# 15.3: Hidden Variables - Bell’s Inequality and Aspect’s experiment


Consider extending the experiment described above to the case of analysers at arbitrary angles which detect all photons. We define measurables a$$(\theta )$$ and b$$(\phi )$$ as +1 if the photon is aligned with the analyser and -1 if it is opposed. What, then, is the ensemble average value of $$P(\theta , \phi ) = \langle a(\theta )b(\phi )\rangle$$? Clearly, if a$$(\theta )$$ and b$$(\phi )$$ are uncorrelated $$P=0$$, but since they come from a common source, this is not the case: their wavefunctions are sometimes referred to as ‘entangled’.

If the photons start out with ‘hidden variable’ polarization $$\chi$$, then it is easily shown that:

$P_{HV} (\theta , \phi ) = \frac{1}{2\pi} \oint \left( \cos^2 (\theta − \chi ) − \sin^2 (\theta − \chi ) \right) \left( \cos^2 (\phi − \chi ) − \sin^2 (\phi − \chi ) \right) d\chi = \frac{1}{2} \cos 2(\theta − \phi ) \nonumber$

Meanwhile if the wavefunction collapses at the first measurement, taken arbitrarily as A:

$P_{QM} (\theta , \phi ) = \frac{1}{2\pi} \oint \left( \cos^2 (\theta − \chi ) − \sin^2 (\theta − \chi ) \right) \left( \cos^2 (\theta − \phi ) − \sin^2 (\theta − \phi ) \right) d\chi = \cos 2(\theta − \phi ) \nonumber$

In 1982, to test this Aspect carried out measurements on $$^{40}$$Ca decays using two different angles for both $$\theta$$ and $$\phi$$. The quantity he evaluated was:

$S(\theta_1, \phi_1, \theta_2, \phi_2) = P(\theta_1, \phi_1) + P(\theta_2, \phi_2) + P(\theta_2, \phi_1) − P(\theta_1, \phi_2) \nonumber$

Where he chose the values which give the largest $$S$$: $$\theta_1 = \phi_1 + \frac{\pi}{8} = \theta_2 + \frac{2\pi}{8} = \phi_2 + \frac{3\pi}{8}$$

The hidden variables theory suggests the result should be $$S=\sqrt{2}$$, while the wavefunction collapse suggests $$S=2\sqrt{2}$$ with perfect measurement devices. Imperfections in the measurement will reduce the measured correlation in each case. Aspect measured $$S = 2.697 \pm 0.015$$, confirming the quantum prediction.

The apparent complexity of Aspect’s experiment is needed to eliminate sources of error due to detector, analyser and source imperfections.

There is an apparent contradiction between quantum mechanics and relativity, in that the interpretation of quantum mechanics requires instantaneous collapse of the wavefunction. There is no measurable quantity for which the two theories give different predictions. “Teleportation” can transport a quantum state arbitrary distances, but it doesn’t transfer information instantatneously.

Most of the wavefunctions we have solved are from Schrödingers equation, which treats time and space in different ways. For a properly relativistic approach, they should be equivalent. This discrepancy between quantum and relativity is easily resolved: the Dirac equation provides a fully relativistic wave equation for which the Schrödinger equation is a low energy approximation. A nice thing about the Dirac equation is it can only be solved by spinors: as with quantization the observed physics turns out to be the only way to solve the mathematics.

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