[math-fun] quantum theory foundational issues, my theory of how they should be resolved
1. Re: NYTimes: quantum theory w/o observers (meekerdb) There's a good paper by Schlosshauer and Camilleri that discusses the historical debate over the Heisenberg cut and how the question has mostly been answered by the idea of decoherence.
http://arxiv.org/pdf/0804.1609v1.pdf
I say 'mostly' because pushing the Heisenberg cut all the way to include conscious observers, as in Everett's interpretation of QM, is still a little controversial because we lack a good theory of consciousness. You won't learn anything useful from the NYT discussion (Callendar's essay is good, but it's short and assumes a lot of background). The best current ideas are decoherence and Everett's many worlds. Read Schlosshauer's review papers:
http://arxiv.org/pdf/0804.1609v1.pdf
or for more depth
http://arxiv.org/pdf/quant-ph/0312059.pdf
Brent Meeker
--ok. First of all, the "decoherence program" seemed to me inadequate. This is that every physical system is contained in an "environment" and since it interacts with environment it effectively is being "measured" (no real measurement is needed though, that concept is claimed really not to exist). So for example, why do you not experience being in a superposition of living in Paris and Tokyo? Because some cosmic ray, or air molecule, or whatever (environment) interacts with you in Tokyo only, thus "measuring" the fact you are in Tokyo or not. Sounds great... except why wasn't the cosmic ray in a superposition of being in Paris & Tokyo? This doesn't solve the problem of why the world is not weird; it merely pretends to solve the problem by already assuming the answer. ("It's turtles all the way down.") I once wrote a short paper on "why quantum computers won't work" whose argument rested on 4 axioms about the nature of decoherence, which may or may not be true. Read it: "Argument against quantum computers (or against certain decoherence models)" http://rangevoting.org/WarrenSmithPages/homepage/whywontwork2.ps Recently Scott Aaronson read this paper and pointed out things are not as simple as my paper made it appear... for example, some of the axioms are not really well defined. (What is the "location" of a "qubit"?) Second. What if the physical system is "the whole universe"? I have an idea about that, but it is only very slightly mentioned in the papers Meeker cited. It is this. There is a part of the universe we can perceive, and there is a part we cannot. For example, we have a hard time perceiving dark matter and gravitons. Nevertheless, there are interactions between parts A and B. So my proposal is, since we can only see part A, part B acts as the external environment from our point of view. Now what is needed to make this work, is that almost always (or at least, "often"), when parts A and B interact, they never do so again. Similarly to, if you interact with that cosmic ray, it flies off into outer space, never to interact with you again. This causes it to carry away "measured information" permanently. If you could keep interacting over & over again with that cosmic ray, it'd get entangled with your position, and hence would no longer be "measuring" your position in the sense of Von Neumann measurement. (There is an interpretation of Von Neumann measurement involving randomization of the phase angles of off-diagonal terms in density matrix; this randoms need to be independent.) So with the cosmic ray, the problem is, the whole universe might get entangled with your position, so the whole damn universe would not be capable of "measuring" it, so the whole argument would become doubtful. That's why the cosmic ray explanation is inadequate. Now, to return to my "parts A & B" explanation, I have proven, that gravitons after they are created almost never interact with anything. Ever. For the rest of time. See this paper by me: "From time-irreversibility in gravity to measurement in quantum mechanics" http://rangevoting.org/WarrenSmithPages/homepage/absorber.pdf http://rangevoting.org/WarrenSmithPages/homepage/absorber.ps "Almost never" means the interactions occur for less than any fixed positive constant fraction of gravitons. "Anything" means matter. My point is this. The gravitons form a second, shadow universe, or external environment, for the parts of the universe we can see. There is no issue that this external environment might get entangled with our part and hence unable to "measure" our part, because of my "almost never interacts again" theorem -- the first time our part interacts with a graviton, it is a virgin, and that graviton (now entangled), never comes back to haunt us (re-interact, non-independent randomness). So effectively via gravitons, there is an inexhaustible resource of new, non-entangled, random qubits in part B of the universe, which is able to keep decohering our part A. If you examine the numbers, this effect alone is large enough (even though gravity is an extremely weak interaction) to explain why the world appears classical to a human observer. So that is my theory.
