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The Standard Model of Particle Physics

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In a previous article1, I've used a conjunction of points to form implications between those points, and these implications were represented by the transition amplitudes of quantum mechanics. And in the next article2, I used transition amplitudes to describe virtual particles, and these virtual particles were used to represent spacetime itself. These virtual particles come in pairs consisting of a particle and an antiparticle. They appear together at one point in space, they travel independently for some distance and meet up at a different point at the same time.  For a virtual pair, the particle has a phase angle that is 180° out of phase with the antiparticle so that when they meet up again, they add in superposition and cancel each other out leaving no observable effects as is noticed in empty space. Mass, energy, and motion were described as a real particle stepping through a series of cancelations with the antiparticles of virtual pairs.

The particle/antiparticle cancelation is possible because the conjunction of points, pqr..., that allows construction of the implication (p q) also allows construction of the reverse implication (q p). And the transition amplitude for (p q) is the complex conjugate of the transition amplitude for (q p). Then it was shown in article2, that for discrete values of time, these complex conjugates could be made 180° out of phase so that they cancel. And for all the phenomena so far described these particle/antiparticles behave as if they were electrons and positrons. For the positron is described as an electron moving backwards in time, and a negative value of time in the exponent of a transition amplitude gives a complex conjugate of the positive value of time.  (I use italic words to label my interpretation of my formulism and un-italic words to refer to the real thing.)

So far it seems I've described the necessity of what appears to be electrons and positrons. But there are quite a few more particles that have yet to be recognized in this formulation. The Standard Model of particle physics describes about 17 fundamental particles. There are bosons and fermions, quarks and leptons; the fermions have 3 generations of mass. There is the Higgs boson, and the charged particles have their antiparticle. The table to the left below shows how these are arranged. The diagram to the right shows how they interact with each other. These particles make up all the atoms and molecules we see. And the trick will be to find them hiding in the logic of this formulation. The attempt here will be to show that the combinatorics between the Standard Model and this formulism seems to match for at least the first generation particles.

File:Standard Model of Elementary Particles.svg     


Previously, I suggested here that first quantization gave implications between points and was responsible for the U(1) symmetry of the electromagnetic force. And then this was iterated to give implications of implications, and this was responsible for the SU(2) symmetry of the Weak nuclear force. And this was iterated again to give implications of implications of implications, and this was responsible for the SU(3) symmetry of the Strong nuclear force. This all seems a bit abstract, and I'd like to flesh this out a little more. The question is, can we recognize the particles of the Weak force in the second iteration to implications of implication? Can we recognize the particles of the Strong force in the third iteration to implications of implications of implications?

In the first quantization, the conjunction, pq, gave us the implication, (p q), which I call an electron, and the same conjunction also gave us the reverse implication, (q p), which I call a positron for brevity. When iterated, the conjunction of any two implications, (r s)(t u), gave us (r s)(t u) and (t u)(r s), and we might wonder if somehow these correspond to particles of the Weak force. But the implication, (r s), could represent an electron or a positron, depending on whether it travels forward or backward through time. And (t u) could represent an electron or positron. So there are four possibilities: an electron electron, or an electron positron, or a positron electron, or a positron positron. 

The implication between points gave us what appears to be electric charge since it gave us particle and antiparticle. So now let us suppose that the implication between electrons and positrons give us the weak isospin charge of the Weak interaction. It might be said that this weak isospin charge is holding the electrons and positron together in the particles of the Weak interaction. Then notice these Weak force particles have both electric and weak isospin charge, which is true of the Standard Model W+, W-, Z0 bosons of the Weak interaction.

We have four possibilities of combining electrons and positrons in the Weak force particles described above. But two of them result in the same particle. The Weak force particle represented by electron positron, has a net electric charge of zero since the electron has an opposite charge from the positron. But the positron electron particle also has a net charge of zero. And by symmetry I assume both of these would have the same mass. It would appear that this is the Z0 bosons of the Weak interaction and is its own antiparticle. When an electron and positron get close enough there are these two ways of connecting them with the Weak force. There is no preference to which way they are connected with an arrow (implication) since both ways result in the same mass and charge. So the two possibilities would exist in superposition.

