> And such a wave function doesn't split just once but commonly many times. So, the one, poor photon as one simple wave function is soon in dozens of pieces all moving away from each other usually never to come together again. After a billion years, the pieces are still moving away from each other.
> Then, presto, bingo, one part of that wave function enters our telescope and hits our detector. Then all the dozens of pieces of that wave function, scattered across a billion light years, have to disappear, instantly, since with two telescopes we can never get two detections from the one photon.
> Well, that's tough to believe.
There are a dozen different possibilities for what the photon has done - a dozen different universes, if you like. As long as the photon never interacts with anything else, this bundle of a dozen universes can act much the same as a single universe. Certainly if you're just looking at a telescope, all dozen versions of that telescope are behaving the same, so they behave like a single unified instance.
When the photon hits the telescope in one universe, the perspective shifts: the telescope is now "inside" the 12-way split, either the photon went one way and the telescope recorded it or the photon went another way and the telescope didn't. And when we look into the telescope, we again shift from looking at the superposition from outside to being part of the superposition on the inside.
Our language isn't great for talking about this, because it's not a binary is/isn't thing - entanglement is a continuous phenomenon. If you imagine just two photons each in one of two states, then it's easy enough to imagine: photon A is in a superposition of states 0 and 1, photon B is in an independent superposition of states 0 and 1 - each photon has two possible "local universes", but because they're distant and noncommunicating, to each one the other looks like it's in a superposition. And it's easy enough to imagine: photon A is in a superposition of states 0 and 1, photon B is in a superposition of states 0 and 1, but they're entangled such that we know the sum is 1 - here there are two possible "global universes", and we see a superposition between them, but ecah particle knows the other is in the same universe - if A is an 0 then it knows B is a 1 and vice versa. The part that's hardest to imagine is that it's also possible to be somewhere in between these two states: A and B can be partially entangled such that e.g. there's a 75% chance that the sum is 1.
> So, maybe the universe is filled with tiny pieces of wave functions condemned to go on forever. Maybe that's the dark matter?
No. The evolution of quantum systems is unitary and conserves energy.
> There are a dozen different possibilities for what the photon has done - a dozen different universes, if you like. As long as the photon never interacts with anything else, this bundle of a dozen universes can act much the same as a single universe. Certainly if you're just looking at a telescope, all dozen versions of that telescope are behaving the same, so they behave like a single unified instance.
Well that's interesting you bring it up this way.
If what you said is possibly true, then light could be a carrier of things like multi-dimensional spin, or possibly where dark matter really comes from (inter-dimensional interference).
I know Greg Egan has discussed the possibility of infinite orthogonal dimensions that energy can leak into. And by energy conservation, the result would be things like dimensional rotation and other effects we cannot yet perceive.
One theory is that the EM-Drive uses a rudimentary version of this effect. We're all still awaiting the results.
> If what you said is possibly true, then light could be a carrier of things like multi-dimensional spin, or possibly where dark matter really comes from (inter-dimensional interference).
That's a total non-sequitur. What on earth are you talking about?
> No. The evolution of quantum systems is unitary and conserves energy.
I'm guessing that all the tiny parts of the wave function basically can't interact and collapse so just continue on forever. I don't see a conflict with unitary or conservation of energy. But these tiny pieces of wave function should continue to play their role in general relativity, gravitation, etc. That's my wild guess.
To pursue wild guesses, my guess is need a solid foundation in the physics. So, I tried again with quantum mechanics. I got a famous text. It claimed that the wave functions form a Hilbert space. Tilt! No way! They can be points in a Hilbert space, but a Hilbert space is complete which from some simple examples of convergence mean that the space has to have a lot of points (wave functions) that are not differentiable and not even continuous!
Then in QM I got to where they differentiate a Fourier transform. Okay. They are doing a differentiation under the integral sign. But, can't always do that. So, I get out a book with details, right, W. Rudin, Real and Complex Analysis, that is darned, fully, careful about such things. He has a nice chapter on the Fourier transform and does differentiate it. But he also justifies the differentiation by using the dominated convergence theorem which he also proves with great care in his early chapters on measure theory. Okay, that case of differentiation under the integral sign does work. So, then, back to the physics. But the physics lectures (an MIT thing) were never clear at all about such differentiation, were not even clear about their use of the Fourier transform, and never mentioned the dominated convergence theorem. Bummer. Further, soon it became clear that physics guy was going to be sloppy: He said that a wave function is differentiable. Okay. Then, seemingly amazingly, he went on to add that it is also continuous. Of course it's continuous: Every differentiable function is continuous.
So, for the unitary stuff, okay, I studied unitary top to bottom, side to side, from Halmos, in group representation theory, for the polar decomposition, etc. I'm eager to see the role in quantum mechanics. But I suspect again I'll have to get the math from math books, not the physics books!
I want to be careful about unitary, and the claim that no information is destroyed, etc. Careful. Really careful. Much more careful than that.
