Dear Ray,

I have thought about the Feynman lecture on why there must be anti-particles and I wanted to clarify my own understanding of it, in case that helps you in your readings. My broad understanding of the first part of his lecture is the following: he intends to show that the marriage of QM to relativity demands antimatter, or rather it demands that for every particle there must be a particle of the same rest mass and of opposite charge. In the second part he intends to show a definite connection between spin and statistics relating to various particles. I definitely understood the first part better than the second, but only after several readings and quite a lot of personal investigations which came from a completely different angle to end up being helpful did I understand anything useful in the first part of his lecture.

The first thing I seem to remember is that an electron, say, has a definite tendency to move faster than light. Although I have not ever read this following argument anywhere, I have conjured an ‘explanation’ for this idea which I think is probably okay for the large part. Or rather, it is almost certainly not completely right but I feel certain that it is bound to be correct in its essentials. It would certainly seem to account for the potential a small particle has to go faster than Cm/s when relativity alone says that definitely cannot happen and it would seem to account for it without any real controversy which I can easily detect. Itlooks good to me, but who the hell am I?

Either way, usual caveats aside, I strongly suspect that this faster than light motion is a direct consequence of taking the truth of the uncertainty principle and generalising its implications. I mean to say that faster than light motions are a definite consequence of uncertainty taken literally and then you go applying the idea to all instances in the laws where you try to determine exact values for position and energy. From Heisenberg, you cannot say where a particle is exactly. Neither can you say exactly when you saw it, nor can you guarantee its exact energy to within the limits laid out by the constant of Planck.

In the loosest scripture of Heisenberg you supposedly can know a position of a particle to infinite degrees of precision, but you then know nothing at all about the momentum of the particle and your uncertainty is made infinite. I’ve read that very thing in science essays, but it’s utterly wrong and the law of Heisenberg tells you it’s utterly wrong.

In reality you are also dealing with physical objects and you just cannot resolve the position of any point in space exactly, if only because of the wave nature of light. You’re stuck with the problem of not being able to resolve two different points on a line, (x) and (x+dx), without using a photon of a smaller wavelength than the distance dx, to do that resolution with. That comes from the first lecture, Quantum Behaviour, in volume III of the Feynman Lectures - the exact same lecture is also to be found as lecture 37 in volume I and he mentions this resolution problem in the second volume when treating light phenomena because it is also a classical problem to resolve two spots and not do a great job of that.

Even then, the very best you can say is that the two spots are no further apart than the wavelength of the photon you used to resolve them and no photon has zero wavelength so that there must be some amount of uncertainty about where the point is located if Heisenberg is correct. So perfect resolution of one point or two points in space is just not possible for that reason alone and it’s definitely wrong to say that you can somehow know the exact position of anything at the cost of your knowledge of the momentum. You can only ever know nothing useful about the momentum, and still be left with some fundamental indeterminacy about the position as well.

Just try letting D x = 0 and still have D p ^. D x ≥ h, somehow still holding true. Cannot be done. The minute D x = 0, the product D x . D p also = 0 according solely to the properties of multiplications of zero. So there is no question that indeterminacy as implied by Heisenberg means that you can never know exactly position or momentum, you are stuck with indeterminacy about both at once. To say there is complementarity in uncertainty is mostly right but partly misleading because it seems to imply that perfect knowledge of momentum or position is still possible, when that is not true. More true is to say you must know something about both position and momentum at the same time, but you are limited in that by a complimentarity principle of doubt about both at the same time as well. The difference between the possibility of some kind of infinite precision and the total impossibility of any degree of infinite precision is definitely not trivial.

To make things far worse than they first appear, the nature of spacetime itself becomes utterly chaotic at extremely small scales according to the strongly related law of uncertainty in action of a field of energy, you get uncertainty now rearing its head in the form of,

D E ^. D x/C ≥ h.

