cabrera, of multiple different types [1]
Conclusive proof for nonexistence is trickier. How would we have known if there was no Higgs boson? The answer is that the Higgs boson plays a particular theoretical role, and decades ago it was already known that it wouldn't be able to play that role if it were heavier than 1000 GeV. Over the years, as more and more evidence was gathered through high-energy experiments, the range of allowable masses consistent with theory was narrowed down. A few months before the Higgs was positively identified at the LHC, enough evidence had been collected to show that the Higgs could not be heavier than 140 GeV. And if your collider is powerful enough so that it should be able to produce 140 GeV particles, and you still don't find the Higgs, then the theory must be wrong. So that's how we would've known.
But when it comes to magnetic monopoles things are trickier, because there are several different theories that predict their existence, so we don't know exactly what role they're supposed to play in particle physics and we can't accurately predict their properties. So, in the immediate future, we can keep building more and more powerful particle accelerators, and if we still don't find any monopoles, it might just be because they're still not powerful enough, and there could be theories that predict their existence at even higher energies. Wikipedia tells me that they probably can't be heavier than 1017 GeV, because if they were, then the universe's energy density would be too high, and it would have already collapsed. However, 1017 GeV is way beyond any energies we can plausibly achieve during this century, so it may be a long time before monopoles can be conclusively ruled out.
[1] Of course, those experiments haven't been working thus far, so they're not going to suddenly start working, but if, say, we discovered some hitherto unknown material in a meteorite or something, and placed it near one of these apparatuses and it suddenly started registering monopoles, we'd be forced to consider the possibility that the material was emitting magnetic monopoles. Several different types of experiments would have to be performed, so we'd know it really was due to monopoles and not some interfering effects, as it's unlikely the same interference would yield the same apparent signals in multiple different types of experiments.