It’s not obvious to me that the non-linear effects of probabilities equal to amplitudes would be more noticeable than those of amplitudes equal to squared amplitudes. Perhaps most probability amplitude would be on very “broken” worlds with no atoms, but let’s set that aside and imagine that there are physicists doing experiments to try to discover QM.
First of all, in a two-slit experiment, the wavy peaks and troughs of probabilities would be shaped differently. This makes QM no more and no less noticeable.
You might think a more noticeable effect would be the non-locality. Under amplitudes, but not squared amplitudes, probabilities depend on far away actions outside your light cone. But this would not be possible to discover by experiment. If a physicist on the moon measures the spin of an electron at some angle at the same time as you measure some entangled electron on earth, the probabilities of spin-up vs spin-down that you observe are not just related to those two electrons, but rather all the behavior of all the electrons (and other particles) in the universe, including the two physicists themselves, in ways that do not cancel out as they do with squared amplitudes.
I think this is an important and underrated point. Any aspect of the true Born function which isn’t squared amplitude appears to agents living in the universe like incomprehensible noise.
Let me give an example. If the true Born probabilities were the squared real component of the complex amplitude, and ignored the imaginary component, then we would have no way of telling that universe apart from the one we currently assume we’re living in. (Yes, the usual Born probabilities should be massively favored for Occam reasons. But nevertheless I think it’s useful to have an understanding of what “elbow room” we have to modify the Born probabilities without contradicting observations, should that ever cause the overall theory to be simpler.)
A less jokey example would be quaternions rather than complex numbers.
It’s not obvious to me that the non-linear effects of probabilities equal to amplitudes would be more noticeable than those of amplitudes equal to squared amplitudes. Perhaps most probability amplitude would be on very “broken” worlds with no atoms, but let’s set that aside and imagine that there are physicists doing experiments to try to discover QM.
First of all, in a two-slit experiment, the wavy peaks and troughs of probabilities would be shaped differently. This makes QM no more and no less noticeable.
You might think a more noticeable effect would be the non-locality. Under amplitudes, but not squared amplitudes, probabilities depend on far away actions outside your light cone. But this would not be possible to discover by experiment. If a physicist on the moon measures the spin of an electron at some angle at the same time as you measure some entangled electron on earth, the probabilities of spin-up vs spin-down that you observe are not just related to those two electrons, but rather all the behavior of all the electrons (and other particles) in the universe, including the two physicists themselves, in ways that do not cancel out as they do with squared amplitudes.
I think this is an important and underrated point. Any aspect of the true Born function which isn’t squared amplitude appears to agents living in the universe like incomprehensible noise.
Let me give an example. If the true Born probabilities were the squared real component of the complex amplitude, and ignored the imaginary component, then we would have no way of telling that universe apart from the one we currently assume we’re living in. (Yes, the usual Born probabilities should be massively favored for Occam reasons. But nevertheless I think it’s useful to have an understanding of what “elbow room” we have to modify the Born probabilities without contradicting observations, should that ever cause the overall theory to be simpler.)
A less jokey example would be quaternions rather than complex numbers.