The more you try to say something about a position x with respect to some other position x+dx, the more field (momentum) energy can (and so it must sometimes) fleetingly exist at and around that point x as dx goes to zero as a limit. Heisenberg says this ultimate ratio is not actually possible, while stating clearly how and why it is not possible.

I found this very thing turn up when playing with the famous Buffon needle problem which gives a stochastic measure of p . I solved the classical needle problem using a simple argument of integrals and trigonometry, and feeling a little too pleased with myself, I then got interested in the same problem now modified to accomodate Heisenberg’s principle of uncertainty and that took weeks to muddle through before realising that I could understand the problems far, far better than I could offer any useful ideas about a self consistent mathematical solution. I feel like I have actually partly ‘solved’ the quantum problem , but unmathematically. And it is that problem of no easy solution which helped me to understand that Feynman lecture to any useful degree.

I also thought at the time that I was being very original in taking this quantum Buffon approach but I wasn’t, as a search on the internet soon comfirmed. All the same, I partly solved this problem as well and found what others had already seen when they did exactly the same thing to Buffon and that was definitely a good piece of training for me to have done.

As you probably already know, the solution of Buffon’s classical problem says that if you draw lines on a sheet of paper, where all the lines are parallel and evenly spaced, then if you get a needle which is smaller than the perpendicular distance between the two lines, you can calculate the probability that the needle will cross a line on any individual throw of the needle on to the page. Or rather, if you actually do the experiment with real needles and real lines, you can use your results and a derivable formula of probability to get a Monte Carlo method for estimating p by knowing how many trials you have done and how many of them had the needle crossing a line... Write the problem in terms of complex numbers and you can see e in all your solutions as well. So it’s definitely a cute problem and it’s definitely going to allow you to measure e from experiment as well!

When you go to analyse the Buffon needle problem and add the idea of a very small needle and parallel lines which are also very close together, you find that you cannot say too much about when a quantum needle crosses an imaginary line because of the uncertainty principle. In fact you cannot always be sure if your needle is longer or shorter than the gap between your two lines for the same reason. The more you know about that gap, the more you have to say the gap can be moving in some way and you couldn’t be sure where this gap is moving to. Or if it has a spin.

The needle’s ends also tend to not stop moving when you try to resolve where the thing ‘landed’. The tiniest of needles will always retain an indefinite momentum and it can spin as well. You can’t say where either end of the needle is without changing where it was. It becomes a probability distribution which is completely time independent, so you have to say that the thing behaves as though it has some momentum and therefore not one single position at all, not even for a given instant of time where you’re seeing it.

I decided that this not ultimately a property of some physical needle, but a property of spacetime if only because the idea of an arbitrarily small needle is obviously wrong and not so physical, it was after all, an imaginary needle made of purely mathematical points in a Heisenbergish field where you find even parallel lines go haywire. So the points on an imaginary needle tend to move about with some indeterminate momentum, true of each point on the needle. When you try to find out if a part of the imaginary needle also crosses some imaginary line it gets far worse as was already implied.

An imaginary ‘straight’ line also moves and changes position when you try to know too much about it or any points on it. I treated this as a problem about a gap of spacetime rather than a distance between two parallel lines and found that the more you know about a gap in spacetime, the less you can say about where it will be at some time later, or even what size this gap now has.

The closer together you draw two lines, the less they behave like Euclidean lines with ordered points and the more they behave like an imperfect gas where what you don’t know dominates over what you do know and the system of ordering points is itself an unworkable scheme.

Even if you ignore that devastating effect, and you say lines somehow do have well behaved points, you still cannot say which line is which from an analysis of two points and a gap. Not without giving the gap a definite likelihood of being somewhere else. And so you can appreciate that these two lines can swap and change over between two consecutive measurements, as well, without you knowing they did that, so that the spacings between any two lines is also far from absolute and it now implies an indeterminacy in the energy found between those two lines and it means that you cannot say which line is which when you consider distances that are also extremely small.

