Polarized light

We describe Bell's result using polarized light. There are two motions associated with a wave. Light travels in one direction and the field strength changes as light passes through a fixed point. The field strength change can occur in any direction perpendicular to the direction of motion. We say light is polarized in the direction that the field level changes. See Figure 6.2. Each photon or particle of light has an angle of polarization. We say a source of light is polarized when most of the photons are aligned in a single direction. There are many ways light can become polarized. Light reflected at a shallow angle is polarized to some degree. That is why polarizing sun glasses can reduce glare.

An ideal polarizing filter only allows that component of light to be transmitted that is parallel to the axis of polarization of the filter. If the angle between the axis of polarization of light and the polarizing filter is $\theta$ then the amplitude of the transmitted light is $\cos(\theta)$ . See Figure 6.3. If a a single photon encounters a polarizing filter it must either completely traverse the filter or be completely blocked. It cannot split into any smaller units of energy. However the classical relationship must hold in a statistical sense. The probability that a single photon will traverse the filter must be such that statistically the predictions of quantum mechanics and classical physics will agree.

The strangeness of quantum mechanics makes it difficult to describe these experiments coherently. On the one hand the photon does not have a definite polarization until and unless it is detected. Yet we need to talk about the angle between the photon polarization and the filter. There is no good way to deal with this.

Consider the experiment illustrated in Figure 6.4. We have two polarizers at a $90^\circ$ angle. This blocks all light since the light coming out of the first polarizer behaves as if it is polarized at a $90^\circ$ angle relative to the second polarizer. $\cos(90^\circ) = 0$ . Now insert a third polarizer between the two existing polarizers at a $45^\circ$ angle relative to both of them. The amplitude coming out of the second polarizer is proportional to $\cos(45^\circ) = 1/\sqrt(2)$ . The amplitude coming out of the third polarizer is $\cos(45^\circ)^2 = 1/2$ .

Thus it would seem that the second polarizer changes the angle of polarization of the photons that passed through it Otherwise nothing would make it through the third polarizer. But we cannot assume this as we shall see shortly. Quantum mechanics gives a probability distribution in configuration space that evolves in a deterministic fashion. It says nothing about what is happening in physical space between observation. It only gives the probability of making a given observation.

Many physicists believe it is not meaningful to talk about what is happening in physical space between observations. Of course that does not prevent them from doing so. Its almost impossible not to but one has to be careful about taking such talk too seriously. At best it is metaphor and intuitive guide. Bohm succeeded in giving a consistent theory that talks about what the particle is doing between measurements[7]. However any casual talk about what is happening to the particle between measurements if taken too seriously is almost certain to lead to wrong results.

Consider a single quantum event that creates a pair of photons. Conservation laws require a correlation in properties like polarization for the elements of such pairs. The probability that both will pass though a pair of polarizers is $\cos(\theta)$ where $\theta$ is the angle between the polarizers. Note this says nothing about the polarization angle of the photons. That does not exist until it is observed!

In quantum mechanics it is as if once one of the photons traverses a polarizer the other becomes aligned with that polarizer. Before either particle traverses a polarizer neither particle had a polarization angle. Afterwords they have perfectly correlated polarization angles. This is they type of talk that we just said can be so misleading. Yet quantum mechanics predicts that the detection of one of the photons must influence the detection of the other as if something like this happened.

**Figure 6.2:** Polarized waves
$\includegraphics[scale = .7]{../xpolarized}$ Wave moving along the axis and polarized horizontally or parallel with the axis. $\includegraphics[scale = .7]{../ypolarized}$ Wave moving along the axis and polarized vertically or parallel with the axis.

**Figure 6.3:** Polarizing filter
$\includegraphics[]{../filterin}$ The input wave is traveling along the Z axis. Its polarization angle is $30^\circ$ relative to the polarizing filter which is aligned with the Y axis. $\includegraphics[]{../filterout}$ The output wave is polarized along the Y axis aligned with the filter. Its amplitude is $\cos(30^\circ)$ times the input amplitude.

**Figure 6.4:** *Adding* a polarizing filter allows light to get through
$\includegraphics[]{../twopol}$ With polarizers at a $90^\circ$ angle no light gets through. $\includegraphics[]{../threepol}$ Add a third polarizer in the middle at an intermediate $45^\circ$ angle and half the light gets through.