My Particularly Troublesome Version of the EPR Paradox
In quantum physics, a wave function instantaneously collapses once an observable is measured. For instance, if an electron's angular momentum in the z direction is measured, then the angular momenta in the x and y direction immediately become indeterminate due to the Heisenberg Uncertainty Principle. If two particles are entangled and described by a single wave function, such as two particles that are products of a previous particle that decayed and whose combined spin is known due to conservation of angular momentum, and an observable is measured on one of them, the wave function instantanously collapses on the other, no matter how far apart the particles are. Experiments can and have been done to show instantaneous transmission of information (such as which direction the measurement took place along). But special relativity says that there is no such thing as a universal "instant" for all observers in all frames of reference. So, suppose Observer A is watching a space ship pass by at a very fast constant velocity. Observer B is in the middle of the ship. Two light beams are emitted, one from the front of the ship (the edge of the ship farthest along the direction of motion according to Observer A), such that Observer B observes they have been simultaneously emitted at the same instant. Observer A sees that the light beam from the back of the ship takes longer to reach the middle than the light beam from the front, because it must catch up with the ship, while the ship is racing to meet the light beam emitted from the front. Both observers agree on the speed of light, which is universal. But if the light beams reach Observer B at the same time according to Observer B, Observer A must also see this: If Observer B sets up a machine that kills a cat if both beams reach him at the same time, Observer A would obviously agree that the cat is dead (physics thought experiments are a dangerous place for cats). So Observer A must see the beam of light being emitted from the back of the spaceship before the beam of light is emitted from the front of the spaceship in order for the beams to meet in the middle. If a series of clocks is placed along the length of the ship that Observer B sees as synchronized, Observer A will see the clocks that are further forward in the direction of motion as reading fractions of a second less, depending on how far forward they are. So, now we take our two particles that are entangled, and we put one at the front of the ship and one at the back of the ship. Two machines are set up that measure each particle's spins, and the machines each take a measurement along a different direction. Observer B sees the particle at the front of the ship being measured slightly before the particle at the back of the ship, while Observer A sees the opposite, because the back of the ship is further along in time for Observer A than it is for Observer B. So what happens to the wave function? Which axis does it collapse along? It seems natural that Observer B might be fundamentally correct, since he isn't moving with respect to the particles, but suppose we have four particles that are entangled, and the other two are in corresponding positions on the ship that Observer A is on. Observer B sees himself as being still and Observer A as moving, so neither observer is still with respect to the particles. This is not an extremely technical paradox: The mathematics only require an undergraduate level of physics education, but to my knowledge no one has ever resolved this paradox.