On 7/27/2013 3:18 AM, Warren D Smith wrote:
1. Re: NYTimes: quantum theory w/o observers (meekerdb) There's a good paper by Schlosshauer and Camilleri that discusses the historical debate over the Heisenberg cut and how the question has mostly been answered by the idea of decoherence.
http://arxiv.org/pdf/0804.1609v1.pdf
I say 'mostly' because pushing the Heisenberg cut all the way to include conscious observers, as in Everett's interpretation of QM, is still a little controversial because we lack a good theory of consciousness. You won't learn anything useful from the NYT discussion (Callendar's essay is good, but it's short and assumes a lot of background). The best current ideas are decoherence and Everett's many worlds. Read Schlosshauer's review papers:
http://arxiv.org/pdf/0804.1609v1.pdf
or for more depth
http://arxiv.org/pdf/quant-ph/0312059.pdf
Brent Meeker
--ok. First of all, the "decoherence program" seemed to me inadequate. This is that every physical system is contained in an "environment" and since it interacts with environment
But the weird effects of QM are observed in systems that are (temporarily) isolated from the environment. The isolation doesn't have to be perfect. The weird effects are observed in statistics and so their deviation from non-weirdness shows up as a matter of degree.
it effectively is being "measured" (no real measurement is needed though, that concept is claimed really not to exist).
So for example, why do you not experience being in a superposition of living in Paris and Tokyo? Because some cosmic ray, or air molecule, or whatever (environment) interacts with you in Tokyo only, thus "measuring" the fact you are in Tokyo or not. Sounds great... except why wasn't the cosmic ray in a superposition of being in Paris & Tokyo?
Yes, that's right. Decoherence only goes part way in explaining the classical appearance of the world. It can explain why the density matrix of system+observer becomes diagonalized *when you trace over the environment* assuming the environment has a certain randomness. But tracing over the environment is a mathematical operation with no obvious physical or operational meaning. And even after you get to a diagonalized density matrix there's a question of how to explain the apparent realization of only one outcome. I'll read your papers later. Brent
This doesn't solve the problem of why the world is not weird; it merely pretends to solve the problem by already assuming the answer. ("It's turtles all the way down.")
I once wrote a short paper on "why quantum computers won't work" whose argument rested on 4 axioms about the nature of decoherence, which may or may not be true. Read it: "Argument against quantum computers (or against certain decoherence models)" http://rangevoting.org/WarrenSmithPages/homepage/whywontwork2.ps
Recently Scott Aaronson read this paper and pointed out things are not as simple as my paper made it appear... for example, some of the axioms are not really well defined. (What is the "location" of a "qubit"?)
Second. What if the physical system is "the whole universe"? I have an idea about that, but it is only very slightly mentioned in the papers Meeker cited. It is this. There is a part of the universe we can perceive, and there is a part we cannot. For example, we have a hard time perceiving dark matter and gravitons. Nevertheless, there are interactions between parts A and B. So my proposal is, since we can only see part A, part B acts as the external environment from our point of view.
Now what is needed to make this work, is that almost always (or at least, "often"), when parts A and B interact, they never do so again. Similarly to, if you interact with that cosmic ray, it flies off into outer space, never to interact with you again. This causes it to carry away "measured information" permanently. If you could keep interacting over & over again with that cosmic ray, it'd get entangled with your position, and hence would no longer be "measuring" your position in the sense of Von Neumann measurement. (There is an interpretation of Von Neumann measurement involving randomization of the phase angles of off-diagonal terms in density matrix; this randoms need to be independent.)
So with the cosmic ray, the problem is, the whole universe might get entangled with your position, so the whole damn universe would not be capable of "measuring" it, so the whole argument would become doubtful. That's why the cosmic ray explanation is inadequate.
Now, to return to my "parts A & B" explanation, I have proven, that gravitons after they are created almost never interact with anything. Ever. For the rest of time. See this paper by me: "From time-irreversibility in gravity to measurement in quantum mechanics" http://rangevoting.org/WarrenSmithPages/homepage/absorber.pdf http://rangevoting.org/WarrenSmithPages/homepage/absorber.ps "Almost never" means the interactions occur for less than any fixed positive constant fraction of gravitons. "Anything" means matter.
