Bell’s Theorem

This is a paper that I wrote in September of 2017 about Bell’s theorem, a very impactful and interesting discovery in physics. I wish I understood more about it. In explaining Bell’s theorem, I also elucidated some basic concepts in physics such as locality and how light works. I apologize for referencing YouTube videos, the faux pas of citation, but, hopefully, you can excuse this singular error and enjoy the content of the paper. Want to talk more about Bell’s theorem? Head on over to the forum.


John Stewart Bell Was born on 28 June 1928, or 6/28/28, in Belfast, Northern Ireland. He died of cerebral hemorrhage at 77 on 1 October 1990 at Belfast. Bell worked for CERN as a particle physicist, but accomplished his most important work in his off time as a hobby: developing his theorem. Bell’s thesis was contradictory to Einstein and was that reality must be nonlocal. This has been supported by many experiments after him, but has been challenged by others, and remains controversial (1). To understand his wonderful and amazing discoveries, one must comprehend also some crucial underlying physics that the following paragraphs first explain.

            Realism as a physical theory is not defined with invariance. It is concerned with the essence of scientific knowledge. Scientific realists believe in the epistemic, having faith in information an observer receives through scientific processes (2). Bell’s theorem contests this.

            Locality states that no information or particles can travel with superluminous speed. Bell’s Theorem contests this also (3).

            Now an explanation for light waves must be provided. It is helpful to first break down the term electromagnetic field. It is a word combing the terms electric field and magnetic field. An  electric field can be imagined as a plane with many vectors on it, each representing a point in space. These are force vectors exerting a force on any charged particle in space, in the direction of the vector and proportional to the length of the vector and the charge of the particle. Now imagine another vector field like the previous one. This represents the magnetic field. Only when a charged particle is moving across it, does it act with force perpendicular to the direction of the particle’s motion and the magnetic field, with a strength proportional to the length of the magnetic field vector, the particle’s charge, and its velocity. Maxwell’s equations describe the interplay between these two fields. When an electric field is circular, i.e. the vectors point in such a way to form a circle, a magnetic field will increase in strength perpendicular to that plane. And conversely, a loop of magnetic field created a change a change in the electric field perpendicular to the plane of the loop. The result of this is electromagnetic radiation, electric and magnetic fields oscillating perpendicular to each other and to the direction of propagation (4).

            The electric and magnetic field components are most easily described separately for mathematical purposes. And it would make the mathematical representation even more convenient if the light represented were horizontally polarized. Polarization refers to the direction that a field is oscillating in. For example, vertical polarization describes a field oscillating up and down (4).

            The electric field component of electromagnetic waves can be mathematically modeled by a cosign function, a variable t for time, a variable a for amplitude, a variable p for phase shift (or where the function is at time t), and a variable f for frequency. That would look like this: a(cos(360ft+p)) (4).

            Every valid wave in a vacuum solves Maxwell’s equations. They are linear equations comprised of combinations of derivatives that mathematically modify the electric and magnetic fields so that they equal zero. Every valid wave in a vacuum gives zero when entered into Maxwell’s equations. Therefore  a valid wave, 0, plus another valid wave, 0, gives yet another valid wave, 0! The third valid wave in physics is called a superposition, or sum, of the original two waves. The superposition of two waves depends for its characteristics on the amplitude and phase shift of the original two vectors. If the original two waves have different phase shift, instead of oscillating up and down or left side to right side, the superposition will oscillate in an ellipse. If the two original vectors have the same amplitude and are ninety degrees out of phase with each other, the superposition will oscillate circularly, in what is known as circular polarization (4).

            Every wave can be described  as a superposition of two oscillating vectors, one on the vertical axis, and the other on the horizontal axis. Although, all waves could also be described with respect to perpendicular diagonal axes. The attitude of the perpendicular axes you choose is known as your choice of basis. Depending on the application, it might be more convenient to choose one basis over another (4).

            What has been presented above is the classical understanding. Most of it translates directly in the quantum world. Classically, the energy of a wave is considered to be the square of the amplitude. Theoretically, this should result in an infinite number of possible energy levels for waves. This seems intuitive, but physicists now know that the energy of a wave is always a discrete multiple of the smallest possible unit of energy. Imagine a staircase. Each step represents an energy level. Every wave is on a specific step and nowhere in between. The height of each step represents the smallest possible increase of decrease in the energy of a wave. This smallest amount is known as Planck’s constant, or h. Every wave has an energy equal to an integer multiple of h times its frequency. This means that there is minimum energy level that a photon can have and if it somehow loses energy at that level, it ceases to exist (4).