That’s all okay and consistent with QM if you say there’s an energy field of some kind called spacetime, where we’re trying to discuss distances between points in this energy field with a high degree of determinacy - you find the law of Heisenberg should apply to that field and you find that you cannot tell which way around your lines are located, nor the gap between them, nor if that gap is longer or smaller than your needle. It’s extremely complicated by now, by all of these outrageous factors you have to consider if you want some mathematical solution, it will not look nice and simple for field gravitation where the field comes associated with these kinds of intrinsic uncertainties at every point in the field. Determinism of a field is dead when points behave like invisible billiard balls and carry a definite likelihood of also possessing a field energy.

Indeed you see how you must now find that you cannot say that a line is ever straight either, because the points on the line are of themselves in a kind of perpetual motion and you’re not clear on how aligned one point is with two others, for any brief instant of time.

So in order to locate any point too well, you give that point an indefinite momentum to not be where you thought it was, some time later.

Hence magnified spacetime is necessarily totally hyperactive and energetic over very short times if the principle of Heisenberg is right when it is applied to a field of energy. And the smallness of the Planck constant is a definitive statement about the behaviours of fields containing any energy at all and how they behave in a short period of time. Or any field of any kind since for any field we are always interested in position and location of the arbitrary point.

This was Einstein’s finding too, that Heisenbergish fields always have associated energies by virtue of having a location at all! Although I have no idea how he found it and it seems unlikely he used a variation on Buffon’s problem because that’s not really needed: it’s an unnecessary complication which produces even more dreadful complications.

But it’s still a finding with a pedigree and I was very full of myself indeed after ‘rediscovering’ it in this way, a way which also involves clear indeterminacy about the Monte Carlo value of ‘quantum p and e’.

Suppose now that you have a classical electron moving at v = 0.999 x Cm/s, this is the case where you momentarily pretend to know where the thing was when you first saw it and where you suspect it might possibly have ended up and you pretend that you know these measurable quantities to some infinite degree of precision.

Suppose that you next introduce the measure of uncertainty about initial and final conditions. You are compelled to note that it might actually travel farther than a classical light wave in the same time due to that uncertainty about its beginning and ending points and the indefinite time of travel. Furthermore there is a clear correlation between how much you don’t know about the momentum and the amount you do know about the beginning and ending points and if you know the beginning and ending points very well, the only possible source of your uncertainty (assuming there’s no uncertainty about the mass, which there certainly should be – but we hold that constant for now) will be the time. Only that result for well known distance and indeterminate time can lend a higher obscurity to the momentum if we hold the mass temporarily constant, since velocity is by definition, not so well known but the distance is. How else can you know nothing much about dx/dt , yet know dx very well? Well obviously the only way is to know bugger all about dt.

Because it cannot be said that the electron did NOT travel faster than Cm/s, you are compelled instead to say, it does have a small and natural potential to go faster than Cm/s because there’s a small but definite chance that it really did! Your measurements could not prove it didn’t.

It’s true of photons as well. It’s true of all wave-particles over short enough distances.

Again, this is not any deep and approved argument from the textbooks so it could be wrong in some way invisible to me, other than the assumptions about constant mass. It’s largely from my own internal reasoning about Quantum Buffonery, but it would definitely make some clear and definite sense of the situation as described by Feynman and that remains true even if there is intrinsic indeterminism about the mass separate from time and distance. If Heisenberg is right, if that finding is assumed true generally, then we have a spacetime which is necessarily chaotic and which should behave as a perfect gas where a short enough time is akin to a gas pressure since it is an energy and so also a temperature equivalent.

The shorter the time, the higher the pressure can potentially become, and so by the Born statistical interpretation, should sometimes be! In quantum physics, as they say, if it can happen then it must happen sometimes. Probabilities just are what they are, statements about chance and also inevitability.

Either way, this one idea of superluminal speeds due to uncertainty about points in spacetime now ‘married’ to relativity principles and so also conservation (symmetry) laws, is enough says Feynman, to guarantee the existence of anti-particles.