My point is this. The gravitons form a second, shadow universe, or external environment, for the parts of the universe we can see. There is no issue that this external environment might get entangled with our part and hence unable to "measure" our part, because of my "almost never interacts again" theorem -- the first time our part interacts with a graviton, it is a virgin, and that graviton (now entangled), never comes back to haunt us (re-interact, non-independent randomness). So effectively via gravitons, there is an inexhaustible resource of new, non-entangled, random qubits in part B of the universe, which is able to keep decohering our part A.
If you examine the numbers, this effect alone is large enough (even though gravity is an extremely weak interaction) to explain why the world appears classical to a human observer.
So that is my theory.
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But the weird effects of QM are observed in systems that are (temporarily) isolated from the environment.
... Is the double-slit experiment isolated from the environment? http://en.wikipedia.org/wiki/Double-slit_experiment Best, É.
On 7/27/2013 9:10 AM, Eric Angelini wrote:
But the weird effects of QM are observed in systems that are (temporarily) isolated from the environment.
... Is the double-slit experiment isolated from the environment? http://en.wikipedia.org/wiki/Double-slit_experiment Best, É.
Yes, and the double slit experiment with buckyballs provides an excellent example because it allows varying the degree of isolation and causing the interference pattern to fade in or out. You heat up the buckyballs you are directing at the screen with the two slits. So the buckyballs can radiate IR photons with provides which-way information. So when they are cold you see the interference pattern and as you heat them up it fades away. And you don't have to "see" the IR photons or otherwise detect them; they just go away to be absorbed the lab walls. Brent
Warmed-over nanometre buckyballs through a double-slit !! I had no idea. That's staggering --- yet somehow seems to make the whole business more immediate and comprehensible. [Maybe I never really believed in electrons anyway ...] To a physics tyro, that web page looks impressively well done. WFL On 7/28/13, meekerdb <meekerdb@verizon.net> wrote:
On 7/27/2013 9:10 AM, Eric Angelini wrote:
But the weird effects of QM are observed in systems that are (temporarily) isolated from the environment.
... Is the double-slit experiment isolated from the environment? http://en.wikipedia.org/wiki/Double-slit_experiment Best, É.
Yes, and the double slit experiment with buckyballs provides an excellent example because it allows varying the degree of isolation and causing the interference pattern to fade in or out. You heat up the buckyballs you are directing at the screen with the two slits. So the buckyballs can radiate IR photons with provides which-way information. So when they are cold you see the interference pattern and as you heat them up it fades away. And you don't have to "see" the IR photons or otherwise detect them; they just go away to be absorbed the lab walls.
Brent
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I don't understand what's going on here. 1) Are you really "directing" buckyballs (or radiation) at the screen, or just filtering out all but what goes through (one or both of) the two slits? 2) I would've guessed that there'd be more, not less, infrared radiation as the temperature increases -- at least up to a point. No? 3) Also, what does "which-way information" mean and how do IR photons provide that? 4) Finally, why does the interference pattern fade away as you heat up the buckyballs? Thanks, Dan On 2013-07-27, at 4:31 PM, meekerdb wrote:
Yes, and the double slit experiment with buckyballs provides an excellent example because it allows varying the degree of isolation and causing the interference pattern to fade in or out. You heat up the buckyballs you are directing at the screen with the two slits. So the buckyballs can radiate IR photons with provides which-way information. So when they are cold you see the interference pattern and as you heat them up it fades away. And you don't have to "see" the IR photons or otherwise detect them; they just go away to be absorbed the lab walls.
On Sat, Jul 27, 2013 at 11:59 PM, Dan Asimov <dasimov@earthlink.net> wrote:
I don't understand what's going on here.
1) Are you really "directing" buckyballs (or radiation) at the screen, or just filtering out all but what goes through (one or both of) the two slits?
Buckyballs are usually made with something like an arc welder, so you filter out all but a few going in the same direction at the start. Then you put the double slit at some distance from the source.
2) I would've guessed that there'd be more, not less, infrared radiation as the temperature increases -- at least up to a point. No?
Yes---the hotter the particle, the more radiation.
3) Also, what does "which-way information" mean and how do IR photons provide that?