Obviously, the electron electron particle has a net negative charge, so it seems to represent the W - particle. The positron positron particle has a net positive charge and seems to represent the W+ particle. And again by symmetry I assume they have the same mass. But the actual Standard Model W+ and W- particles have an electric charge of +1 and -1, respectively. So how can the W+ and W- particles I describe have a charge of +/-1 when they each have 2 of the same charge?

One possibility is that each of the electrons in the W- particle is not allowed to engage in virtual particle trading with virtual pairs that might take it far away. If these electrons could arbitrarily shift position to anywhere, then the Weak isospin force between them would be meaningless because they could no longer be identified as a particle since their constituents would fly away. So if they are prevented from interacting with far away virtual pairs, then their effect on nearby electric charges is diminished. Or perhaps there is something to accounts for their charge and mass in the math of superposition, since there are two ways to connect two positrons or to connect two electrons.

The photon also needs to be explained in this view. The photon has two properties that suggest that it is a composite of and an electron and a positron. First, the photon has a spin of 1, whereas the electron and positron each have a spin of 1/2. If the photon were a composite of an electron and a positron, then their spins would add to 1, as we observe of the photon. And second, the photon has a charge of zero, which could be the equivalent of the negative charge of the electron plus the positive charge of the positron. The only property contradicting this composite is that both the electron and the positron each have a mass of equal value, but the photon does not have mass. Usually a composite particle adds up the masses of its constituents. But the photon might be explained by the way the electron/positron propagate.

Normally, positrons and electrons so close to each other that they can be considered a single particle will very quickly approach each other and cancel out. But as explained here, if they are travelling at the speed of light, then time for them slows to a halt so that they will never meet. And if they are travelling at the speed of light, which is the fastest on average that particle trading can occur, then there will be no particle trading in arbitrary directions especially in reverse. Then since mass was defined in terms of particle trading in arbitrary directions, if they travel at the speed of light, then there is no arbitrary virtual particle trading, so they have no mass.

And ...

Next, if we iterate the formulism again, we get the conjunction, [(r s)(t u)] [(t u)(r s)], and this implies that [(r s)(t u)] [(t u)(r s)] and [(t u)(r s)] [(r s)(t u)]. And we might wonder if these new implications correspond to the particles of the Strong force. Previously, (r s) represented electrons and positrons, and (r s)(t u) represented the W+, W-, Z0 bosons. And now we have a new implication between these bosons. This give us new implications listed below:

  u d u d
  W+ W+  =  (p p) (p p ,  +4 u    
  W+ W-  =  (p p) (e e) ,       0      
  W+ Z0  =  (p p) (e p)  ,  +2 d    
     =  (p p) (p e)  ,  +2 d    
  W- W+  =  (e e) (p p)  ,       0      
W- W-  =  (e e) (e e) ,  - 4 u
  W- Z0  =  (e e) (e p)  ,  - 2 d    
     =  (e e) (p e)  ,  - 2 d    
  Z0 W+  = (e p) (p p) ,  +2 d    
     = (p e) (p p)  ,  +2 d    
  Z0 W-  = (e p) (e e)  ,  - 2 d    
     = (p e) (e e)  ,  - 2 d    
  Z0 Z0  = (e p) (e p) ,       0      
     = (e p) (p e)  ,       0      
     = (p e) (e p)  ,       0      
     = (p e) (p e)  ,       0      


Where e stands for electrons and p for positrons, and each e in a line is different from another e in the same line, same with the p.

The first implication between points gave us electrically charged particles, e and p. The second implication between e and p gave us the Weak isospin charged particles, W+, W-, and Z0. And the third implication between the W+, W-, and Z0 is expected to correspond to the Strong color charged particles, called quarks and labeled here as q. If they do, then these quarks will participate in the Strong force interactions, the Weak force interactions, and the electromagnetic interaction as is observed of the Standard Model quarks with the forces thereof.

The table above shows the preliminary charge of these particles. The math involved with their superpositions might change the actual value of the charge. And the u, d, u, d, designation in the table is a preliminary attempt to correlate these with the up or down flavor based on relative charge. I was hoping that there would be three different particles that would result in the same charge; then I could label them as different color charge. But there is only one u, one u, four d, and four d. But perhaps there is some way of getting three of each using superposition. Or perhaps the color charge is seen only in the way the quarks interact with each other. I'll add to this if I get any ideas. Use the Comment button below if you have any thoughts on the subject. Thanks.