QG Theory Leading to GR in the Classical Limit / Dynamics of Complex Waves

0:00 Thank you. you want to say during the presentation. So what I've done is find a theory of quantum gravity that becomes general relativity in the classical limit as it leads to. Then... Although I suspect you're not using the phrase quantum gravity as people hear. Well, you mean, did I start with the Einstein equation of quantitis? Is that what you're suggesting there? No, I didn't start with the Einstein equation and do that. The quantum theory I produce is in fact of the same form as quantum electrodynamic, with electric charge replaced by mass. and that isn't quite the same as can't be quite the same as general relativity because for general relativity you've got a tensor, you've got the matter stress energy tensor and for quantum electrodynamics you've got the current, which is the vector so if you're trying to say the two are analogous, you can only do it to a certain extent So what I do in addition to that is consider conglomerations of matter and produce a statistical argument. And only then do you get the general relativity. So that's what I mean there. So there's that quantum theory of gravity, which one of the classical limits is general relativity. theory to derive the amper-comatorial hierarchy, and that is the whole number of bits of the

2:30 four members, plus also fractional bits. And I get the most common one, the most well-known one, is the fine structure constant, of course. I get that to, I think it's about two outs in the seventh figure. In other words, it's not as good as Clive has done for getting experimental results. But on the other hand, I get a parameter that refers to the strong interaction and weak interaction as well. And also, I have some thoughts on the gravitational one. So anyway, that's that. And what I then do is use the members of the ANCA hierarchy to, and I find expressions. So to get those expressions, I'm using the quantum gravity theory. I'm using the standard model of particle physics. And I find, I calculate expressions I calculate about 13 of the 18 arbitrary parameters in the standard model, and that's what some of this is here. I start here, that's the electron, and although it's exact, I've taken it as an exact thing there, I can also calculate that. I'll count as to how. This is the mu meson, it's mass, and alpha there is the fine structure constant, and nu is one over the member of the ampers series which attaches to the weak interaction. Here, this is the pi meson, same thing again, fine structure constant. That gamma is the one that is the member of the ampers series that attaches to the strong interaction. Here we've got the mass of the Tormeson, and that doesn't use any new things. Then the mass of the proton, first expression for it. That was introduced. These are all relative to the mass of the electron, incidentally. So it's this expression times the mass of the electron. Now, if you take the top one, the MP times the bracketed quantity, MP is not the master proton, is it?

5:00 No, it's a parameter that I set initially by looking at the electron. So that parameter is multiplied by something which is very slightly bigger than 1. That's right, yes. But to make all the expressions nice and simple, we'll get them all on an overhead, I didn't bowl with that factor, and I just, these are relative to the mass of the electron, all these here. And the column is in MEVs. The numbers are actually masses in MEVs, not negatives in the electrons. Yes, and they're simple, not all of them are MEVs. Oh, and not all masses, yes, sorry. Some of them are MEV, some of them are in other units, kilogram, second, centimetre, and... And loads of noughts, one way or another. Precisely, I was going to come on to that. The powers of ten also agree, I think. I'm pleased to hear. I might hear that. So, if I were to come by all the small print, I would have to add more small print to those masses that are up there, but those are what I get, anyway, as the estimates. This is an expression that depends on the mass of the Z and W bosons, and since we know the mass of one of those and in terms of the mass of the other, we've got all that. The second expression for the mass of the proton, which is how I make the whole thing bite its own tail and derive the mass of the electron, because we've got two expressions for this, so we can find out what are the m e, the mass of the electron that we're multiplying that by. This down here is the fourth member of the AMBER series, and to say, and I derive an expression... That's the 10 to 39 thing, is it? Yes, yes. That's the one. That's the one. And that is quite, I think, usually known as the Planck mass. I derived an expression for that, just as I derived those other expressions, in terms of the quantum gravity and the equations of the standard model, quantum electrodynamics,