"Which-way" information is information that can be used to detect which slit the particles went through. An IR photon emitted when the buckyball is near a slit will heat up the material around the slit it goes through more than the material around the other slit.
4) Finally, why does the interference pattern fade away as you heat up the buckyballs?
Because the buckyballs get entangled with the slits, so the balls hitting the screen are in a mixed state (the slit information is "traced out") rather than a coherent state.
Thanks,
Dan
On 2013-07-27, at 4:31 PM, meekerdb wrote:
Yes, and the double slit experiment with buckyballs provides an excellent example because it allows varying the degree of isolation and causing the interference pattern to fade in or out. You heat up the buckyballs you are directing at the screen with the two slits. So the buckyballs can radiate IR photons with provides which-way information. So when they are cold you see the interference pattern and as you heat them up it fades away. And you don't have to "see" the IR photons or otherwise detect them; they just go away to be absorbed the lab walls.
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-- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
Thanks a lot, Mike. That definitely makes things a lot clearer (though not clear per se). But I'm fundamentally still confused with whether it's just radiation you see on the screen, or whether the buckyballs themselves ("going in the same direction") are moving toward the screen. (And hence whether the interference pattern is just one of radiation or of carbon molecules.) --Dan On 2013-07-27, at 11:11 PM, Mike Stay wrote:
On Sat, Jul 27, 2013 at 11:59 PM, Dan Asimov <dasimov@earthlink.net> wrote:
I don't understand what's going on here.
1) Are you really "directing" buckyballs (or radiation) at the screen, or just filtering out all but what goes through (one or both of) the two slits?
Buckyballs are usually made with something like an arc welder, so you filter out all but a few going in the same direction at the start. Then you put the double slit at some distance from the source.
2) I would've guessed that there'd be more, not less, infrared radiation as the temperature increases -- at least up to a point. No?
Yes---the hotter the particle, the more radiation.
3) Also, what does "which-way information" mean and how do IR photons provide that?
"Which-way" information is information that can be used to detect which slit the particles went through. An IR photon emitted when the buckyball is near a slit will heat up the material around the slit it goes through more than the material around the other slit.
4) Finally, why does the interference pattern fade away as you heat up the buckyballs?
Because the buckyballs get entangled with the slits, so the balls hitting the screen are in a mixed state (the slit information is "traced out") rather than a coherent state.
On Sun, Jul 28, 2013 at 12:31 AM, Dan Asimov <dasimov@earthlink.net> wrote:
Thanks a lot, Mike. That definitely makes things a lot clearer (though not clear per se).
But I'm fundamentally still confused with whether it's just radiation you see on the screen, or whether the buckyballs themselves ("going in the same direction") are moving toward the screen. (And hence whether the interference pattern is just one of radiation or of carbon molecules.)
The buckyballs themselves hit the screen; the interference pattern is interference in the matter waves, not the light emitted. -- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
See also this page: http://www.univie.ac.at/qfp/research/matterwave/c60/ On Sun, Jul 28, 2013 at 12:36 AM, Mike Stay <metaweta@gmail.com> wrote:
On Sun, Jul 28, 2013 at 12:31 AM, Dan Asimov <dasimov@earthlink.net> wrote:
Thanks a lot, Mike. That definitely makes things a lot clearer (though not clear per se).
But I'm fundamentally still confused with whether it's just radiation you see on the screen, or whether the buckyballs themselves ("going in the same direction") are moving toward the screen. (And hence whether the interference pattern is just one of radiation or of carbon molecules.)
The buckyballs themselves hit the screen; the interference pattern is interference in the matter waves, not the light emitted. -- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
-- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
On 7/27/2013 11:31 PM, Dan Asimov wrote:
Thanks a lot, Mike. That definitely makes things a lot clearer (though not clear per se).
But I'm fundamentally still confused with whether it's just radiation you see on the screen, or whether the buckyballs themselves ("going in the same direction") are moving toward the screen. (And hence whether the interference pattern is just one of radiation or of carbon molecules.)
--Dan
On 2013-07-27, at 11:11 PM, Mike Stay wrote:
On Sat, Jul 27, 2013 at 11:59 PM, Dan Asimov <dasimov@earthlink.net> wrote:
I don't understand what's going on here.