If you allow for time to go backwards and consider two charges and their forces and signs in reverse time, then the existence of a single charged particle and relativity gets you two charges for one. The reason is (I think) as follows. Assume as true the idea that an electron goes faster than light from point A to point B.

Suppose also that you are positioned such that you are definitely ‘between’ those two points so that light from A and light from B take exactly the same time to reach you by the classical law of waves, so that A, B and your position form a perfect equilateral triangle.

Ray

Suppose also that you have two clocks, one clock at A and another at B, such that in your position you will always see both these clocks as registering exactly the same time at the same instant. Note that the times you see on these two clocks can easily be made to appear asynchronous, simply by taking one step closer towards one clock and one step away from the other. Indeed that must hold true if light has a finite velocity. The time you read on the clocks depends on where you are standing and their distance apart from you and so each other also.

Suppose now that each second, on the second, (rest frame and no gravity acting) both clocks send out a strong light signal in all directions. Then if you really are equidistant from both clocks, these two signals will arrive at your position at precisely the same instant.

Suppose also that we have constructed the arrangement so that the length of any one side of our equilateral triangle is exactly C metres so that the instant you intercept a photon from both clocks, another two new signals are being created, which you will not see until exactly one second later. We also know that because the triangle is an equilateral triangle that the instant you receive a signal from A, the clock at B receives the same signal and vice versa. So an observer standing at B sees clock B running precisely one second ahead of clock A, and that too is a reversible situation because of how we arranged the clocks and observers.

Now suppose that at the same instant, say 12:00.00 midnight, both clocks send out a light signal and coincidentally an electron at A is clobbered by a high energy photon. So this electron also moves from A to B, but since it has a potential to do it, it goes from A to B faster than the photons do.

Clearly, if this is at all possible, the electron will definitely reach B before the co-incident light signal from A arrives at B. The electron and the photons both left A at the same time, we stipulated that as given, but the electron moves a greater distance in a given period so that it definitely reaches B before the light signal from A which also carries the information about the scattering of the electron from A. That is to say that the electron moves faster than the wave of information about its own motion. In relativity that can mean going back in time, and if we define time in a certain way, it can definitely mean going backwards in time!

An observer stationed at B, then, will first intercept the electron at B and only then watch it leave A as the light from that event arrives on a ‘delay’. An observer at B concludes that the electron arrived at B before it was sent from A. That is to say that an observer at B sees the electron as arriving at B and next being at A. So Mr. B sees the process happen in reverse, just as you would expect if the particle moved backwards in time by running film of the motion, backwards through a film projector.

Einstein’s principle of relativity is very strict about causality and his entire relativity theory assumed that causality, somehow, had to remain preserved given a finite speed of light for all observers. And that you could get back to a kind of definite idea of ‘simultaneous events’, but you had to abandon old ideas of simultaneous events and embrace new ideas instead. People say the idea of simultaneous events is dead. It is not dead, it is refined and developed so that we still know exactly how to state the idea and Einstein takes great pains to establish that this is still possible so that this very point, that you can say when somebody sees two things as simultaneous which someone else does not agree, constitutes a large part of his 1905 paper on the relativity principle. To say simulteneity is dead is to simplify the thing to a point of being content free. Simulteneity is well and truly alive, it’s just not like we used to think of it. That’s one discovery of Einstein.

He declares that one observer can see different times for some event than another observer and that this is not any paradox. And he also says that you can see two events in a different order than some other observer, but that you will only see order reversed under special conditions whereby one event did not cause the other.

Suppose you watch two stars explode, one independent of the other so that they appear to you to blow up at the same time. Einstein says it is possible to find a place near you where you see them not go up at the same time because you are further from one star and closer to the other. He says in this instance the fact that you can see them happen at the same time implies they are not causally connected such that the blowing up of one star influenced the blowing up of the other.