7:30 the electro-weak and also the electro-weak interaction but the unbroken electro-weak symmetry as well and the equations of quantum chromodynamics so that anyway is that part so what I've got on overheads now is a standard run through of the earlier part what this quantum theory of gravity actually is, where it comes from and I've got that, and I've also got with me a second talk, which is how you use the quantum gravity theory to derive the amper hierarchy in detail. So, have you any thoughts about what you'd like me to do first of those two? I think, well, the quantum gravity... Yeah, that in a way comes first, so perhaps that would be a good idea to start at the beginning. a brief summary of this one. So it's a theory of quantum gravity that leads to general relativity as a classical limit. It's the same as single particle QED theory for Schwarzschild metric and relativistic generalizations because the Schwarzschild metric being a solution of general relativity, you can throw it around in space any way you like, in space and time, you know, and it's still true. So that actually, so then what we get is that the electric charge becomes a matter charge and then the quantum QED current becomes the matter tensor for the Schwarzschild metric, which I've said a little bit about before. Then I use statistical considerations using lots of particles of matter to lead to the full theory. So I have sort of two bites of it before I get to general relativity. So, in fact, you do something like Eddington did, which was in his book on general relativity, was to do the Schwarzschild thing, and then say, well, lots of particles are lots of beans, and then gradually move it over to the continuous matter thing.

10:00 Right, yes, I haven't read that book, but I obviously should have done it. Is that the most recent thing? I mean, that's not, I mean, that's ages old, that's 1926 or 28 or something. Right. But the thing is that the transfer to continuous matter seems to be a difficult problem. Anyway, that's how he does it. Right. So you appear to be doing the same thing, only with a different theory. Well, that's very interesting. I ought to look at that. See how nearly the two are the same. Now, my starting point is to show that I'm using throughout the single particle QED theory. Now, if you want the field equations, it is open to you at any point to derive them by canonical second quantization and canonical quantization or the Feynman method. So it doesn't, you know, if you really want those, you can still get them from what I've done. Right, so I start by showing that the electron can have a compatible, is compatible with a classical spin. And this is... Just a very simple piece of algebra, putting the Dirac equation in another form, and then showing that by making the wave function a matrix, you actually get, in addition to the old method of doing a relativistic transformation on it and you also get one in which the which the spin vector does behave like a four vector so it's

12:30 a sort of second quantization in which you represent the psi as it were as a you say this is this is a measurable and so we're going to represent it by a matrix is that is that the sort of thing you'll do no I just I just I'm up and represented by a matrix I mean oh I see I'm not sure how good I am at doing these things Oh I see. The wave function, those are linear combinations of the high spinners. So, it's simple enough. And if you, I've got a set of references, you've got that up. It was published in Foundations of Physics in 2000. So, yes, I looked that way a bit fast, I mean, an explanation as to when this theory might apply, and the electron actually has a spin, which is a four vector, might be when it's going in a circle, now that would answer me straight into, um, into considering, unsurprisingly, That's restricted in the details that they're considering Bohr's model for the atom. And that's what I use. And I derive Bohr's model for the atom from the equations of QED. So it's done on for something that you park on ad hoc, as it were. It actually comes out of the quantum electric dynamics. You have to transform space a little bit to get there, but it does. Right, so when you've got the electron travelling in a circle, you can consider it as doing this because it's in a potential, like the one that the proton generates.

15:00 It's now a pixel photon. We're not going to have the source move at all, but the electron is going around like this in a circle, and you can find out what its energy levels are, and they are the principal energy levels, which you would expect in a bound state between two-point particles, following an inverse square ball for the force. Well, okay, this is cylindrical symmetry rather than the usual spherical symmetry you do and if you do cylindrical symmetry, you've got two dimensions left over, space and time So I've got another release of course in those two dimensions which of course we're looking at from a different point of view and that nets you the foreign structure that you would expect so you've got it sort of orbiting in the other plane but the other plane isn't just space it's space and time, yes that's right and in fact what happens is that time when you want it orbiting in a circle you can have time going along the radius of that circle So when you do your transformation, you start with it at right angles and you get it twisted around in a circle. You've got some slight linguistic trouble which is inevitable here. You talk about travelling in a circle, but in the case of the space and time one, that word is a bit awkward, isn't it? What I do is I, from that point of view, I park time back in here and I reuse one of those dimensions as time for the second one. Yes, I mean if you consider the Schrodinger model rather than the Bohr model, so that it's waves, there's no such thing as time anymore because the wave is just locked in and that is what the energy levels are telling you. This is the pattern you have to have in order that I can be an eigenstate or whatever.