1) Are you really "directing" buckyballs (or radiation) at the screen, or just filtering out all but what goes through (one or both of) the two slits? Buckyballs are usually made with something like an arc welder, so you filter out all but a few going in the same direction at the start. Then you put the double slit at some distance from the source.
2) I would've guessed that there'd be more, not less, infrared radiation as the temperature increases -- at least up to a point. No? Yes---the hotter the particle, the more radiation.
3) Also, what does "which-way information" mean and how do IR photons provide that? "Which-way" information is information that can be used to detect which slit the particles went through.
Right.
An IR photon emitted when the buckyball is near a slit will heat up the material around the slit it goes through more than the material around the other slit.
No. It's not a matter of heating the slit. If you were "watching" with the right instrument you could see where the buckyball was going with enough resolution to say which slit it would go through.
4) Finally, why does the interference pattern fade away as you heat up the buckyballs? Because the buckyballs get entangled with the slits, so the balls hitting the screen are in a mixed state (the slit information is "traced out") rather than a coherent state.
The buckyball is entangled with the IR photons, that's why they provide information about it. Brent
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On Sun, Jul 28, 2013 at 1:34 AM, meekerdb <meekerdb@verizon.net> wrote:
On 7/27/2013 11:31 PM, Dan Asimov wrote:
Thanks a lot, Mike. That definitely makes things a lot clearer (though not clear per se).
But I'm fundamentally still confused with whether it's just radiation you see on the screen, or whether the buckyballs themselves ("going in the same direction") are moving toward the screen. (And hence whether the interference pattern is just one of radiation or of carbon molecules.)
--Dan
On 2013-07-27, at 11:11 PM, Mike Stay wrote:
On Sat, Jul 27, 2013 at 11:59 PM, Dan Asimov <dasimov@earthlink.net> wrote:
I don't understand what's going on here.
1) Are you really "directing" buckyballs (or radiation) at the screen, or just filtering out all but what goes through (one or both of) the two slits?
Buckyballs are usually made with something like an arc welder, so you filter out all but a few going in the same direction at the start. Then you put the double slit at some distance from the source.
2) I would've guessed that there'd be more, not less, infrared radiation as the temperature increases -- at least up to a point. No?
Yes---the hotter the particle, the more radiation.
3) Also, what does "which-way information" mean and how do IR photons provide that?
"Which-way" information is information that can be used to detect which slit the particles went through.
Right.
An IR photon emitted when the buckyball is near a slit will heat up the material around the slit it goes through more than the material around the other slit.
No. It's not a matter of heating the slit. If you were "watching" with the right instrument you could see where the buckyball was going with enough resolution to say which slit it would go through.
Sure; but the material around the slit itself is such an instrument. The material is an observer if it and the buckyball exchange photons. Same with every other bit of matter in the room, though it becomes harder to correlate the photon with path information as you get farther away.
4) Finally, why does the interference pattern fade away as you heat up the buckyballs?
Because the buckyballs get entangled with the slits, so the balls hitting the screen are in a mixed state (the slit information is "traced out") rather than a coherent state.
The buckyball is entangled with the IR photons, that's why they provide information about it.
Yes, and the IR photons that interact with the material around the slit get entangled with it, so then the material is entangled with the buckyball. It's similar to the teleportation protocol: the buckyball and the IR photon form an entangled pair, then the photon interacts with the slit material (or any other matter it hits) analogous to a Bell measurement. The result is that the buckyball and the material form an entangled pair---not maximally entangled, but as the buckyball gives off more and more photons, the state depends more and more on the shared information with the other matter. When you trace it out by only looking at the position of the buckyball on the screen, the interference disappears. -- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
On 7/28/2013 8:56 AM, Mike Stay wrote:
An IR photon emitted when the
>buckyball is near a slit will heat up the material around the slit it >goes through more than the material around the other slit.
No. It's not a matter of heating the slit. If you were "watching" with the right instrument you could see where the buckyball was going with enough resolution to say which slit it would go through. Sure; but the material around the slit itself is such an instrument. The material is an observer if it and the buckyball exchange photons. Same with every other bit of matter in the room, though it becomes harder to correlate the photon with path information as you get farther away.