Now consider the case where one star blows up and some time later the explosion reaches a second star close by, a binary companion and this somehow induces this second star to blow up. Einstein says this interaction happened and now you, me and everybody else, no matter where we stand, we will all definitely agree that the same star went first and the same star went second. There is no place we can stand where we see the order reversed.

Relativity is partly a principle which maintains the notion of causality in a universe with a finite velocity for light, a velocity which all observers measure as the same irrespective of their initial motions. So relativity says you cannot see causes and effects differently to another observer, you cannot and will not ever see effects precede causes was the original statement and QM modifies that and it says, where it seems effects really do precede causes because of Heisenbergish interferences, everyone must agree that the effect preceded the cause in this one instance.

So modern SR says backwards causality is okey dokey and a-okay and just fine with mother nature. So long as we retain the condition that everyone agrees "I too saw that happen faster than light, just like you did!"

That means that even positioned where you are, away from A and B, causality in the modern sense now requires that you too should also conclude the same behaviour of the electron as if you stood at B. Similarly it demands that if you stand at A, you should register the same events in the same sequence as if you stood at B. Ie, if we interpret this as time reversal, which seems consistent with some intuitions about how time might appear in reversals of film reels, everybody has to see this as a time reversal for basic causality principles to be upheld, as per modern relativity theory. If an electron goes from A to B faster than light, we all see the below,

Ray

Indeed the only ‘simple’ way known to reconcile causality with faster than light motions of electrons is to say, as Feynman did, that when an electron moves faster than light it also, always, goes backwards in time. That way you too can be allowed to see the electron arrive at B before it left A and the principle of causality is modified so that effects and causes can be reversed, no problemo, but still preserved is the idea that observers agree about ordering of causally connective events, so that all observers can agree on what they saw first and what they saw next even when effects do apparently precede causes.

That implies that when the electron arrives at B, it has not gone forwards in time, but backwards instead so that it arrives at B some time before the clock at A, as seen from B, reads 12:00.00, which is when we stipulated it left A.

So everything about reverse time is consistent because this electron which left at the stroke of midnight might arrive along with the wavecrest corresponding to the A time of 11:59.59.9999, with some some definite degree of uncertainty about the time of arrival, but enough certainty to assure that it arrived before A’s midnight. It went backwards in time while still travelling and that demands that B says it arrived at A some time after it was registered at B because they will see it at A at the stroke of A’s midnight, but see it at B before the stroke of A’s midnight. That is to also have demanded that it looks like it goes from B to A slower than light even when it went from A to B faster than light. In the process a charge did something, it changed signs.

It may not be true that causality principles really are upheld, so that switching from B’s position to your position is a matter of assuming the relativity principle is okay to get away with, but if you assume the opposite and reject relativity and reversible time and instead you assume that the order of events for one particle interaction can somehow be disagreed upon by two observers, you also do not end up able to predict the existence of antimatter. A problematic fact which is clearly not so easy to explain away through co-incidence because antimatter is as real as anyone could hope for and there are infinitely many ways not to find it as a consequence of some other theory of wave particle duality.

But I have given no reason to believe that you should want to reverse the sign in the charge from electron to positron and I must redress that right now. Supposing you can actually reverse time, you then consider next what happens to a charged particle pair when time is imaginarily reversed. Attraction becomes repulsion and vice versa if we watch a movie of charges interacting in reverse. We would say in this case that unlike charges seem to repel, like charges now go against their own nature and attract – the total opposite of reality of course, but that’s the effect of reversing time only. So if we reverse time only we get the forces coming out wrong, but systematically wrong and that system of wrongness is a clear indication of what we would need to do to make it right again.

