## Electrons and Quarks.

These ideas about electrons and quarks are probably completely wrong, but I still find them amusing. I originally wrote this as just an example of a 'crackpot theory'. One motivation for this 'theory' was an idea I read that the entire universe could be made from a single electron (Wheeler 1940.). This one particle starts at some point in time, travels forward to another point in time, then travels backwards through time again, and repeats this sufficient times to account for all the observed electrons in the universe. An electron travelling backwards in time is identical to a positron travelling forwards in time, so all the positive charges are also accounted for. The positive charges in the observed universe are mostly protons rather than positrons (this was Feynman's objection to Wheeler's idea), and protons are believed to be made from quarks (plus a few other things such as gluons), so explaining these particles becomes a problem (Wheeler suggested that the positrons were in some way hidden inside the protons), and that is the purpose of the following ideas. The 'physics' in this article is over simplistic, and should not be taken too literally.

The model of a proton used here is similar to the idea that the proton is made from three quarks, but the currently accepted version arising from quantum field theory is somewhat more complex, involving a proton structure which includes many virtual quark-antiquark pairs and gluons in addition to those three quarks needed to give the proton its charge and spin. Gluons, in common with a few other predicted particles, have not been directly observed, so may turn out to be less essential to an 'alternative theory', but even so there is fairly conclusive experimental evidence for their existence. so I will try to return to this problem later.

One of many unanswered questions in physics is 'why is space 3-dimensional?'. To look for clues we could make a list of laws or principles which will only work in 3 dimensions. One example is a simple property of vectors representing a velocity or flow. In two dimensions, if we chose a reference frame in which two vectors are perpendicular, then we can find an infinite number of other reference frames moving at different velocities relative to the first in which the two vectors are also perpendicular. For N arbitrary vectors in a space of N dimensions, then for any value of N greater than 3 there is not in general any reference frame in which these vectors are all perpendicular to each other, although we could of course choose a special set of N vectors which just happened to be perpendicular. It is only in three dimensions that we find something unique. For any three distinct velocity or flow vectors it is possible to find exactly two reference frames in which they are all perpendicular, and the two results are a mirror image of each other.

Suppose a particle in its rest frame can be represented in some way as a set of such perpendicular vectors, or more in line with one version of quantum mechanics as a superposition of many such sets, then it follows that it is only in three dimensions that the particle will certainly have a unique rest frame. But what about the second, mirror image set of vectors? Apart from the mirror image, we could also add a time reversal to our vectors and still leave them perpendicular. It is conventional to define operators T and P (for 'parity') to represent time reversal and mirror reflection, and the result of combining the two is believed to be given by PT = C, where C is the 'charge conjugation operator', or more simply represents electric charge reversal. The result of this operation on an electron is to change it into its anti-particle, the positron. i.e. a mirror reflection plus a time reversal changes an electron into a positron. Our second set of vectors then represents a positron, in its own separate rest frame.

There is another single reference frame in which something unique happens. There is one frame in which both sets of vectors add up to zero. For one set in this frame there is cancellation of the three vectors if they are added as if they were forces acting on a body, but as flow vectors they will not cancel in the same way. Although the average flow is zero in all directions there is still a nett outward flow from the origin of the vectors. Only the combination of the two mirror image sets, with a time reversal applied to one of them will give a total cancellation. So, we have a frame in which 'nothing exists' and an electron and positron each at rest in their own frame, and each an exact time reversed mirror image of the other.

Onwards to another property of electrons, the spin. What is significant is the value of the spin, which is half a unit. All known particles have spin zero or some multiple of half a unit, and the importance of this is that they have different behaviour depending on whether they have an odd or even number of half units of spin. Particles such as the photon, with spin 1 are known as 'bosons' while the electron, quarks and neutrinos with spin 1/2 are known as 'fermions'. An example of the difference in behaviour is that if one photon is in a given state then this increases the probability of another photon entering the same state. This phenomenon is responsible for the laser, in which many photons are in the same state, giving 'coherent light'. Electrons on the other hand are prohibited from having two in the same state, and this is why in an atom all the electrons do not simply fall to the lowest energy level, but form 'shells', some at higher energy levels, giving the familiar variety of chemical properties depending on which levels are full, or have vacancies.

