Bell's inequalities

In classical physics one can set up light waves that are linearly polarized with any given orientation. And if these hit polarizing ("anti-glare") filters whose orientation is off by an angle θ, then the waves transmitted will have intensity Cos[θ]². In quantum theory the quantization of particle spin implies that any photon hitting a polarizing filter will always either just go through or be absorbed—so that in effect its spin measured relative to the orientation of the polarizer is either +1 or -1. A variety of atomic and other processes give pairs of photons that are forced to have total spin 0. And in what is essentially the Einstein–Podolsky–Rosen setup mentioned on page 1058 one can ask what happens if such photons are made to hit polarizers whose orientations differ by angle θ. In ordinary quantum theory, a straightforward calculation implies that the expected value of the product of the two measured spin values will be -Cos[θ]. But now imagine instead that when each photon is produced it is assigned some "hidden variable" ϕ that in effect explicitly specifies the angle of its polarization. Then assume that a polarizer oriented at 0° will measure the spin of such a photon to have value f[ϕ] for some fixed function f. Now the expected value of the product of the two measured spin values is found just by averaging over ϕ as

Integrate[f[ϕ] f[θ - ϕ], {ϕ, 0, 2 π}]/(2 π)

A version of Bell's inequalities is then that this integral can decrease with θ no faster than θ/(2π) - 1—as achieved when f = Sign. (In 3D ϕ must be extended to a sphere, but the same final result holds.) Yet as mentioned on page 1058, actual experiments show that in fact the decrease with θ is more rapid—and is instead consistent with the quantum theory result -Cos[θ]. So what this means is that there is in a sense more correlation between measurements made on separated photons than can apparently be explained by the individual photons carrying any kind of explicit hidden property. (In the standard formalism of quantum theory this is normally explained by saying that the two photons can only meaningfully be considered as part of a single "entangled" state. Note that because of the probabilistic nature of the correlations it turns out to be impossible to use them to do anything that would normally be considered communicating information faster than the speed of light.)

A basic assumption in deriving Bell's inequalities is that the choice of polarizer angle for measuring one photon is not affected by the choice of angle for the other. And indeed experiments have been done which try to enforce this by choosing the angles for the polarizers only just before the photons reach them—and too close in time for a light signal to get from one to the other. Such experiments again show violations of Bell's inequalities. But inevitably the actual devices that work out choices of polarizer angles must be in causal contact as part of setting up the experiment. And although it seems contrived, it is thus at least conceivable that with a realistic model for their time evolution such devices could end up operating in just such a way as to yield observed violations of Bell's inequalities.

Another way to get violations of Bell's inequalities is to allow explicit instantaneous propagation of information. But traditional models involving for example a background quantum potential again seem quite contrived, and difficult to generalize to relativistic cases. The approach I discuss in the main text is quite different, in effect using the idea that in a network model of space there can be direct connections between particles that do not in a sense ever have to go through ordinary intermediate points in space.

When set up for pairs of particles, Bell's inequalities tend just to provide numerical constraints on probabilities. But for triples of particles, it was noticed in the late 1980s that they can give constraints that force probabilities to be 0 or 1, implying that with the assumptions made, certain configurations of measurement results are simply impossible.

In quantum field theory the whole concept of measurement is much less developed than in quantum mechanics—not least because in field theory it is much more difficult to factor out subsystems, and so to avoid having to give explicit descriptions of measuring devices. But at least in axiomatic quantum field theory it is typically assumed that one can somehow measure expectation values of any suitably smeared product of field operators. (It is possible that these could be reconstructed from combinations of idealized scattering experiments). And to get a kind of analog of Bell's inequalities one can look at correlations defined by such expectation values for field operators at spacelike-separated points (too close in time for light signals to get from one to another). And it then turns out that even in the vacuum state the vacuum fluctuations that are present show nonzero such correlations—an analog of ordinary quantum mechanical entanglement. (In a non-interacting approximation these correlations turn out to be as large as is mathematically possible, but fall off exponentially outside the light cone, with exponents determined by the smallest particle mass or the measurement resolution.) In a sense, however, the presence of such correlations is just a reflection of the idealized way in which the vacuum state is set up—with each field mode determined all at once for the whole system.