The next step in our consideration is a more detailed look at spacelike correlations and hidden variables. In order to do this, we will use measurements at three angles in the x-y plane (with the direction of travel along the z axis. Using the transfer between Cartesian and polar coordinates where (x=1,y0) is radius 1 at 0 deg, and (x=0,y=1) is radius 1 at 90 degrees, we will consider three measurements. at 0 degrees, 37 degrees, and 74 degrees.
We will also need to consider what is required for a local hidden variable theory to be valid for spacelike correlations. It should be clear that such a theory cannot include signals sent from one measurement to the other...since we are specifically considering two measurements that are spacelike from each other. For this reason, hidden variable theories had assumed that these hidden variables were set initially, at the point that the two spin 1/2 particles seperated. Using such a theory, the spin that would be measured at any given angle can be considered pre-set. We don't know what the variables are, but we can posit model of underlying mechanics that includes a number of possible variables that would determine what the measurement of spin would be. So, with a local hidden variable theory, a two-particle system would have pairs of preset spins in all directions. For all angles for which particle 1 is pre-set to be measured in the up direction, particle 2 is pre-set to be measured in the down direction. We can check this at all three angles by setting both measuring devices in the same direction. We could look at one million pairs at 0 deg, one million pairs at 37 deg, and one million pairs at 74 deg. We find that, in each direction, we have half a million times that particle 1 is measured up and particle 2 is measured down, and half a million times that particle 1 is measured down and particle 1 is measured up.** With this, we can see that, when both particles are measured in the same direction, there is perfect anti-correlation between the particles. Thus, using the hidden variable theory, we can make the following conclusion: Whenever particle 2 is measured in the up direction, particle 1 has been preset to be measured in the down direction. Whenever particle 1 is measured in the down direction, particle 1 has been preset to be measured in the up direction. As a shorthand, we will call both the actual measurement of particle 1 as up and the measurement of particle 2 as down "particle 1 is up", even though we are really saying "particle 1 was preset to be up" in both cases. OK, having set this up, let us consider measurements in different directions. First we measure one particle at 0 deg, and one at 37 deg. We find particle 1 being up at 0 and 37 degrees 450,000 out of a million, down at 0 and 37 degrees 450,000 times, and down at one angle and up at the other 100,000 times. Thus, we can say that there is a correlation (two up measurements or two down measurements) 90% of the time, when one measurement is at 0 degrees and one at 37 degrees, and anti-correlations 10% of the time. We repeat the experiment at 37 deg. and 74 deg, and find the same results. (Since the assignment of 0 degrees is arbitrary, it would be shocking if we didn't.) So, now we will predict the % of time that there are correlations between 0 deg. and 74 deg. When particle 1 is up at 37 degrees, it is up at 0 deg. 90% of the time and down at 37 deg. 10% of the time. From our measurements at 37 deg. and 74 deg, we know that when particle 1 is up at 37 degrees, particle 2 is also up 90% of the time. If the anti-correlation between measurements at 0 degrees and 74 degrees is maximized, we have the following: up at zero degrees, up at 37 degrees, up at 74 degrees 40% of the time up at zero degrees, up at 37 degrees, down at 74 degrees 5% of the time down at zero degrees, up at 37 degrees, up at 74 degrees 5% of the time Using similar logic, we also get the following results down at zero degrees, down at 37 degrees, down at 74 degrees 40% of the time down at zero degrees, down at 37 degrees, up at 74 degrees 5% of the time up at zero degrees, down at 37 degrees, down at 74 degrees 5% of the time Adding these up we get correlations between the measurements at 0 degrees and 74 degrees at least 80% of the time, and anti-correlations no more than 20% of the time. We can also calculate this number from QM as cos(74 deg)^2 = 64%. This is far less than the minimum predicted by hidden variable theories. What we requried in order to make these predictions are: 1) The ability to count 2) The distributive law of logic: if A & (B or C), then (A & B) or (A & C). 3) Local hidden variables. Either one of these must be dropped or QM is wrong. But, Aspect and other experimentalists have confirmed the predictions of QM for spacelike correlations. Therefore, one is required to drop one the the three assumptions listed above. While philosophers had toyed with "quantum logic" without the distributive law, this proved problematic if one wishes to keep the logical/mathamatical toolkit of physicists. Thus, it is generally accepted that local hidden variable theories have been falsified. Dan M. ** Obviously, there are statistical errors, it is really half a million +/- 707....But, .17% error will be seen to be far less than the differences we will see. Also, there will, due to experimental error, be some glitches, when both are measured up, one isn't measured, etc....all of which need not be considered in this idealized experiment but are discussed and accounted for in real experimental papers. _______________________________________________ http://www.mccmedia.com/mailman/listinfo/brin-l