17:30 Now, I can imagine that if you went into T plus X, there might be something rather similar. But you have to be careful with the T. It's not the quantum T. It becomes some kind of spatial T. And you have another T as well. Yes, yes. That's right. So it's in its, if you like, it's in its own space, you know, there's another space there. But the space is, and I'm going to come back to that, because there's a series of these, and it turns out, and that's very important, because it applies to general relativity as well. Right, so having taken hold of the Dirac equations, Maxwell's equations, and derived the Bohr atom, then I go the other way about, and I take Bohr's equations as the axioms, and I derive QED theory, that is, the Dirac equations and Maxwell's equations. and that is very handy for me because now what I need to do is show that Bohr's equations hold somewhere and then I can say, well right, in that case so will QED so what I do next is prove Bohr's equations hold for the Schwarzschild metric in general relativity Now, right, well, I go to essentially what is a relativistic version of Newtonian gravity, and I show that that leads to Bohr's equations. Well, yes, I can believe all this because I've been studying what happens if two particles are in outer space, as it were, and there's no matter around, and so they have to be their own coordinate system. And you've only got one coordinate, except possibly you may have a time coordinate as well. Now, if you said, I want to do this for a Schwarzschild space

20:00 rather than a Minkowski space or even a Newtonian space, which is what I've been doing, you could do it. It would simply mean certain distortions, wouldn't it? And also certain Yes, I mean, I'm simplistic about that. I get the equation invasion and work with that, rather than very directly using the curvature of space. I mean, I do use the curvature of space, yes, but I work with a metric, and I don't work very much with the, well, I work with metrics, but I don't work with the actual Einstein equation itself. Well, can I just ask a very quick question about, you say, a relativistic version of Newtonian gravity. Now, that suggests a certain theory to me, just tell me if I'm right or not. It would be a version in which, for one thing, you took account of the increase of mass with velocity for a thing going around in an orbit, and secondly, instead of the inverse square law, you'd have, which satisfies a pluses equation, you would perhaps have some other relativistic equation that you'd satisfy. Am I on at all the right track? Yes, I think that's right. Can you repeat the first one again? Yes, you're taking account of the increase of mass with velocity. Yes, that's right. That's a very good point. I wanted to say that. There's always been a sort of question mark over whether there's a relativistic increase in mass with something going around in a circle as opposed to going along in a straight line. And when I do the transform that has the electron from going in a straight line, you transform space, and it gets curved around, and it comes around in a circle like that. And you can show the Dirac equation works for this space, and we know that the Dirac equation does embody the Lorentz transformation in itself,

22:30 which justifies it. Yes, though he doesn't. he doesn't do and this makes me suspicious is that he he says there is a central force which he describes by means of a potential but he doesn't say that thing producing the central force has a finite mass so and this is slightly connected with the increase of mass thing because if it had a finite mass it would be itself doing a little bit of relativistic right but but I suppose that's a quibble in a way because you're not you're not too worried about that because as and as in the case of Schwarzschild you're saying well really the thing that's moving is a test particle and it's light and I'm not In a way, I get round, I sort of leave the source out of account all together, because I do a little bit of space-time engineering, and the electron without a force would be going in a straight line. Now, without a force, it travels round in a circle. And then I say, oh, well, this is very handy. This must be a very good model for what you've got when you've got a central attraction there. That's what you mean when you say you've been spatially. Now, I still haven't got the general relativity proper quite yet. I've got my metric and I've got Bohr's equations, but I haven't got the metric I need for saying it's the Schwarzschild's solution of general relativity. So what I do next is general relativity has things to say about the properties of a self-gravitating thin sphere. and inside the sphere mass acts like charge and Bohr's equations hold

25:00 outside we get the Schwarzschild symmetric and that's just a duration, I've used references for it and so on, that you can look up by a gentleman called Kuchar if that's how you pronounce his name he's Czechoslovakian so we've got a thin sphere of matter and we've got in the centre well we want obviously no force and in fact we get something like Bohr's equations holding and on the outside we get the Schwarzschild solution that's what general relativity has to say then what I do so this situation here represents what I've just described I put in another circle so there's one circle here, one circle there, one sphere here, one sphere there. In the center goes the original mass, the original one, we put the original inside with mass acting like electric charge. Between the two reams, we put our Newtonian solution, and on the outside, we put the Schwarzschild solution. all the equations you find that that that's what you get. Now if you've got the, if you say Schwarzschild is outside then actually the Schwarzschild radius is likely to be inside and in fact not not really exist at all because you say you're changing the nature then the nature of the beast once you get inside Now, is it true that the gravity is very weak between the two spheres and that's why you use Newtonian, or what is the reason? No, it's more abstract than that. If you have a mass in space, a ball, a massive ball, it will generate a Schwarzschild metric. Now, if instead of the massive ball, you have a hollow sphere, and the sphere has mass, then it will generate the Schwarzschild metric outside. And nothing inside.