It can be shown that the fundamental difference between bosons and fermions is that rotating a boson with spin 1 through 360

^{o}returns it to its original state, while a fermion with spin 1/2 must be rotated through 720^{o}to get back to its original state.

(A spin 2 particle needs only a 180 degree rotation to return to its original condition, and a spin 0 particle remains unchanged by any rotation. There may also be fundamental particles with spin 3/2, but these are predicted by 'supersymmetry' which may or may not be valid, and these particles may be unobservable in practice even if they do exist. The same could be said for the proposed spin-2 graviton, which also may be impossible to observe directly. The recently discovered Higgs particle is expected to have spin 0. Composite particles could have practically any number of half units of spin, for example the delta particle has 3/2)

We are familiar with the idea that if we turn round through 360^{o}then we are back where we started, but there are simple examples of situations where two such rotations are needed. Consider a solid object, such as a pencil, and tie two strings to its ends, and tie the other ends of the strings to another pencil, as Fig.1a.With just the two strings connected spin one of the pencils through 360

^{o}so that the strings become twisted. Without rotating either pencil further it is possible to untangle the strings by looping one back over one of the pencils.

Next add a third string between the centres of the pencils as in Fig.1b and try the same thing again. It can't be done! The strings cannot be untangled without rotating the pencils or disconnecting the strings. What may seem surprising is that if the rotated pencil is turned another 360^{o}in the same direction, making a total of 720^{o}then although it seems that the strings have become even more badly tangled, it turns out that by various operations of looping strings back over the pencils it is actually possible to get back to the untangled original state.There are other simple situations in which a 720

^{o}rotation is needed to return to the start, for example supporting a glass of water on the palm of your hand and by twisting your arm and wrist rotate the glass without spilling the water. To return to the starting point needs the glass to rotate through 720^{o}. For obvious reasons the three string example is of some interest. The three perpendicular vectors referred to earlier could lead to this sort of result if they started at one particle, and ended at another. This ties in with the time reversed version representing a positron. The vectors need to flow out of the electron, then flow into the positron. So, we have something explaining the spin related behaviour of the electron, and an explanation for the existence of anti-particles. We could add the possibility that a photon has only two of these vectors, so that it is a boson, and furthermore it has no rest frame.This sounds promising, but we have yet to find any physical interpretation of these 'flow vectors'. It is tempting to try to relate them to the electric field. Obviously (?) the field of an electron is not just a set of three vectors, but as I suggested earlier it would fit in better with quantum theory to suppose that there is a superposition of many such sets. It seems likely that an electron should interact with the electric field of every other charged particle in the universe, a total of something like 10

^{80}particles, so there should be 10^{80}sets of vectors, which, depending on ones favourite interpretation of quantum theory, either exist in parallel universes, or exist only when an observation forces them to do so, or whatever.....Another property of electrons is their electric charge. This is not, however, believed to be the smallest unit of charge. The quarks have charge of 1/3 or 2/3 times the electron charge, so we could say that the electron has three times the smallest known unit of charge. Once more we have the number 3. But if the number of vectors referred to earlier is related to the electric charge, then this suggests that quarks have only one or two of these vectors, which then fails to account for them being fermions like the electron. A way round this problem will be mentioned later.

The quarks have a different sort of charge known as 'colour charge', which comes in three varieties, conventionally referred to as red, blue and green, and interact primarily via the 'strong force'. There are four known forces occurring in nature. The weakest is gravity. The next strongest is the 'weak force' which is a short range force involved in such phenomena as beta decay, and is unique in that it does not conserve parity, i.e. it is not symmetric with respect to a mirror reflection. Next is the electromagnetic interaction, and finally the strong colour force.

It is interesting that each force can be matched to an operation on space-time. Gravity is related via General Relativity to the geometry of space-time. The weak force is related to mirror images, i.e. x <=> -x or y <=> -y or z <=> -z or all three together. I have used <=> to denote an exchange of coordinates. There is a link between electromagnetism and time reversal, i.e. t <=> -t , and with with mirror reflection, which together reverse electric charge. So what is left for the colour charge to be related to? There is only one simple and obvious possibility left, and sure enough there are three different but similar forms. These are x <=> t , y <=> t , and z <=> t .

