On Bell’s Inequality

Bell’s 1964 paper¹ has spawned a large literature of theoretical and experimental studies. Mermin’s 1981 paper² provides a simple and clear presentation of Bell’s work. We will analyze Bell’s inequality from Mermin’s paper. Mermin designed a device to explain the basic ideas. At the appendix of the paper, he first described the structure of the device.

The device exploits Bohm’s version of the Einstein-Podolsky-Rosen experiment. The two particles emerging from the box are spin-½ particles in the singlet state. The two detectors contain Stern-Gerlach magnets, and the three switch positions determine whether the orientation of the magnets are vertical or ±120° to the vertical in the plane perpendicular to the line of flight of the particles. When the switches have the same settings the magnets have the same orientation. One detector flash red or green according to whether the measured spin is along or opposite to the field; the other uses the opposite color convention. Thus when the same colors flash the measured spin components are different.

Then he explained the physical principle of the expected experimental results.

It is a well-known elementary result that when the orientations of the magnets differ by an angle θ, then the probability of spin measurements of each particle yielding opposite values is cos² (½θ). This probability is unity when θ = 0 [case (a)] and ¼ when θ = ±120° [case (b)].

Numerous experimental results are consistent with this “well-known elementary result” in the quantum theory.

In the main body of the paper, Mermin provided a different interpretation of the device based on predetermination, an interpretation favored by Einstein. From this interpretation, the probability is higher than 1/3, not ¼ predicted from quantum mechanics. The experimental results are not consistent with this interpretation.

The interpretation is presented in Mermin’s paper in great detail and precision. The interpretation is based on the assumption that there was a one-to-one correspondence between two colors, which represent two quantum states. Bohm³ made a very specific comment on one-to-one correspondence in discussing the paradox of Einstein, Podolsky, and Rosen. He noted,

In fact, in quantum theory, one makes a quite different, but equally plausible, hypothesis concerning the fundamental nature of matter. Here, we assume that the one-to-one correspondence between mathematical theory and well-defined "elements of reality" exists only at the classical level of accuracy. For at the quantum level, the mathematical description provided by the wave function is certainly not in a one-to-one correspondence with the actual behavior of the system under description, but only in a statistical correspondence. (P 620)

This is the main difference between the classical theory and the quantum theory.

The experimental results are consistent with “a well-known elementary result” in quantum theory. But there is also a need to introduce the concept of “instantaneous action over the distance” to interpretate the data. Like many others, I still don’t feel comfortable about the concept. Is the reality really that strange? Or is there a more coherent theory to be developed?

References

1.    Bell, J.S., 1964. On the Einstein Podolsky Rosen paradox. Physics Physique Fizika, 1(3), p.195.

2.    Mermin, N.D., 1981. Bringing home the atomic world: Quantum mysteries for anybody. American Journal of Physics, 49(10), pp.940-943.

3.    Bohm, D. 1951, Quantum Theory, Prentice Hall