Heavily physics checks itself with experiments and, then, feels free to be sloppy with the math. But for considering wild stuff such as for goofy situations for wave functions, two particles entangled, the Einstein-Podolsky-Rosen (EPR) "spooky action at a distance", symmetry giving a conservation law, across billions of years and light years, I want the math correct right down to Bourbaki and axiomatic set theory -- no more of this sloppy stuff!
The sloppy math is why I left physics and continued with math. Now I'm ready, with enough solid math, to return to parts of physics, but just out of curiosity! I'll continue with the MIT QM lectures to get out of them what is there I don't know, but often I will have to be holding my nose over how he does the math. It's old for me: My ugrad physics prof kept telling me "Let's don't get hung up on the math". Well, I did and would say "Let's get the math straight. If we are going to differentiate under the integral sign, then let's hear about the dominated convergence theorem. And, by the way, for this stuff about integrating over unbounded sets, for that mostly we have to junk the Riemann integral and use the Lebesgue integral, so, let's do that. Let's start with sigma algebras, etc., something I never heard about in physics class!"
So, I can't push my wild guesses without more physics, and there I want to be careful about the math!
While a careful review of the foundations is valuable, and there's always a chance to find something others have missed, honestly it sounds like you're missing the physical intuitions a lot more than the maths. When I read what you wrote, thinking about unitarity came later; the immediate instinctive thought below even the level of language was "wrong, the wavefunction doesn't do that". I guess it depends what you want to do, but I'd suggest getting used to working with QM in the context of physical experiments, calculating energy levels and so on, and only looking to the foundations once you've got a better sense of how the formalism is supposed to work and what problems it's trying to solve.
I know about unitary in math. Just what QM does with it I don't know yet!
I'm eager enough to further develop my intuitive understanding. My ugrad physics prof said I had "a good feeling for physics" -- that was after I blew away everyone else in his freshman physics class!! :-)
Yes, being careful can yield new results in old fields. I've published two papers that apparently did that, one paper in optimization and the Kuhn-Tucker conditions and one in mathematical statistics.
In physics I'm not really trying to publish tricky papers at the core of the foundations, but I do get the impression that quite a lot about QM is still a bit fuzzy to nearly everyone.
So, I'd like to clean that up, as much as I can, both intuitively and mathematically.
Going into the details about how to get approximations, etc. about the orbitals of the water molecule, likely relevant enough for a physics student, seems a bit of a detail I'm willing to skip over. And generally I'm willing enough to take the results of the more famous experiments at essentially face value although might try to differ on the explanations.
My ugrad physics prof pushed hard on the MM experiment, Young's double slit, and Fabry-Perot, and it's amazing how close those remain to challenging current topics.
> I do get the impression that quite a lot about QM is still a bit fuzzy to nearly everyone.
I have the opposite experience; QM is really very clean, coherent, elegant, and well-understood - far more so than GR, I found. The people who want it to work like classical physics get themselves (and anyone who listens to them) awfully confused, but that confusion goes away, and you get a much better appreciation for what QM does if you put aside the "big questions" for a while and do some concrete computation. At least that's how it worked for me.
> Then, presto, bingo, one part of that wave function enters our telescope and hits our detector. Then all the dozens of pieces of that wave function, scattered across a billion light years, have to disappear, instantly, since with two telescopes we can never get two detections from the one photon.
> Well, that's tough to believe.
There are a dozen different possibilities for what the photon has done - a dozen different universes, if you like. As long as the photon never interacts with anything else, this bundle of a dozen universes can act much the same as a single universe. Certainly if you're just looking at a telescope, all dozen versions of that telescope are behaving the same, so they behave like a single unified instance.
When the photon hits the telescope in one universe, the perspective shifts: the telescope is now "inside" the 12-way split, either the photon went one way and the telescope recorded it or the photon went another way and the telescope didn't. And when we look into the telescope, we again shift from looking at the superposition from outside to being part of the superposition on the inside.
Our language isn't great for talking about this, because it's not a binary is/isn't thing - entanglement is a continuous phenomenon. If you imagine just two photons each in one of two states, then it's easy enough to imagine: photon A is in a superposition of states 0 and 1, photon B is in an independent superposition of states 0 and 1 - each photon has two possible "local universes", but because they're distant and noncommunicating, to each one the other looks like it's in a superposition. And it's easy enough to imagine: photon A is in a superposition of states 0 and 1, photon B is in a superposition of states 0 and 1, but they're entangled such that we know the sum is 1 - here there are two possible "global universes", and we see a superposition between them, but ecah particle knows the other is in the same universe - if A is an 0 then it knows B is a 1 and vice versa. The part that's hardest to imagine is that it's also possible to be somewhere in between these two states: A and B can be partially entangled such that e.g. there's a 75% chance that the sum is 1.
> So, maybe the universe is filled with tiny pieces of wave functions condemned to go on forever. Maybe that's the dark matter?
No. The evolution of quantum systems is unitary and conserves energy.