But how can we interpret these exchanges of coordinates? One obvious choice is to say that they transfer a particle through the 'light barrier' to become 'tachyons' or faster than light particles. This is not especially appealing, to try to claim that quarks are electrons travelling faster than light, but this would not be directly observable because quarks are confined within composite particles such as the proton. The three coordinate exchanges carried out in sequence would convert an electron to an 'up' quark with charge -2/3, ( or more precisely an anti-quark, because the up quark has positive charge ), then to a 'down' quark with charge -1/3, and then to a particle with zero charge. I originally guessed that this zero charge particle is a neutrino, but later a different possibility is suggested, that some composite particles contain such a particle internally.

There may be some slight mathematical justification for the 'faster than light' quarks.

Normally we represent the square root of minus one as the letter i. Then there are two square roots, plus and minus i, but there are other alternatives. One of these rather neatly fits the faster than light theory of quarks. This is an example of 'non-commutating' maths. What this means is that a x b does not equal b x a. In this case in fact (a x b) = - (b x a). This is called 'anti-commutating'.The relevant equation which describes the square roots of minus one was discovered by William Hamilton, an Irish mathematician, while trying to solve the problem of how to describe mathematically a rotation of a solid object in three dimensional space.

A rotation could be described by a vector pointing along the axis of the rotation, and perhaps with a magnitude proportional to the angle of rotation. Although this contains all the information necessary to completely define the rotation we run into trouble when we want to add together two rotations round different axes. Not only does vector addition give the wrong answer, but the correct answer is different when the rotations are carried out in reverse order. This requires non-commutating maths. The following equation is used in the description of rotations, and gives a set of three square roots of minus one:

I

^{2}= J^{2}= K^{2}= IJK = -1.This is the equation discovered by Hamilton.

There is another useful representation which helps in some practical calculations, and this involves the replacement of conventional numbers by 2x2 matrices. Instead of the square root of minus one we must now look for the square roots of minus the unit matrix:

-1 0 0 -1 The three square roots are: 0 -i 0 -1 -i 0 -i 0 1 0 0 iThis representation is particularly useful for working out products such as IJ or JK, and what we find is that such products do not lead to any new numbers. IJ for example is equal to K. The only numbers we need to worry about then are the real numbers, the conventional square root of minus one, i, and our three new numbers I, J and K.

The application which interests me is to the square root in the Lorentz transformation equation for the space coordinates x, y and z. For x the equation is:

x' = ( x - vt ) / sqrt ( 1 - v^{2}/ c^{2}).

The important feature of this is that for any value of v greater than c the square root is of a negative number. So how are we to interpret a space coordinate multiplied or divided by the square root of a negative number?The possibility which I am suggesting is that the square root is different for the three directions x, y and z, and it is the three roots I, J and K described above which we need to use. A transition through the 'light barrier' in each direction gives a space transformation which could then represent a transition to a new dimension. There are not necessarily just the three new dimensions given by I, J and K, because there are also the multipliers iI, iJ and iK which could represent new dimensions. There are therefore 6 new dimensions, giving a total of 10 when we include the normal 3 space plus one time dimension. The faster than light transformation also changes time t to it, so including this gives 11 dimensions in total. The transition to the dimensions represented by I, J and K occur only unobserved within composite particles such as the proton, so there seems no necessity for these extra dimensions to be observable, or for any relevance outside of particle physics. The physical significance of the dimensions represented by iI, iJ and iK is not clear. These are actually three square roots of +1, and are also, in their matrix representation, the Pauli spin matrices.

One objection is that the three space directions are entirely arbitrary, and we could choose any three perpendicular directions to be x, y and z. This is not a problem because the three square roots, and the matrix representations also have the same arbitrary nature. The three matrices can be thought of as components of a vector, and it can be shown that they transform in the same way.

A problem left unresolved earlier was that if an electron passing through the light barrier was left with only two components of its charge, then why would it, in becoming a quark, still be a fermion according to the mechanism suggested which involves three 'strings' related to the charge. Rather than saying that a component of the field reduces to zero, the present interpretation suggests that it gets rotated into another dimension, so all three 'strings' still exist.

