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.

Matter wave

In this post, I will explain how to built a 4-D torus that would tell why a matter wave wavelength depends on its speed (De Broglie wavelength).

1) Take a slinky. Make a mark along the slinky. I cut out an inch of tape paper and attached it to each spiral.  Then, I marked one side of the tape paper, this represent "arrows".
2) Make a clockwise turn around the slinky edge and glue the tail arrow with the head arrow. This makes a torus. The twisted path of the arrows is very noticeable.
3) Take another slinky. Do the same as before but a counter-clockwise turn.

Heuristically, these slinkies have to be together. It is too difficult to see anything when they are together, so I draw them separately. You have to arrange them in the way shown in Figure 1. The clockwise-turned slinky is at the right of the Figure. The movement of its arrows is clockwise. The counterclockwise-turned slinky is at the left of the Figure. The movement of its arrows is counterclockwise. This makes that the wavelength left on flatland depends on the object speed.

Figure 1 the two slinkies are arranged as shown. The clockwise turned is at the right of the Figure. This arrangement allows to have a wave that just depends on the speed of the object. This occurs because of the clockwise and counterclockwise movements.

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

Figure 2 electric field vectors imprinted in the plane. The wavelength will depend on the speed of the object and will use the two dimensions of the plane to travel.

On Fig 2 undistinguishables states occur at 0, 180 and 360 degrees. Notice that the printed arrows close a circle when the particle arrive to 180 degrees. Thus, an observer in the plane may judge that one complete round has been made. However, the intersection at 180 degrees is different to the original state at 0 degrees. After another round, the printed pattern is exactly like the state at the origin. This is a characteristic of a spin 1/2 particle.

Particle-Wave duality

After I read Mikio Kaku's "Hyperspace", I felt that the quantum weirdness was about the lack of room to explain it. I will explain the particle-wave duality based on this principle -"Lack of Room"

In order to increase the room, we need more dimensions. Lets start with four. To do that, our familiar three-dimensional word has to be flattening out. This produces a plane, which now contains our familiar three-dimensions. This plane exists at present time, all the time. This means that the future is above and the past is under it.

This new time dimension is like the other three ones. But observers in the plane cannot detect anything in it. However, a significant portion of the physical object is there. A physical particle symmetrically exists in this wider room, half of it in the past and the other half in the future. The only part of the particle that an observer can detect is the intersection at the present time. This exercise shows you the need to understanding the intersection of a higher dimensional body, into a lower dimensional space.

To do that, I begun to use the tools suggested in "Hyperspace" and Thomas Banchoff's "Beyond the Third Dimension: Geometry, Computer Graphic and Higher Dimensions" to imagine a four dimensional 4-D object. One of the tools is slicing. You take a 3-D solid and make it travel through a 2-D surface (a plane). This will produce slices of the object in the plane. Then, you make a revolution of every slice. This revolutions will make back a 3-D object. The shape of these 3-D objects will depend on where the slice occurred and how the revolution was performed. The point is that those 3-D objects are the intersections of the 4-D object by a 3-D plane. Thus, the sequence of the intersection of a hypersphere ( 4-D sphere) by a 3-D space will be a 3-D sphere which diameter increases, it reached a maximum and then decreases. This occurs until the sphere disappeared. Figure 1.

Figure 1 the 3-D sphere is sliced by the 2-D plane. This produces a circle that increases in diameter, reaches a maximum and then, decreases until it disappeared from the 2-D plane. This same exercise with a 4-D sphere (hypersphere) and a 3-D plane (normal 3-D space) will produce a 3-D sphere following the same pattern as the circle in the plane.


Now imaging a donut. That donut is only intersected in the middle. Haft of the donut is in the past and the other half is in the future. You can only see the present. If you follow this, you will get just three shapes: 1) when the donut is cut through its equator. Regardless the orientation of the revolution, you will get two concentrical spheres. 2) when the donut is cut perpendicular to its equator. You will get two separated circles. Here you can do two different revolutions of the two circles. One through a symmetrical edge that join them, which will give two separated spheres. Another through an edge perpendicular to the previous one, which will render another donut. You can see this last intersection in http://www.bekkoame.ne.jp/~ishmnn/java/hypertorus.html, see Figure 2.

Figure 2 the intersection of the donut through its equator will give two concentric circles. The revolution of these circles will give two concentric 3-D spheres (bottom of the Figure). The intersection of the donut perpendicular to its equator will give two circles. The revolution performed as depicted in the Figure gives two 3-D spheres or a torus. The structures at the far right of the figure are what have been found for the deuteron (JL Forest et al., 1996).

Now, the reader may ask - Why am I going through all this? Who cares about these higher dimensions? Fair enough. We need evidence indicating that this would be relevant. After a while, I found that the structures just described occur in the atom nucleus and they are shown at the far right of Figure 2 (J.L. Forest et al. 1996 "Femtometer Toroidal Structures in Nuclei") and in the chemical bond ( "Laplacian of the electron density of the urea molecule"), see Figure 3.

Figure 3 laplacian of the electron density for urea molecule. Core attractors are observed in all the atoms; oxygen, carbon and nitrogen. It is not observed for hydrogen. The hydrogen bond shows a two- separated sphere structure. The lone pair on the nitrogen consists of two toruses, part of these two toruses are fused on the side of the nitrogen opposed to the carbon atom.

How does this have anything to do with the particle-wave paradox? I think it is obvious that we are trapped in a 3-D plane, trying to observe a 4-D torus, which is impossible. The evidence suggests that the deuteron and the electron are 4-D toruses. This is the particle!

To have more room opens the possibility of a 4-D object (the particle) to leaving a 3-D wave when it moves through the 3-D plane! This 3-D wave is described by the wave equation.

Furthermore, now we can conceive a non-moving particle, i.e. when the wave function collapses.

Is the paradox resolved?

In the next Post, I will describe the 3-D wave.