27:30 And nothing inside. And general relativity will tell you that. I mean, that's the Cavendish screening, basically. So that's what I start with. And that is what general relativity says is perfectly okay and is actually a solution of general relativity, etc, etc, etc. Then what I do is make the single sphere into two spheres. And I'm going to assume they're both thin, they're both practically touching each other, the same radius. And the inside, right inside here, is going to be what was on the inside of my thin sphere according to general relativity. Which is nothing much. Precisely, yes. And in the gap between the two spheres, which is this black thing, I put the solution I'd previously got with my Razzuristic Newtonian type solution which isn't a solution of Jeremiah's Riddling simply isn't the right solution on the outside we do get again something to do as we get to the Schwartz trial solution so I have to ask Why do you say it's Newtonian in there if you feel that it isn't the right solution or perhaps I was misunderstanding the language? Well, for one thing, it's a tachyon type solution. Oh, I see. And when I say it isn't the right solution, it isn't the solution found by general relativity, and which I'm suggesting is sort of curtailed by leaving this one out. Can I just ask what sort of continuity conditions across the surfaces you employ? Well, if we have the original with general relativity, then there's some quite careful questions about that. I have to admit, I think things would join up. Whether we're ones then getting the derivatives joining up, you know,

30:00 with that little jagged bit or something, I haven't honestly checked. Yes, it's a very good question. It's not one I'd considered and can answer off the card. It does me do it because I was for many years involved with a man who was doing something not like this in electrodynamics. and he refused to put the junk conditions in the two cases and when you insisted on them in one, he'd do it and then it was discontinuous in the other and vice versa and so you could get all sorts. So it needs doing, but anyway, okay. Don't continue. I'm just wondering what I've got in the next one. yeah I'll put this one I'll put this next one up if you remember that this is what we're dealing what we're dealing with and I basically get a series which I can iterate but to go back to that question of what happens when we reach the boundary Basically, what we get is that we can address any one of these spaces as curved if we want to, or we can address any one of these spaces as having, if you like, that tide in it, a moving space. You've got, no space can be curved. I'm introducing the idea of a moving space, and whether you have a curved one or whether you have a moving one is optional. the sort of motion picture of it, it's going to change every time you get to a spherical boundary, that's what the relative velocity should be, so that there's going to be a jump. Just to say what I can about um, um, Clive's questions. Question. So, what I then find is that outside we get, um, we get the Schwarzschild metric, we get the metric that we want, and that we know, and we find out that Bohr's equations actually hold for that too. So that we can uphold Bohr's equations and on the outside, which is where I wanted to try and get to.

32:30 I'm not making a very good sharing of describing this. I've got the paper with me, but whether it will actually help if I sort of delve into that at the moment, I'm possibly not. Right. When you say Bohr's theory is already relativistic, it only can be made relativistic if you say, yes, and this rotate, this mass that's going around in a circle, yes, yes, and that's what you mean. Once you've got a solution of the equation of QED, it's going to be relativistic. So I've taken a detail via general relativity, and of course I have to do this because there have been a lot of good experiments performed that shows that general relativity brings home the bacon, as it were, in lots and lots of circumstances. So you have to be quite sure that we're not contradicting it. But general relativity doesn't come into the description apart from that, that it summarises, if you like, it's a pendium of a lot of experimental results which have come out to be what they have. and you can find out exactly what they are by cracking a handle with general relativity. So, right, to go back to the self-rabbitating spheres, we managed to get a progression from one surface to two surfaces and we can go on. As you will be unsurprised to see, we can have lots and lots of these things.

35:00 Four orbits are different because some of them are inside the spheres, is that right? I mean, if you're completely outside the central mass, then it should be very much the same as the ordinary theory, but as you move in, it gets scrambled in some way. Well, yes. I mean, I don't think it would, it wouldn't work to take general relativity and try to produce the whole of the solutions, because, well, you could, you could start, you could start with any, any, any set of two. I mean, you could have, you could, for general relativity, you could have this one and this one and this one of green, or you could have this one, this one and this one of green. But in between, you get a tachyon solution, which in general relativity, at least in its standard form, wouldn't be too happy about. So I have said this already, so I won't leave it up for very long. So what we've got out of all this is a possible contender for a theory of quad gravity. We don't know whether nature does seem this way or not. There might be other ways of getting a theory. I don't know what there are, and obviously it's a difficult thing to do, But, nevertheless, that's the status of it. And as for the... I don't feel I've done getting a general relativity part of it very adequately here now. I mean, if we have some time, I'll go back to that. Or otherwise, I can suggest you read the paper to see whether you think that it follows logically. But I'm saying that that does work. Right, so we'd like to know whether our theory is the right theory of quantum gravity.