The only way to reverse time and still have like charges repelling and unlikes attracting in a backwards movie, which is what we surely want to see come out if relativity is preserved again, this time in a brand new way, is to say that whenever you reverse time, you must also change the ± sign on one of the charges, but not both. That means we need to film a completely different movie and reverse the wrong movie! That will restore the situation, we can keep our two laws of likes repelling and unlikes attracting in reverse time, but the cost of doing so was to reverse the sign on a single moving charge out of two and at first glance we have violated relativity because now we have said that we can arbitrarily reverse the sign of either charge. But causality says this is wrong. We have no choice in some sense about which particle we change because we are assumed to be inside this same picture and that declares a rest frame as implicit. Relativity instead demands that we know which particle moved faster than light and which one did not when we look at the movie.

Ie we really do watch a completely different movie in reverse, where we changed one of the charges non-arbitrarily and filmed it forwards and then run it backwards and we definitely get back to the right interactions and the forces are all the same everywhere we look and we cannot tell the damned difference between movie A going forwards and movie B going backwards in time.

Later it was of course discovered that parity is also violated, so that to do a time reverse of particles in motion you have to film some other movie C, change the sign on one charge (the one you see going faster than light) and also invert the ± signs on your co-ordinate axes. You film it in forwards in a mirror, then run it backwards through a projector, and then you retain your ability to run time back and forwards and keep the forces all looking identical. I can’t remember if I stole the movie and mirror analogy from that particular Feynman lecture, but I stole it it from him, no question.

Parity inversion is an operation supposedly needed fully for only a few special cases involving particle decays, but it represents the generalisation of the situation so my guess is that it is actually of enormous significance for all laws involving charged masses.

Hence Feynman’s expression CP = T refers to a ± time reversal being equivalent in the laws, to changing the ± signs of charge and parity glued together, since one implies the other, always. And it includes the idea of preserving causality in the limited sense. Now we are reversing time, changing a charge and filming (or watching) everything in a mirror.

That means that if the time reverse theory of Feynman really is useful, if causality is preserved, if the idea of the invariants is okay, then just as Dirac predicted, so too Feynman could use his ideas to predict that there definitely should be found instances in nature where we see a particle which has the same rest mass as the electron with the same magnitude of charge too, but the charge is the opposite sign. And because the electron was arbitrarily chosen in the example, and because all particles have a quantum (wave) character, so too for any other fundamental, charged particle, the idea remains equally true or untrue.

It is also to say that if a particle seems to go faster than Cm/s to one observer, all observers will agree that the same particle went faster than Cm/s. So it’s truly okay to reverse the sign on one charge but not the other and still be okay with modern relativity. In fact the "inverted opposite" is also the case, to remain okay with any relativity principles in light of Heisenberg and Feynman, we can only reverse the sign on one charge and not both (unless both charges move faster than Cm/s). Where both charges run faster than Cm/s, or slower than Cm/s and parity is not violated, then we have a simplifying and special symmetry where the time forwards and time reverse are the same movie, A.

An interesting feature is the creation of charged particles from collisions in which the ‘lost’ collisional energy creates new particles by E=MC². The time reverse of particle creation is particle annihilation. Also since energy is mass, we may suggest a principle (probably naïve) which says that any energy type will always be able to convert to any other energy type and so deduce that annihilation and creation of particles into other energy forms must take place because energy is conserved and yet mass is not always conserved.

So either way, taking that experimental fact of annihilation and creation and now adding in the local conservation of energy and the local conservation of charge, Dirac’s theory expected that a particle and its antiparticle can and must annihilate with a release of energy into several different quanta so as the two opposing charges vanish, a conversion of energy procedes and the time reverse of this is a gamma ray producing a particle pair.

The local conservation of momentum adds that at least two photons must be created in that annihilation. You cannot have only one photon created, since that would violate the momentum conservation law, but you can have two or three photons created from an annihilation because that means that the centre of mass of the system can stay in the same place, or retain the same initial translational motion. So the theory of time reverse also predicts that photons can collide by collapsing at a single point in spacetime.

