This post is a little follow up for the series of posts I wrote about faster than light travel. In one of the posts on this subject I brought a simple example of how in Quantum mechanics faster than light travel is possible, despite it being impossible according to relativity. The example I talked about in that post was very simple, and it was easy to explain why this is indeed what happens. But this example talked only about faster than light travel on a very small scale. In this post I want to talk about another example, which is far more complex but it shows that faster than light travel is possible also on large, even cosmic distances.

Lets consider the following situation. Suppose you have two balls, one is pink and the other is green. However, the color property of the balls is quantum - both of the colors are in superposition, so both of the balls are pink and green in the same time. But if you will measure one of them, the superposition will collapse to one of the options. What is interesting is that if you will measure one, you will cause the other one also to collapse, because you now know its color as well so it is no longer in superposition.

There are no such balls in the real world, however it is possible to create particles with all the required properties. I don't want to talk about a specific example in this post so we will agree that the balls stand for some object that have a quantum property which we will call color.

Now, lets suppose that you create two such ball in the laboratory and give one of these balls to your friend. Lets label this ball A. You friend happens to be an astronaut and he flies to the moon with this ball. When he gets there, you measure your ball (B) and discovery that it is pink. This measurement causes your ball to collapse - it is no longer in superposition of ping and green, but it also causes the second ball to collapse, in the exact same instance.But even light travel to the moon in over a second. So, something changed in the second ball, A, without a reason to this being in its "Cone of light". This again means that the information of the measurement traveled faster than the speed of light.

This is known as the Einstein paradox. He originally presented it in an attempt to prove that quantum mechanics is incomplete. He claimed that it is not correct that quantum processes are probabilistic - "God doesn't play dice". In this mind experiment there is nothing impossible from the position of quantum mechanics, but allowing faster than light travel we allow time travel, and give place to a lot of other paradoxes. He offered a solution, to add a unknown property lambda which we cannot yet measure but that decides the outcome of the measurement. This works because quantum mechanics says that from all the properties of the object we now about we cannot deduct its state (in our example the ball color) so until we measure it, the object is in superposition. But if we allow for such lambda to exist we get that there is no superposition the balls are always the same color - there is no longer probability involved, all is determent from the beginning.

Interestingly, Einstein was wrong. It took some time but eventually a test that checks if such lambda exist was performed. The test was an experiment that returned a value, we will cal it S. If S is less or equal to two, then there is lambda. If S is bigger there might be lambda, but faster than light travel (in the case of the Einshtein paradox) is possible. To be even more specific, if quantum mechanics is correct . However, in physics to show that something is equal exactly is nearly impossible, so the main point here is to check if S is less than two or not. This experiment was performed a lot of times. In the beginning the equipment wasn't sensitive enough, but after a few decades the result was that . Since this is large than two, the case was closed, faster than light travel is possible. It is still unknown if there is lambda. It turned out that we can design a theory with lambda and without it, and they both will work always. They both manage to explain all the results of all the experiments conducted until now.

The Geometric McKay Correspondence (Part 1)

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