37:30 This one seems to be a bit bigger than Maslow, there they are. Right, so we've got two theories here which are really rather similar. We've got gravity and we've got electromagnetism. And they seem to be working in similar ways. or that at least you seem to be able to show that they could have this connection that doesn't quite say how any more about how gravity and electromagnetism are related you say alright there's similar ways well how exactly are you related and this is what one needs to do to produce that is to break The symmetry between mass and charge that holds for gravity and does not hold for electromagnetism. And this, I'm quite sure that ancient theories are not so ancient, but I'm not the first to have done this. Let's put it this way. What we say is to break that connection between mass as a ponderable matter and mass as charge, permeativity or permeability of the vacuum. And I also, for good measure, assume that mass could be negative, but that doesn't actually alter what the precautions you finally get. And what we find is that when we do this, we get a distinction between mass and charge, and we get a more general set of scaling orbits, the sort that I was showing you before, you the sphere within a sphere within a sphere, the Russian doll's thing. So the next thing I did was tackle the Feynman series using the version of QED theory in which the electron spin is a classical four vector.

40:00 And the Feynman series became a simple geometrical progression. and I think I've probably got something on the next I'll go on to that on the next side and what we find when we get this progression is that it's equal to the sum over the more general scaling orbits we found earlier in other words we've got solutions in these spheres stacking up like that and in order to do the Feynman the last bit of getting the calculations of quantum Electrodynamics correct, the Feynman series, we can do by simply summing over those orbits. But there's a caveat that I'm coming to. because I know that Pierre has also done this basically find it because this series you can sum, so you get a finite result from that before the Feynman series. And what we can do with the series, if we want to get it straight away, you can alter, say, Maxwell's equations, and so we don't have to do this sum as well. However, if we do that, remember we've got a single particle version, we can't then use canonical quantization or particle methods as this stands to get a field theory and I myself have used other ways of treating multiple particles in which we see the sequences of several orbits so just to put that in there so what Pierre has done I haven't done exactly the same things here so in the little theory I've been talking about in which you use the permittivity of vacuum, you get to the set of scaling orbits for electromagnetic ratios of mass and charge of a particle, and the charge on the source of the mass and the charge on the particle and on the source are fixed. And you do get a prediction out of the theory on that. Right, so we're now going back to the overhead I showed you when we

42:30 started and we find out we can get the masses of them particles in the standard model and they just fall out of the theories but do you put any of these pure numbers in you presumably put some of the amphotype pure numbers yes that's right yes the one I showed on the on the coupling constants of what goes in and only the coupling constant the speed of light planets constant and the gravitational concept and that's all right so I suppose you probably haven't noticed that I've said this but I did say I used the the equations of the standard model, as well as the theory of gravity. And the equations of the standard model, as I mean here, are the unbroken electroweak theory, which is an SU-2 symmetry, and the SU-3 symmetry, quantum chromodynamics. And I have, in fact, solved these equations, and I'm in the middle now, writing up some papers in which I solved these equations, to show that there's a family of solutions for QCD and the unbroken-electro-weak interaction where they're actually behaving like like quantum electrodynamics. And nobody seems to be bothering about that. It's a bit surprising because I wouldn't expect if somebody would have said I couldn't do that. However, since you don't mind, I think I might be able to. I'm deeply suspicious of the whole standard model anyway, so I don't mind what you do with it. Right. Well that's been a pity really, because I've been carefully keeping to certain sets of equations and so on. Other people in this room may take an entirely different view. Right. OK, so here's one of these series that I've been talking about. I don't know if that's visible at the back, possibly not, it's rather small, it's 14 point, which probably isn't.