Either way, conservation of momentum says the centre of mass of the entire system cannot remain in the same frame of motion if a single photon is emitted when the positron and electron collide. Now there’s also neutrinos to worry about and they were perhaps too conjectural for Dirac to ever worry about. But a high energy neutrino will have the wavelength of a gamma ray anyway so it will freely interact with matter in collision. I think neutrinos might be bogus since you can call them a photon and say they have an equivalence to a rest mass and there you have one less particle and no other problems. Neutrino detectors could be detecting photons and there’d be no way of telling the difference so far as I can see. Neutrinos need to have the same properties as a photon in QM, because a photon has a tendency to travel slower than light, that implies it has a mass when it does so. Again there are posited whole classes of neutrinos so that idea is very likely wrong for reasons I don’t really appreciate.

Either way, Feynman then goes on in his lecture to talk about the connection of intrinsic spin to Bose and Fermi statistics and I have understood virtually nothing of that part of his lecture. It is very likely that I have made some large errors in the alternative account I have given of the first part of the Dirac memorial lecture because as I said, I do not understand the lecture so well as yet and I completely made up large parts of the argument for why a particle is expected to go faster than light. So this "explanation" hopefully gives a better indication of what I don’t understand, than what I do. I know some deep things are missing from my world view because spin is something I have yet to actually even partially understand as a consequence of other things.

Also, I’m not sure that anybody but Feynman really understood all of what he was saying, because nearly all of that particular lecture was extremely technical and the parts which are less technical are clearly insane. I mean the mathematics and theory look good to my crude eyes from the rudimentry thinking contained in this essay, but that nature is actually like this or that she should care how we label our axes when we run movies backwards is hard to fathom, impossible to believe but a definite outcome of finding ways to write laws in terms of invariants.

This much I do know about Bose and Fermi statistics and intrinsic spin. Firstly and most obviously, that spin was a prediction of the Dirac equation and was found true (beforehand ?) by the Stern-Gerlach experiment. If I had to guess at how this was done theoretically, I would say it probably comes from what happens when you write the laws of photon absorbtion and emission for an electron without a classical mass at rest. As we saw with Buffon, a tiny needle will spin and two parallel lines need not remain parallel. If we say there’s a mass equivalent to a gap, but we can’t say too much about the mass without being somehow wrong as well, you get a change in the original deBroglie matter wave, a definite modulation if you assume it is a harmonic wave in some way and that other properties of the old unmodulated classical electron wave are still more or less dependable.

It might be that this modulation represents something like spin, so that they call it spin. What spin is, is not so clear to me no matter WHAT the hell they call it. It is a harmonic modulation of a statistical wave. How you set about calling that spin and getting away with it is somewhat beyond me and I strongly suspect that spin is a very bad and wrongful analogy to have made for the reason that the modulation translates to a change in the probability amplitude and not a real and definite motion of some definite particle. That is clearly the connection between spin and statistics qualitatively. The fact that the spin is equivalent to a change in the amplitude for the probability is the only possible connection to care about. But how idea of a spin which is a statistical modulation and not a physical one, how that goes on to also give Fermi and Bose stats is totally beyond me.

Having worked very hard on relativity, I naturally tried very hard to get out my own grasp of the modern theory of the electron by trying to find what is known to be okay using the laws which were known to be okay such as deBroglie and Heisenberg. The best I could manage was to find that emission and absorbtion of an electron must have two possible solutions due to the existence of the square root appearing in a sum.

That says nothing to me as yet about any stable state as should be found when the electron is near the nucleus of the atom. If E_M is the rest mass energy of the electron, then an electron which emits a photon and changes states as a consequence, gives the photon an energy and frequency given by,

hf = E_M - ±√ (E_M - E_M v²/C²).

Either way, all lepton particles possess a definite tendency to have half integral spin and that this spin is something which influences the behaviours of other half integral particles in the same locality. How local is local enough, I do not know. Nor am I so clear on how one particle ‘knows’ the state of some other particle without any direct interaction, so I assume with some safety that the interaction is indirect and has to do with the field (wave) aspect of probability amplitudes and not the particle aspect of being clobbered by something with a mass energy equivalence.

