Spin in Flatland!

What is Spin?

When you find something new and unexpected, it is natural to associate that with some other phenomenon that you are more familiar with. This is the case according to the invention of Spin, which was the interpretation of the Stern-Gerlach experiment's results.

The Stern–Gerlach experiment consists of sending a beam of particles through an inhomogeneous magnetic field and observing their deflection. The results show that particles possess an "intrinsic angular momentum" this is most closely analogous to the angular momentum of a classically spinning object, "but that takes only certain quantized values" (see http://en.wikipedia.org/wiki/Stern-Gerlach_experiment).

For example, tiny current loops will have an associated magnetic moment. Thus, they could be thought of as tiny magnets. If you send these tiny magnets through an inhomogeneous magnetic field, they will land into a detector screen according to its initial orientation. Thus, each tiny current loop would be deflected by a different amount, producing some density distribution on the detector screen. This is not what is observed in the Stern-Gerlach experiment, which means that the "Spin" phenomenon detected has nothing to do with current loops, magnets and/or spinning objects. It is something else.....

Lets say that these particles are as shown in the post "Matter Wave". The particle is a toroid made with a current. Always, this  toroid intersects at present time in a symmetrical way. Half of the toroid is in the past and the other half in the future. This current produces an internal magnetic field. If the toroid flips 180º, its internal magnetic field is inverted. As a consequence, the particle can switch between these two structures. This would be the origin of the 50:50 percent probability for either structure to show up, see Figure 1.

Figure 1 Present time flat particle intersection and its 180º flip structure. Notice that, after this transformation, the internal magnetic field gets inverted. This is the origin of the 50:50 percent probability to get either structure.

I took disk magnets and arranged them as suggested in Figure 1. I pass them through an external magnetic field. The result of the experiment is shown in Figure 2. Please do this experiment, it is very easy!

Figure 2 Flatland version of the Stern-Gerlach experiment.

I think this is the flatland equivalent of the space-land Stern-Gerlach experiment. Given that the particle can switch between these two structures, there is a 50:50 percent probability to have either configuration.

Thus, the 50:50 percent probability and the "superimposition of states" are needed to account for the two different outcomes of this experiment. This is fully consistent with the predictions of Quantum Mechanics.

This may indicate a way to understand the Stern-Gerlach experiment without a classic analogy.

Self-interference

Now that we have constructed the Matter and Light waves, I would explain a phenomenon where they behave in the same way....i.e. Self-interference. If you send individual photons to a double slit, you will get an interference pattern. An interference phenomenon entitles destructive and constructive interference. When the valley of one wave coincides with the crest of the other (destruction) and when two crests coincide (construction).

The latest self interference experiment makes a weak-measurement of a single photon several times. The average trajectory of the photon can be detected. Please read "Observing the Average Trajectories of Single Photons in a Two Slits Interferometer"(http://materias.df.uba.ar/labo5Aa2012c2/files/2012/10/Weak-measurement.pdf) Figure 1 shows the average trajectories of Single photons. You can see that in three ocations, two trajectories merge from the slits and get together toward the center of the figure. This produces the central maximum in the pattern. The rest of the trajectories diverge towards other maxima, but they don't merge. This means that there is no interference as described in the previous paragraph.

                                           
                                              Figure 1 Average trajectories of a single photon

These results are very similar to Ashfar's experiment (http://en.wikipedia.org/wiki/Afshar_experiment). He puts a grit of wires at the dark fringes, which in Figure 1 are the spots with lower density of trajectories. Ashfar did not find a significant reduction in the interference pattern with or without the grit! This means that there are regions that the photon avoids, i.e. there is no destructive interference.

Back to Figure 1, at the slit position 0 mm, two trajectories merge toward the center of the figure. This occurs several times, at the distances 3500, 5500 and 6000 mm. At each side of the central trajectories, there are no trajectories. These are the places where the dark fringes occur. The rest of the trajectories diverge from the center, move somewhat sinusoidaly and aim toward other maxima.

Every single photon was divided in two at the slits. Thus, every two trajectories belong to a single photon. If there would be constructive and destructive interference, every two trajectories should merge at the detector. This is not happening. Besides the central trajectories, the divided photon continues being divided as it travels toward the detector.

Also, the trajectories deviate from a straight line and correct its optical paths. It looks that the photon is trying to reach a maximum in the interference pattern. Hence, the probability to reach a minima is very low.

Every single photon reaching the detector, is still in two places at the same time. Upon arrival to detection, the photon still has the chance to collapse in one of the two spots. This last process will be stochastic and the photon will leave a mark at either place of landing.

Given that these maxima and minima occur by following an interference law, these results support a model where the components to produce maxima and minima are internal to the particle.  Thus, no merge of the trajectories between slits are required.

Light wave

In this Post, I will explain How to make a photon in flatland! It follows the same instructions as in the previous post, for constructing a matter wave. The difference is that now, there are two turns in each slinky.

A complete different structure shows up! the result is observed in Figure 1. Heuristically, these slinkies also have to be together. It is difficult to see anything when they are together, so I draw them separately. You have to arrange them the way they are shown in Figure 1. Both, the clockwise and the counterclockwise-turned slinky's arrows move clockwise. This is due to the two turns. A consequence of this structure is that the wavelength left on flatland does not depends on the object speed. It is inferred that this object speed is constant and it is the speed of light. It will depend on the intersection speed. Low intersection speed would be a redish photon, whereas, a high intersection speed would be a bluish photon.

Figure 1 The flat photon, the two slinkies are arranged as shown. Both slinkies move clockwise. This arrangement creates a wave that does not depends on the speed of the object.

These arrows are electric field vectors that get printed in the plane. The intersection will look like this:

Figure 2 electric field vectors imprinted in the plane. The wavelength will not depend on the speed of the object and use just one dimension of the two available.

On Fig 2 undistinguishable states occur at 90 and 270. Notice that the printed arrows close a circle when the particle arrives to 360 degrees. Thus, an observer in the plane may judge that one complete round has been made. This is a characteristic of a spin 1 particle.

What is Mass?

In the previous Post, it was shown that the matter wave use all the dimensions of the plane. Now, you can see that the light wave uses just one dimension of the two available in the plane. Since, the photon necesarily has no mass, it looks as requirement that: to have mass the particle intersection needs to use all the dimensions of the intersecting space.

Coming back to spaceland (3-D space), You can see that the deuteron has different toroidal shapes, all of them uses the three dimensions available. The same thing occurs with atoms as observed with scaning tunneling microscopes, You see spheres which uses all the three dimensions. Whereas, the electromagnetic wave uses just two dimensions from the three available. You just see a sinusoidal plane of electric field vectors with another sinusoidal plane of corresponding magnetic vectors at 90 degrees.