George Levy asks recently "Could somebody incorporate complementarity in a thought experiment in the style of Bruno's duplication experiment?"
This is an interesting proposal and I would be glad if someone manage
to present one. Just that it is *because* duplication-like experiment leads
quickly to obscurities and misleading intuitions, *that* modal logic
appears to be a fruitful investment, even if it is not the only one.
As it stand, the comp hypothesis is only a philosophical exercise because it does not reproduce the same phenomenon as QM in particular the phenomenon of complementarity. Therefore, to establish a meaningful relevance between comp and QM we must show that such phenomena can be incorporated in comp.
The following thought experiment is an attempt to illustrate how complementarity can be incorporated into a duplication experiment. This experiment raises some interesting questions regarding the relationship between the scientific MW and the philosopical plenitude.
We are onboard the starship Enterprise, and captain James T. Kirk is asking Chief Engineer Scotty to beam him down planet Alpha where a large contigent of Klingon troops is attacking a human settlement mining Dilithium crystals. There is not enough military personnel aboard the Enterprise to engage the Klingons. Kirk comes up with a brilliant idea.
Resolutely turning toward his science officer he says, "Spock!" is it possible to set the transporter to send two or more copies of an original person?"
It has never been done captain, but there is no theoretical reason why it could not be done" Spock replies with a puzzled look on his face. But that would be a brilliant way to quickly generate an army on the planet surface to fight the Klingons! Each copy will have to be sent to a slightly different location to avoid having the copies materialize inside each other. But this is not a problem. This can be done by setting a grid on the planet's surface and set the transporter to send each copy to a different point of the grid.
"Scotty," Kirk says, could you please modify the transporter to send a thousand copies of myself to the planet's surface. Set up a 1024 point grid on the planet's surface exactly centered at a Latitude of 32° 44' N and a Longitude of 117° 10' W. I want each point of the grid to be spaced by approximately one meter so I won't bump into myself. Send me to the center of the grid and all the copies to the other points of the grid.
"What you're asking me is impossible, Captain" exclaims Scotty. "The transporter will not let me do this!" I can't specify which of you is the copy and which is the real you! The noise level in the transporter is so low that the copies will be as good as the original, and there will be no way to tell which is the copy and which is the original. There is one more restriction. It's the grid. I can't make the grid any size or shape that you want.
"Why can't the grid be any size or shape?" responds Kirk. Explain yourself!
"Excuse me captain" interjects Spock, can I please answer for Scotty?"
"By all means" Spock, Kirk replies, but make it short!
"It's actually very simple." Spock explains, the transporter has a beam width which is determined by the transporter's antenna. Since our distance to the planet is comparatively large, the transporter beam intersects the planet's surface over an area approximately 6.6257 square miles. Changing the shape of the antenna will result in changing the shape of the grid. The grid can be circular, ellipsoidal, square, or rectangular. You can pick any shape you want. However, the area of the beam intersecting the planet is a constant. It must remain at 6.6257 square kilometers. Thus if the grid forms an elongated rectangle with 100 to 1 proportions (25.74x0.2574=6.6257), any one copy of you may find itself anywhere within that surface up to 25.74 kilometers from another.
Spock! Are you telling me that my location on the planet surface will be indeterminate?
Yes Captain. And there is not much you can do about it, except to change the shape of the gird but the overall grid area must remain equal to 6.6257 square kilometers. For the simple case of a rectangular grid, the indeterminacy in your x and y position is complementary. Consider yourself lucky that this positional uncertainty is only confined to the surface of the panet and not to three dimensions. Some of your copies might have found themselves transported into solid rock or 10000 feet above ground.
"Thank you Mr. Spock."
Positioning himself in a transporter Kirk orders, "Scotty energize!" and he is gone to fight his battle.
The end of the story is left to the reader to imagine how the thousand copies of Kirk are transported to the planet, defeats the Klingons and then, in the end, how the copies decide which ones will commit Hara Kiri and which one will be allowed to go back to the Enterprise.
End of Thought Experiment
I am well aware that there is a contradiction if the transporter antenna can resolve a single point of the grid to better than 1 meter square but cannot resolve the whole grid to better than 6.6257 square kilometers. Let's assume that this is due to the peculiarities of the transporter design: because of the very high power levels, the transporter makes use of cosmic string segments to connect each of the grid point to the transporter antenna. Thus, a single cosmic string is attached to a a single grid point and can therefore provide a high resolution for the point. However, the antenna which is used to control the bundle of strings has a resolution of 6.6257 square kilometers. :-)
This thought experiment, attempt to provide a model of how MW relates to the Comp hypothesis. Many questions arise.
1) Why is it that the Plenitude is not directly accessed by QM as explained by comp. Why is there a need for an intermediate MW characterized by complementarity?
2) Why is complementarity two-dimensional? Could it be three-dimentional? or higher?
3) Is the two-dimensionality of complementarity fact-like? Are there other worlds in the Plenitude which have a complementarity with a higher dimensionality?
4) Is the MW only one instance in the Plenitude? How many levels do we have to go from the scientifically determined MW to the philosophically determined Plenitude?
5) Is complementarity anthropically necessary?
This is only a feable attempt in the generation of a physical model to relate comp to the MW. I hope that we can improve on it through our discussions.