It's not clear to me that presence of matter is important. Just the photons radiating away into empty space should be enough. Essentially the EM field is an environmental sink for information just as much as quark fields in baryonic matter. So while you're right about the interaction with the matter of the slit and the room, I don't think that's essential. Brent
An interesting calculation by Ashtekar showing that all the handwaving of all the theoretical physicists in the world has almost surely not generated a single graviton. https://groups.google.com/d/msg/sci.physics.research/d6iyjGe5Zo0/naMHyue5AW4... If stuff is decohering by emitting gravitons, it's got to be terribly low-frequency gravitons. On Sun, Jul 28, 2013 at 1:07 PM, meekerdb <meekerdb@verizon.net> wrote:
On 7/28/2013 8:56 AM, Mike Stay wrote:
An IR photon emitted when the
>>buckyball is near a slit will heat up the material around the slit >> it >>goes through more than the material around the other slit.
No. It's not a matter of heating the slit. If you were "watching" with the right instrument you could see where the buckyball was going with enough resolution to say which slit it would go through.
Sure; but the material around the slit itself is such an instrument. The material is an observer if it and the buckyball exchange photons. Same with every other bit of matter in the room, though it becomes harder to correlate the photon with path information as you get farther away.
It's not clear to me that presence of matter is important. Just the photons radiating away into empty space should be enough. Essentially the EM field is an environmental sink for information just as much as quark fields in baryonic matter. So while you're right about the interaction with the matter of the slit and the room, I don't think that's essential.
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On 8/4/2013 5:01 PM, Mike Stay wrote:
An interesting calculation by Ashtekar showing that all the handwaving of all the theoretical physicists in the world has almost surely not generated a single graviton.
https://groups.google.com/d/msg/sci.physics.research/d6iyjGe5Zo0/naMHyue5AW4...
If stuff is decohering by emitting gravitons, it's got to be terribly low-frequency gravitons.
And of course that means very long wave-length gravitons, which in turn means they don't localize things very well. That's part of the lesson of the buckyball two-slit experiment. The cooler buckyballs still emit some IR photons, but if the photon wavelength is large compared to the slit spacing then the buckyballs aren't localized enough to prevent the interference. Brent
On 7/27/2013 10:59 PM, Dan Asimov wrote:
I don't understand what's going on here.
1) Are you really "directing" buckyballs (or radiation) at the screen, or just filtering out all but what goes through (one or both of) the two slits?
http://arxiv.org/abs/quant-ph/0402146
2) I would've guessed that there'd be more, not less, infrared radiation as the temperature increases -- at least up to a point. No?
Right. As the temperature increases the buckyballs radiate more and shorter wavelength photons which, if they were observed would give the observer some (though not perfect) information as to which slit a buckyball passed through. The interesting point is they don't have to be observed to have the effect of localizing the buckball and maybe incidentally, the buckyballs are not sufficiently localized by other interactions such as gravitons, cosmic rays, or neutrinos.
3) Also, what does "which-way information" mean and how do IR photons provide that?
See above.
4) Finally, why does the interference pattern fade away as you heat up the buckyballs?
See above, although that's sort of like asking why does the world look classical - we're not sure but it seems to be because information necessary to localize things is available in the environment (in this case IR photons). Brent
Thanks,
Dan
On 2013-07-27, at 4:31 PM, meekerdb wrote:
Yes, and the double slit experiment with buckyballs provides an excellent example because it allows varying the degree of isolation and causing the interference pattern to fade in or out. You heat up the buckyballs you are directing at the screen with the two slits. So the buckyballs can radiate IR photons with provides which-way information. So when they are cold you see the interference pattern and as you heat them up it fades away. And you don't have to "see" the IR photons or otherwise detect them; they just go away to be absorbed the lab walls.
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On 27/07/2013 11:18, Warren D Smith wrote:
So for example, why do you not experience being in a superposition of living in Paris and Tokyo? Because [...]
Why would you expect a superposition of experiencing being in Paris and experiencing being in Tokyo to resemble, in any way, experiencing being in a superposition of Paris and Tokyo? In default of some reason to expect that, I don't see why the question arises at all. And I know of no such reason, though I'm not a QM expert and maybe there is one that I've never encountered. -- g
participants (7)
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Dan Asimov -
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meekerdb -
Mike Stay -
Warren D Smith