Another point relating to the three string idea is that the example I gave of three strings tied to a pencil has little relevance to an electron. There is no reason to believe that an electron can be represented as an extended solid object to which anything could be tied. To get round this objection one idea is to abandon strings in favour of two dimensional surfaces. The strings are then replaced by tubes, and the requirement for continuity of the surfaces when we have two or three tubes joined together may supply the rigidity needed to produce behaviour analogous to the pencil and string example. The representation of photons as loops could then lead to the idea that the tubes are actually a flow of virtual photons, which is then not so far removed from the standard theory of electrons where interactions take place primarily via the exchange of virtual photons.

## Particle Diagrams.

This section adds little to what has gone before, it is merely a way of visualising elementary particle structure as predicted by the theory. I find it useful myself as a way to think about the problems involved, and one slightly unexpected consequence, a new but unobservable particle, is revealed, which leads on to ideas about the mu and tau mesons. The idea is to represent the 'light barrier' as a circle, and put everything happening above light speed inside the circle, and everything under light speed outside the circle. The basic particles from which everything is constructed are drawn as smaller circles, and part of these circles may be inside the 'light circle'. An example may clarify what I mean:

FIG.1.This is a diagram of the neutral pion, conventionally regarded as a composite particle made from a down quark and a down anti-quark. These quarks have charge 1/3 of an electron charge, so in the present interpretation two of the three strings or tubes are inside the light barrier and one outside. Those outside are of opposite sign, so they cancel to give a neutral particle, and we can simply join them together to give flow out of one and into the other. Note that the particles are labelled u, d etc only to show the connection to the conventional theory.

FIG.2.The positive pion is conventionally an up quark and a down anti-quark, but now we find something different happens when we try to draw our diagram. There is a flow inward along three strings and the only way to terminate these is to add another internal particle existing entirely unobservable above the light barrier, i.e. entirely in the proposed extra dimensions. This diagram also shows that for a charged particle there are three strings leading out into normal space, as for a single electron or positron.

FIG.3.Next is a three quark particle, the proton, possibly the only stable composite particle.

FIG.4.The neutron again needs an extra internal particle to terminate the strings.

FIG.5Finally another variation is the delta minus particle, which needs to have at least two internal particles.

FIG.6The decay of the delta minus into pion and neutron requires 7 particles, so this alternative version fits better. The conversion of energy into particle and anti-particle pairs however means we don't necessarily need to conserve particle numbers, and such creation of up-quark and anti-quark will also match this decay.Although I describe the internal particles as unobservable, this is only while they remain in the other dimensions, which in most cases is not very long, because the particles decay. Only the proton may be stable, and this has no internal particle. The neutron is only stable within a nucleus, and all others decay rapidly.

My first reaction on drawing the neutron diagram was to think this matches the normal theory of neutron decay in which the neutron emits a virtual W minus particle which then decays into an electron and a neutrino. The internal particle can therefore be identified as the W particle I thought, but of course there is one big problem with this, which is that the internal particle is positively charged, so it is not easily interpreted as a negative W particle. A better idea is that the neutron decay just involves moving the light barrier in Fig.4 so that the internal particle becomes a second u quark and the d quark at the top returns to the normal 3 dimensions as a free particle, but with spin 1, so not an electron, it is the negative W, which decays into an electron and a neutrino, each with spin 1/2.

The internal particle of the neutron has spin half in it's own rest frame, but is required to not contribute a spin component to the composite particle, this being perhaps a consequence of existing entirely in the other dimensions. This is also required for the positive pion which has spin zero because the two quarks have opposite spin, and the third internal particle must then have zero spin in the external reference frame. This could be proposed as an explanation for the existence of the neutrino, which is needed to make the total spin correct after decay, effectively carrying away the excess spin from the 'hidden' internal particle.

It is worth noting that the charged pion and the neutron with their internal 'hidden' component decay via the weak interaction. The charged pions with this component have a weak interaction decay, but the neutral pions without the internal component decay via the electromagnetic interaction. Having internal particles does not guarantee a weak decay, it's a bit more complicated than that. For an even number of internal particles the spins can cancel so there is no need for a neutrino and so no weak decay. There are also some particles which we may expect to have a single internal particle such as the delta++ which decays primarily into a proton and positive pion. The pion still has a single internal particle, so here also there is no need for a neutrino. The charged pion will however then decay and release the internal particle in a weak decay into a muon plus neutrino. I am not certain whether the delta can decay directly into proton, muon and neutrino, I can find no reference to such a decay, but don't see why it would be prevented. My only guess is that the observed decay is much faster, so the slower weak decay has an extremely low probability of happening first.

