To what extent does the existence of topological order in condensed matter systems favor/disfavor different interpretations of quantum mechanics? For example, does it falsify spontaneous collapse of wave functions (such as in GRW theory)?

Review of topological order by one of the participants. GRW is a theory of the spontaneous collapse of the wave function.

Perhaps a good way to phrase the question is to ask if models exhibitting topological order can be used to constrain theories involving spontaneous collapse of the wave function.

Usually you use topological order in order to avoid effects of decoherence or losses. So, it will be automatically immune to such spontaneous collapse theories.

Macroscopic superpositions of objects are more likely to test such theories.

Is there a fundamental limit to how big of quantum superpositions we can have?

We began to discuss topological order, but the discussion inevitably returned to the quantum gravity discussion from earlier. There is a question of whether or not space can be emergent. Suppose time is absolute and space is emergent. The party line, in some sense, then, is that geometry emerges from entanglement. The argument goes like this. You assume quantum mechanics is correct and time is absolute. Then, in the limit of a large number of correlated systems, the math for handling entanglement begins to have a geometric flavor, the degrees of freedom appear to encode geometry.

Here we discussed Penrose's proposed connection between wavefunction collapse and gravity. The idea is this. You put two non-gravitational objects in a superposition and then you couple them to gravity. Depending on where the objects are, the gravitational field will be different. So that means that, since the positions are entangled, the observer should measure a superposition of fields. No one is sure what will happen after that.

It's strange that there's no way to screen gravitational waves. This is very different from photon emission. Gravitational waves can't be reflected. This means it's impossible to remove the decay of a state due to emission of gravitational waves. Does this have something to do with the unavoidable decoherence due to gravity? On the other hand, this is all a discussion of the constraint equations, which  have no dynamics. That's what you'd measure when you did the Penrose experiment. But there's also the gravitational wave degrees of freedom, which are what somebody would quantize. So maybe the Penrose experiment isn't a good test.

Somebody says that measuring a single graviton may, in principle, be impossible because the sensitivity required to make such a measurement would have a high enough energy density that the lab would collapse into a black hole.

Photons fields have vacuum fluctuations, the Cassimir effect. Do gravitons? Sound waves don't have vacuum fluctuations because they are emergent... they're entropic in nature. So perhaps, if spacetime is emergent, we shouldn't expect to need to quantize gravity, because it is entropic in nature... a many-body effect.

Emergence plays a key role: String net model found an emergent U(1) theory but no emergent gravitational modes.
More promising seems to be many body localization.
Think of GRW as a model which is just like the tradiational model for quantum mechanics but has some errors.  Would experiments with topological phases of matter somehow amplifiy these errors? Perhaps, but building systems with well-undersatood topological order is already such a subtle and difficult task that we might not abe able to detect the effect of something like GRW collapse, at least not with current technology.

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