            Energy, then, comes in discrete packets of different sizes, but there is a minimum size and all other sizes are a multiple of this minimum one. Now, different frequencies of light only form when the the right size packet is available. When one arrives, they zoom off with it and a light ray is formed. The higher the energy of the packet, the higher the energy of the wave that will take it away. That’s why yhe hotter a fire is, the brighter it is and why its color shifts towards violet as the temperature increases. In normal ambient conditions, only little packets are available, so humans don’t get blinded or cooked to death! These little packets are called quanta, hence quantum mechanics (5).

            An electromagnetic wave at the minimum possible level is known as a photon. The reason photons in themselves can have different energies is because of the third variable in the equation for the energy of an electromagnetic wave: frequency. A different photon can exist at any possible frequency (4).

            In quantum mechanics, the superposition of two perpendicular oscillating vectors to describe any electromagnetic wave must have a new definition. This is because in classical understanding a photon would be a superposition of two vectors with a fractional modulus, and the quantum understanding knows this to be impossible because photons carry the minimum possible energy at their frequency. Classically, the squares of the moduli of the 2 vector components of each wave, tell what percentage of that waves energy can be found in a given direction. However in quantum understanding, a photon must have all of its energy in one direction because its energy cannot be subdivided. So, the amplitudes of the component vectors give the probability that the photon can be found in a certain direction. If the probability is fifty percent for a given direction, half of the time a photon of a certain frequency will be in that direction, and half of the time it will not (4).

            Now the reader is prepared to delve into Bell’s theorem. Proof of Bell’s Theorem involves the use of what are called polarizing filters. A polarizing filter either blocks light from passing through it, or polarizes it in one direction determined by its attitude. What follows is a description of an easy demonstration of Bell’s Theorem and not the actual experiment which is quite complex (5).

            Imagine one vertically polarizing filter. All light photons oscillating in the vertical direction will be let through one hundred percent of the time. Photons oscillating at a forty-five degree angle from vertical will only pass through fifty percent of the time. Now imagine a second philter is placed on top of the first. As the second filter is rotated away from vertical towards ninety degrees away from vertical, less and less light is let through until at ninety degrees away from vertical, light passes through both filters zero percent of the time, provided the filters are perfect. This is because all light that passes through the first filter is vertically polarized, meaning that it has zero percent chance to pass through a horizontally polarized filter (5).

            Now imagine that the second filter is angled at ninety degrees, vertical being zero degrees. Add a third filter in between at forty-five degrees, and more light passes through than before! Twenty five percent of the light passing through the first filter to be exact. This is because the filter in the middle angled at forty-five degrees lets fifty percent of the vertically polarized light through, and that fifty percent becomes polarized at forty-five degrees. Fifty percent of the light polarized at forty-five degrees passes through the third filter at ninety degrees. This seems natural and intuitive (3).

            Many people  have speculated that quantum mechanics isn’t intrinsically probabilistic as shown by how photons pass through a polarizing filter, but that there are some “hidden variables” that man has yet to grasp that describe a fundamental state photons that actually determines whether a photon will pass through a filter or not, not probability (3).

            Bell’s Theorem rests on what happens when a filter at 22.5 degrees, B, is placed on top of a filter at zero degrees, A, and is below a filter on top of the other two at forty-five degrees, C. Based on previous demonstrations, it would be reasonable to expect that seventy-five percent of the vertically polarized light would pass through B and C, because without B, fifty percent of the vertically polarized light passes through, and 22.5 degrees falls halfway between zero and forty-five degrees. Actually, only fifteen percent get blocked at B and another fifteen percent at C (3)!

            To disprove hidden variable theory, first it is necessary to assume it is true. Imagine 100 photons that do have a mysterious hidden variable that answers these following questions. Would a photon pass through A? Would a photon pass through B? Would a photon pass through C? Assume all photons start out being vertically polarized and therefore all pass through A. Fifteen percent get blocked at B, so eighty-five make it through. Another small about, about fifteen percent of eighty-five, get blocked at C. That is much less than the 50 that would get blocked if B weren’t in the middle. So experiments contradict hidden variable theory (3).

            Except, there’s a loophole: if passing through one filter affects how a photon will interact with future filters, then the phenomenon is easily explainable (3).

            But there is a way to circumvent that loophole. It is called the Einstein Podolsky Rosen, or EPR, experiment that was published May 15, 1935 in Physical Review. It uses entangled pairs of photons to measure the probabilities of photons passing through different combinations of filters A, B, and C at the same point in space. Basically, it proves that it is impossible for reality to be locally real (6).

            However, experiments up until 2015 couldn’t prove this unequivocally due to flaws in the equipment used and the experimental setup. But in 2015, this result became unequivocal with a loophole free experiment (3).

References

  1. https://www.youtube.com/watch?v=i1TVZIBj7UA
  2. https://plato.stanford.edu/entries/qt-epr/
  3. https://www.youtube.com/watch?v=MzRCDLre1b4
  4. https://www.youtube.com/watch?v=zcqZHYo7ONs
  5. https://plato.stanford.edu/entries/scientific-realism/