Anyway, there is still the unanswered question of why there are two other families of particles based on the mu and tau mesons rather than the electron. One idea arises from the neutron decay process just described. One way a mu-meson can be created is from the decay of a charged pion. Looking at the proposed structure of the positive pion in Fig.2 we can see that by moving the light barrier in the same way as for the neutron decay we can arrive at the following possibility:

Fig.6This arrangement has another variation in which the light barrier is shifted the other way, so that the top single string is inside the light barrier instead of the two lower strings. Again, by comparison with the neutron decay we would expect a neutrino to be emitted, and this does in fact occur in reality. The charged pion has just this one decay result, i.e. a muon and a neutrino. The other variation of fig.6 could be interpreted as the tau meson. The tau particle has a greater mass than the muon, so the pion decay would not normally have enough energy to produce the tau. How to interpret fig.6 is not clear, and I am not convinced that this is in any way a possible structure for a muon. The top particle looks almost like a free electron, but perhaps in this arrangement it is bound temporarily to the other two particles. In the case of the neutron decay all three strings of the 'free' electron are attached to the proton, but this is just the normal interaction between two free charged particles. With just one string attached and the other two leading out into space something different may be happening, e.g. there may be something other than an inverse square force variation, so that the particles are more tightly bound. The mu and tau mesons decay rapidly, so this temporary state may be sufficient explanation for their existence.If the suggested structure of the mu and tau leptons was correct what can we do about the associated quarks? It helps to look at an example of a particle decay, for example the lambda zero particle. which consists of 3 quarks, up, down and strange. This can decay into a proton and a negative pion, and working backwards we could arrive at the following structure for the lambda:

The diagram shows a plausible added structure to make the strange quark, plus another internal particle needed to link to the down quark. The next diagram shows how this can decay to give the proton and pion structures. The s quark now loses part of the internal structure to becomes a d quark.

There is so far no mention of energy or mass, but there is at least a clue concerning the question of the relative mass of charged and uncharged particles. We could guess that being charged adds more mass because of the energy of the electromagnetic field, but comparing pairs of particles it is found that sometimes the charged one has more mass but sometimes less. For example the charged pion is heavier than the uncharged, but for nucleons the neutron is heavier than the proton. The proposed structures suggest why this could be so. For both examples the heavier particle of the pair has an additional internal particle, and the decay involves the emission of a charged particle. The neutron can emit an electron to become a proton, but the proton can not emit a positron to become a neutron, only the neutron has an extra internal particle and can emit a charged particle. Similarly the heavier charged pion is the one with an extra internal particle. The decay of a charged pion to an uncharged pion plus electron plus neutrino is quite rare, but has been observed. The most common decay is into a muon plus neutrino, as considered earlier to arrive at a muon structure.

Even that falls apart for some particle pairs such as the kaons, the heavier neutral kaon has a possibly simpler structure than the charged versions. Higher mass particles however can produce particle-antiparticle pairs to decay into more complex collections of particles, for example the charged kaon can decay into three pions, requiring two particle-antiparticle pairs in addition to its simplest expected structure. By including sufficient particle-antiparticle pairs in the particle structures it appears to always be possible to split any particle into its observed decay products, but a few problems remain, for example photons are not as yet included in the descriptions, and the structure of the top and bottom quarks needs some thought. The charm quark is not so much of a problem, the decay of the uncharged D particle into K-plus and pi-minus gives some clue, but this again needs two particle-antiparticle pairs to balance the structures.

## Further Thoughts.

The decays in which a neutrino is emitted require that particle to carry a half unit of spin, but the internal particles involved initially exist entirely in the higher dimensions, and it seems reasonable to assume they have spin in one of these dimensions. So what happens to that component of spin when the particle returns to the 'normal' dimensions? We may assume it is conserved, so the neutrino needs to have two seperate spin components. Normally, in 3-dimensions components in these directions just add to give total angular momentum in some single direction, but the 'light barrier' separating the higher dimensions may keep the angular momentum in those directions separately conserved. Could this explain the three different varieties of neutrino? That second spin component will not be observable in our 3 dimensional world, but the information is still carried along to any future interaction, and may restrict the possible interactions, which require spin to be conserved, so that for example a muon neutrino can only interact with muons, not electrons or tau mesons.