All the same, the correlation ultimately comes down to a statistical expression which relies on projective geometry, (of which I know nothing at all except how to write it in a different way than does Feynman in that lecture) so that somehow, the spin state of particle n is correlated to the spin states of other particles in the locality by two completely opposing laws of chance. Fermi laws involve those particles which do not like finding themselves in identical energy states. Bose laws are for those particles which prefer to exist in identical energy states. The difference is a single ± sign in the one basic law, as I seem to recall. I have seen the different two laws in Vol III but have no immediate recollection of how they were demonstrated there, as being true from other principles.

I know that the Pauli exclusion principle and fine structure splitting ultimately comes down to Fermi statistics, so that the two electrons in a helium atom will always assume different energies from a Fermi interaction which demands that by having electron 1 in state p, we dramatically decrease the probability of finding electron 2 at the same energy to zero, and we can say that electron 2 will not be in state p.

I also read somewhere that these statistical ideas can be extended to the nucleus as well so that a nucleus with an odd number of particles obeys Fermi stats while a nucleus with an even number of particles obeys Bose stats and it might well be in that Dirac memorial lecture where I read it. Among other things this differentiates the nuclear behaviour of two different isotopes with the same net charge on the nucleus such as carbon 13 and carbon 12 (or carbon 14), while leaving the electronic behaviour of those same isotopes effectively unchanged.

This even-odd nature of Bose-Fermi statistics and particles is truly remarkable of course because it suggests among other things, that adding or losing a single neutron can change the entire behaviour of the rest of the nucleus and all of the particles in it so that they all make a definite transition from Bose to Fermi states or vice versa, while leaving the electrical properties of the atom effectively unchanged. All based upon the idea of odds and evens.

I have found no account anywhere of why this is thought to be true, it is merely stated in terse and lofty overtones as though it were an uncontraversial fact students are expected to digest, not attempt to even partly fathom, nor is it presented as the mystery it certainly is, irrespective of your mathematical models. I find that extremely irritating and I strongly suspect that almost nobody knows the deeper origins of the theory which underpins the experimental findings, or that the only people who did understand the deeper theory and were willing to write on it sensibly and respectfully, are now dead.

There is one other thing which is important to note about the discussion on antiparticles and that is that when I made the distances between A and B roughly 3x10⁸ metres, this was for convenience in discussing clocks, not accuracy about particle behaviour. There is no real chance of any electron going so great a distance faster than light the entire time, so far as I know.

The reason is that the effect is only expected to happen for very brief intervals of distance, certainly because of the smallness of Planck’s constant, but what physical interpretation might we try to place on it? Is it possibly due to the interaction of the electron with spacetime itself? Is it possibly due to the interaction of the electron with its own field? Can you assume an interaction between the mass and charge conservation laws, that they are the same thing for a fundamental particle and what might that imply?

I’m not too sure about that part at all, why large distances and so on wash out these quantum effects, but in terms of the mathematics, it definitely has to do with the smallness of Planck’s constant as compared to SI units for time implying a small energy for the equation,

D E ^. D x/C ≥ h.

That is also why in Feynman’s discussion he describes interactions over very small times. So the ideas I came out with are very likely to be at least partly wrong, but the model I used is certain to be wrong - but only (so far as I know) because of the distances and so also the times involved. If I said define the unit second to be 10^-64th of a standard second, we have great big energies which can be potentially associated with our times, through the principle of Heisenberg.

Anyway, that’s enough of me. I hope you enjoy Feynman’s Dirac memorial lecture which contains no such buffoonery (Buffonery?) at all and I hope that you can get more from a second or third reading than emerged from the first. My experiences with all his undergraduate lectures is that they are exactly like this. He expects a lot of the reader, but at the same time he certainly expected far more from himself. That Dirac memorial lecture is swift, rich and economical in terms of providing a giant argument within a small space.

Regards,

Bart