- 808

- 10

**must**be satisfied by locally causal theories. Therefore, it is often concluded that violation of these inequalities by experiments provides very strong evidence (if not conclusive proof, the only doubt being due to imperfect experiments) that reality is non-local.

However, I present a macroscopic example using coins where, Bell's theorem is always violated in the experiments. I would like to stimulate discussion of

1) why this specific example exhibits such behaviour?

2) where is the non-locality, or non-reality, or any other spooky conclusion?

3) what does this mean for the popular belief that Bell's inequalities are applicable to the EPRB experimental results?

Here it goes:

We have three coins labelled "a", "b", "c", if we toss all three a very large number of times, it follows that the inequality |ab + ac| - bc <= 1 will never be violated for any individual case and therefore for averages |<ab> + <ac>| - <bc> <= 1 will also never be violated (where heads = +1 & tails = -1).

**Proof**: For three coins with outcomes ±1, there are 8 possibilities. the LHS for each possibility is always <=1 as illustrated below:

a,b,c = (+1,+1,+1): |(+1) + (+1)| - (+1) <= 1, obeyed

a,b,c = (+1,+1,-1): |(+1) + (-1)| - (-1) <= 1, obeyed

a,b,c = (+1,-1,+1): |(-1) + (+1)| - (-1) <= 1, obeyed

a,b,c = (+1,-1,-1): |(-1) + (-1)| - (+1) <= 1, obeyed

a,b,c = (-1,+1,+1): |(-1) + (-1)| - (+1) <= 1, obeyed

a,b,c = (-1,+1,-1): |(-1) + (+1)| - (-1) <= 1, obeyed

a,b,c = (-1,-1,+1): |(+1) + (-1)| - (-1) <= 1, obeyed

a,b,c = (-1,-1,-1): |(+1) + (+1)| - (+1) <= 1, obeyed

**The Bell inequality Challenge**:

Find a locally realistic situation which violates the above inequality. According to Bell and proponents, this is impossible to do.

**Experimental Violation**:

We have 3 coins labeled "a","b","c", inside a special box. A button on the box releases only two of the coins at random when pressed. The coins must be returned to the box before the next press, therefore only two of the three coins can ever be outside of the box at the same time.

Since we can not toss all three at once, and since the inequality only contains averages of pairs of outcomes, we assume the inability to measure all three simultaneously is inconsequential. We decide to perform the experiment by tossing just the pairs a very large number of times and group the results into 3 runs for the pairs (a,b), (a,c), (b,c) tosses. Using the pairs, we calculate <ab>, <ac> and <bc>. Even though the data for each pair appears random, we find that from our data <ab> = -1, <ac> = -1 and <bc> = -1, which violates the inequality when substituted into the LHS. (i.e 3 <= 1 according to the inequality). This was supposed to be impossible according to the challenge!!! Does this mean "local causality" or realism is false?

**The Explanation**:

Consider the following: Each coin has a programmable bias which can be changed by the box just before it is released but not after. The box has an internal clock which keeps track of the time (t) in seconds. The above scenario [<ab> = -1, <ac> = -1 and <bc> = -1 ] can then easily be realized if the special box operates as follows:

Every time a button is pressed, calculate calculate sin(t) where t is the time read off the internal clock. If sin(t) > 0, program coin "a" to be biased for heads (+1) and coin "c" to be biased for tails (-1). Then randomly pick two of the three coins. If coin "b" is one of the picks, program coin "b" to be biased for tails (-1) if the other pick is coin "a", otherwise program coin "b" to be biased for heads (-1). If sin(t) <= 0 reverse all the signs.

**Conclusion**:

a) The box will always produce <ab> = -1, <ac> = -1 and <bc> = -1, no matter how many times the coins are tossed, (1 or 50 billion).

b) The results will be random

c) The inequality will always be violated no matter how many times the coins are tossed (1 or 50 billion)

d) The box and coins operate in a completely locally causal manner.

e) The result of each toss is non-contextual and predetermined from the moment the coins are produced

f) There are no loopholes in the experiment

f) There is no spooky business happening

g) YET Bell's inequality is violated.

How come?