The role of mathematics in physics is interesting. Originally, mathematical relationships were used to summarise a myriad of observations, thus from Newtonian gravity and mechanics, it is possible to know where the moon will be in the sky at any time. But somewhere around the beginning of the twentieth century, an odd thing happened: the mathematics of General Relativity became so complicated that many, if not most physicists could not use it. Then came the state vector formalism for quantum mechanics, a procedure that strictly speaking allowed people to come up with an answer without really understanding why. Then, as the twentieth century proceeded, something further developed: a belief that mathematics was the basis of nature. Theory started with equations, not observations. An equation, of course, is a statement, thus A equals B can be written with an equal sign instead of words. Now we have string theory, where a number of physicists have been working for decades without coming up with anything that can be tested. Nevertheless, most physicists would agree that if observation falsifies a mathematical relationship, then something has gone wrong with the mathematics, and the problem is usually a false premise. With Bell’s Inequalities, however, it seems logic goes out the window.
Bell’s inequalities are applicable only when the following premises are satisfied:
Premise 1: One can devise a test that will give one of two discrete results. For simplicity we label these (+) and (-).
Premise 2: We can carry out such a test under three different sets of conditions, which we label A, B and C. When we do this, the results between tests have to be comparable, and the simplest way of doing this is to represent the probability of a positive result at A as A(+). The reason for this is that if we did 10 tests at A, 10 at B, and 500 at C, we cannot properly compare the results simply by totalling results.
Premise 1 is reasonably easily met. John Bell used as an example, washing socks. The socks would either pass a test (e.g. they are clean) or fail, (i.e. they need rewashing). In quantum mechanics there are good examples of suitable candidates, e.g. a spin can be either clockwise or counterclockwise, but not both. Further, all particles must have the same spin, and as long as they are the same particle, this is imposed by quantum mechanics. Thus an electron has a spin of either +1/2 or -1/2.
Premises 1 and 2 can be combined. By working with probabilities, we can say that each particle must register once, one way or the other (or each sock is tested once), which gives us
A(+) + A(-) = 1; B(+) + B(-) = 1; C(+) + C(-) = 1
i.e. the probability of one particle tested once and giving one of the two results is 1. At this point we neglect experimental error, such as a particle failing to register.
Now, let us do a little algebra/set theory by combining probabilities from more than one determination. By combining, we might take two pieces of apparatus, and with one determine the (+) result at condition A, and the negative one at (B) If so, we take the product of these, because probabilities are multiplicative. If so, we can write
A(+) B(-) = A(+) B(-) [C(+) + C(-)]
because the bracketed term [C(+) + C(-)] equals 1, the sum of the probabilities of results that occurred under conditions C.
B(+)C(-) = [A(+) + A(-)] B(+)C(-)
By adding and expanding
A(+) B(-) + B(+)C(-) = A(+) B(-) C(+) + A(+) B(-) C(-) + A(+) B(+)C(-) + A(-)B(+)C(-)
= A(+)C(-) [(B(+) + B(-)] + A+B– C+ + A– B(+)C(-)
Since the bracketed term [(B(+) + B(-)] equals 1 and the last two terms are positive numbers, or at least zero, we have
A(+) B(-) + B(+)C(-) ≧ A(+)C(-)
This is the simplest form of a Bell inequality. In Bell’s sock-washing example, he showed how socks washed at three different temperatures had to comply.
An important point is that provided the samples in the tests must give only one result from only two possible results, and provided the tests are applied under three sets of conditions, the mathematics say the results must comply with the inequality. Further, only premise 1 relates to the physics of the samples tested; the second is merely a requirement that the tests are done competently. The problem is, modern physicists say entangled particles violate the inequality. How can this be?
Non-compliance by entangled particles is usually considered a consequence of the entanglement being non-local, but that makes no sense because in the above derivation, locality is not mentioned. All that is required is that premise 1 holds, i.e. measuring the spin of one particle, say, means the other is known without measurement. So, the entangled particles have properties that fulfil premise 1. Thus violation of the inequality means either one of the premises is false, or the associative law of sets, used in the derivation, is false, which would mean all mathematics are invalid.
So my challenge is to produce a mathematical relationship that shows how these violations could conceivably occur? You must come up with a mathematical relationship or a logic statement that falsifies the above inequality, and it must include a term that specifies when the inequality is violated. So, any takers? My answer in my next Monday post.