Enginerd: Linearity of QM can be proven? I didn’t know that. I don’t suppose you’d be able to sketch out the proof, or provide a link to one?
When I say it’s a theorem, I mean that when you look at all known laws of quantum physics, they are linear; so it’s a theorem of current physical models that Q(A) + Q(B) = Q(A + B). And furthermore, this was suspected early on, and used to help deduce the hypotheses and form of the Standard Model.
I don’t mean there’s a simple a priori proof, or a simple proof from experiment like there is for the identity of particles—if such a thing exists, I do not know it.
Silas: Hm, I thought you were going to have a mind-blowing, semi-novel explanation of Heisenberg. Your explanation turned out to be the same that Roger Penrose gives in The Emporer’s New Mind. Am I wrong in this characterization?
I doubt it. I mean, physics is physics is reality. You don’t get extra points for novelty. Coming up with a new way of explaining arithmetic is fine, but that which is to be explained, should still have 2 + 2 equaling 4.
The simplest proof I’ve seen is actually of my own devising, though I’m pretty sure others have come up with it before. Suppose the general framework is right, but there’s some superlinear term somewhere. Essentially, it changes things if any one component gets big—like an absolute amplitude of 1⁄2 or something. And these components can be any of the usual observable variables we’d have—distances between particles, momenta, etc.
Based just on the general framework, the individual components of the wavefunction get very small very quickly. The system decoheres into so many tiny blobs in configuration space that none of them are dense enough for those superlinear terms to be a big deal. And I don’t mean we could prepare a highly concentrated state either. Right there at the Big Bang, the superlinear components could be doing something important, but a few Planck times later they’d be a minor deal, and a millisecond later there’s no conceivable experimental apparatus that one could build that would ever detect their effect.
If there are nonlinearities that do have effects to the present day, they have to have been shepherding the wavefunction in a highly specialized way to prevent this sort of thing from happening. Mangled worlds might do it, but I have low confidence in it.
Cool, I’ve thought of that too. The problem with this approach is that it’s not obvious how to apply the Born rule or whether it must be revised. Apparently Weinberg wrote a paper on something similar, but I’ve never been able to find it.
Hmm. I see what you mean—you can end up with a sort of sleeping beauty paradox, where some branches remain more concentrated than others, and over time their ‘probabilities’ grow or shrink retroactively.
I don’t see that being a fundamental issue of dynamics, but rather of our ability to interpret it. If the Born Rule is an approximation that applies except at the dawn of time, I’m okay with that.
I don’t see that being a fundamental issue of dynamics, but rather of our ability to interpret it. If the Born Rule is an approximation that applies except at the dawn of time, I’m okay with that.
Yeah, that’s what I meant by ‘revised’. I don’t even know if it’s possible to find an approximation that behaves sanely. Last time I thought about this, I thought we’d want to avoid the sort of situation you mentioned, but I’ve been thinking about the anthropic trilemma post and now I’m leaning toward the idea that we shouldn’t exclude it a priori.
Enginerd: Linearity of QM can be proven? I didn’t know that. I don’t suppose you’d be able to sketch out the proof, or provide a link to one?
When I say it’s a theorem, I mean that when you look at all known laws of quantum physics, they are linear; so it’s a theorem of current physical models that Q(A) + Q(B) = Q(A + B). And furthermore, this was suspected early on, and used to help deduce the hypotheses and form of the Standard Model.
I don’t mean there’s a simple a priori proof, or a simple proof from experiment like there is for the identity of particles—if such a thing exists, I do not know it.
Silas: Hm, I thought you were going to have a mind-blowing, semi-novel explanation of Heisenberg. Your explanation turned out to be the same that Roger Penrose gives in The Emporer’s New Mind. Am I wrong in this characterization?
I doubt it. I mean, physics is physics is reality. You don’t get extra points for novelty. Coming up with a new way of explaining arithmetic is fine, but that which is to be explained, should still have 2 + 2 equaling 4.
The simplest proof I’ve seen is actually of my own devising, though I’m pretty sure others have come up with it before. Suppose the general framework is right, but there’s some superlinear term somewhere. Essentially, it changes things if any one component gets big—like an absolute amplitude of 1⁄2 or something. And these components can be any of the usual observable variables we’d have—distances between particles, momenta, etc.
Based just on the general framework, the individual components of the wavefunction get very small very quickly. The system decoheres into so many tiny blobs in configuration space that none of them are dense enough for those superlinear terms to be a big deal. And I don’t mean we could prepare a highly concentrated state either. Right there at the Big Bang, the superlinear components could be doing something important, but a few Planck times later they’d be a minor deal, and a millisecond later there’s no conceivable experimental apparatus that one could build that would ever detect their effect.
If there are nonlinearities that do have effects to the present day, they have to have been shepherding the wavefunction in a highly specialized way to prevent this sort of thing from happening. Mangled worlds might do it, but I have low confidence in it.
Cool, I’ve thought of that too. The problem with this approach is that it’s not obvious how to apply the Born rule or whether it must be revised. Apparently Weinberg wrote a paper on something similar, but I’ve never been able to find it.
Hmm. I see what you mean—you can end up with a sort of sleeping beauty paradox, where some branches remain more concentrated than others, and over time their ‘probabilities’ grow or shrink retroactively.
I don’t see that being a fundamental issue of dynamics, but rather of our ability to interpret it. If the Born Rule is an approximation that applies except at the dawn of time, I’m okay with that.
Yeah, that’s what I meant by ‘revised’. I don’t even know if it’s possible to find an approximation that behaves sanely. Last time I thought about this, I thought we’d want to avoid the sort of situation you mentioned, but I’ve been thinking about the anthropic trilemma post and now I’m leaning toward the idea that we shouldn’t exclude it a priori.