As I mentioned Friday, I’m going to begin a short discussion about this paper on some consequences of special relativity and quantum mechanics on our view of determinism and the Universe. The authors, John H. Conway and Simon Kochen, establish three “axioms” (and a “paradox”) and from these statements establish consequences which have wide ranging implications. All of these measurements and the following discussion regard the behavior of a spin 1 massive particle. Spin 1 massive particles can have three possible measured values of quantum mechanical spin, namely -1, 0, or 1.
The first of these axioms is a consequence of spin statistics known in this paper for reference as the SPIN axiom. If we take three orthogonal measurements and the norm (or square) of that spin value then the only possible value for a spin measurement consistent with quantum mechanics is that two of those squared spin values are 1 and one is 0 (or “101” in the paper for brevity). This leads to a paradox, named the Kochen Specker Paradox. This paradox arrives as follows.
If we were to set aside the more troubling aspect (from a classical viewpoint) of quantum mechanics for a moment and imagine that the values of possible measurements of the spin was known before the measurement was taken. If we then examine the set of 45 degree rotations about any and all possible axis from the original orthogonal axis. Takeing a subset of 33 of these possible axis and then attempt to assign “1” and “0” values for the axis points spread about the surface. If the measurement values were known ahead of time, then a value should be pre-assignable via some function to these nodal points. But it turns out that no such function exists. That is, it is impossible to assign these values consistently throughout all permutations these 45 degree symmetry transformations. Therefore no such function can exist. Yet of course, experimentally it does. Quantum mechanics is very well established experimetnally. This function does not exist yet this is what is observed. Which means that values of those experiments are not preassigned.
The next quantum mechanical conseqence that is used is called the TWIN axiom by the authors. This is the basics of quantum entanglement. If we create two particles “twinned” or created by a particle anti-particle pairing their squared orthogonal SPIN measurements will be the same if the two measurements of the two particles are taken on the same axis.
Finally the last axiom (MIN) isolates a particular peculiarity of special relativity and brings that into the context of this discussion. In special relatively simultaneity is not a clear cut matter as it was in a Newtonian system. An “event” in a relativistic setting is an occurrence, like the (idealized) snap of a finger which occurs at a singular point in space and time for any given observer. In special relativity two events separated in space can be seen to occur in the opposite order in different inertial frames. That is one observers moving past (and by internal frame that means the observer is not accelerating) by in different directions might observe event “A” to occur before “B” while another observer might observe “B” to have occurred before “A”. The MIN axiom basically asserts that our two experimenters measuring two entangled spin one particles SPIN measurement can independently and freely choose the axis by which they measure the particle.