45:00 but here at the top here what I've got are two things they're going to interact as an inverse square law once they're going to mutually go around each other and this table is going to apply to them and this table is Bohr's equations basically quite simply so here we've got the source and the charge and the mass of the source are the same we haven't broken them at all using our intermeability theory of the lacuum. Next down, we have a series of parties. So if you want sort of one instance of it, you've got to take one level, which refers to one place, one position in the set of Russian dolls. So we take the minus one level, so as we are, and then we read down like that. And that's what we get for the whole, that's how these things fit together. a parameter there, three parameters there NS and the other we've got alpha well that can be the fine structure constant but it could in fact be the electro weak one or it could in fact be the strong one or it could in fact be the gravitational one and now we've got another constant in there that we can set any way we like and that's W but when we get down here able to get to the particle charge and the particle mass, if you look at those two, they've got the same constants in, and there's a rather strict way in which the two relate. You do redistribute the charge so does everything have an equal charge at the end of the day, but there's a definite prediction there that you can have one third, but you can have one to two and you've got the velocity we won't bother with what the winding number is the excitation level for the Bohr orbit is the same in all them at all levels and we've got the radius that's the same and Planck's constant and Planck's constant varies with level so at one particular level it's constant just as but it can vary with level. So that pretty well summarizes the whole of it. Isn't that last fact rather a worry, one about transconsum?

47:30 Well, these self-scaling orbits have not been used before by anybody. Okay, so, I see, perhaps I slightly... suggesting that there will be situations in which people will find things constant is different. I agree with that. There's one already that, I mean, if you take the solar system, or the orbits, moons orbiting as far as et cetera, then there's more than one reference for this spalter is in my papers I don't use it very much but since you bring up that particular issue it solar system works as though it's a an approximate ball system and this is because well the studies are quite massive as compared to the planets and this inverse square pretty good well yes the inverse square obviously it's going to get there with that it also gets there with the other equation another one of Bohr's equations which the way Bohr originally had it and he put it in terms of angular momentum which you could only have angular momentum in terms of integer numbers of Planck's constant and you can do that for the solar system, and basically we're sort of thinking of just the sun and one other planets, and off we go. And we found that it approximately obeys this relationship. And it's not exact, as some are normally is, but there's a huge volume of evidence that that's going on. I'm not quite clear what it is you're saying is going on. If I did a bore for the sun and the earth, for example, all I could get in the end is energy levels out of it, which would be very close together. and it wouldn't say anything about H because H was introduced by saying the angular momentum is NH where N is the quantum number, the principal one and you crank the algebra and you get E is proportional to 1 over N squared

50:00 where N is the quantum number and the constant proportionality depends on the mass and the charge, and in this case it would be sort of GMM instead of charge squared, and Planck's constant. Now, how is it that you've got evidence that Planck's constant is different? Well, because it's not different for different planets. Once you say it's the same planets, you can start saying, well, is this planet fitting in level N, you know, is it the excitation level N, or is it N plus 2 or N minus 1 or whatever? And you can then start fitting it and see whether there's any tendons. But aren't the energy levels fantastically close together, so there's no chance of measuring them? What would you take as the energy? We're going to have the sun fixed in the middle. I mean, you know, the kinetic energy of the kinetic and potential energy of the planet. I think, Tony, you have to review what you mean by Planck's concept. Whether you're hanging on to the current meaning or whether in the solar system model there's like an analogy in which Planck's concept prime or something has a new concept. Oh, well, if you're going to do that, that's fine. But what I'm saying is if the Planck's constant is the one we know and love, then, for example, to get the angular momentum of the Earth around the Sun, I have a simply fantastically high quantum number. It's a very big quantum number. I don't see the problem, you work it out. So if it's a very big quantum number, okay, like one with umpteen nobs, the energy level changes at what you actually measure, and so you'd be saying it's Rydberg or something like Rydberg over n squared, sorry, 1 over n1 squared minus 1 over n2 squared, where the n's are ginormous numbers, this is the point. So that you can't... No, they're not. I mean, Planck's constant,

52:30 well, the thing that takes on the role of Planck's constant is not small. Ah, yeah, if something else takes on the role of Planck's constant, I... Oh, I see. Well, that's what I'm saying, because the question was whether it wasn't worrying that it varied. I might just spin on in the other way. And it does... No, it's a different theory. It's got nothing to do with Planck's constant, really. I mean, it's formally the same, but the thing that's called Planck's constant is a quite different number. Yes, what I'm saying is that the solar system is approximately like the system ones I'm talking about here and the ones I talk about to find the masses for the particles, which is Bohr's equations all over again. I put up this other overhead that I showed you earlier. They come out of Bohr's equations. So you have, actually I don't mean Pakes Constantly, I think they're particularly simple. So you have that constant which applies to those, and another one which applies to the solar system, because it's a system at a certain level. That's all I'm saying. You know, it's just a question of whether that's... Yeah, but you mustn't call it Planck's constant. You can call it a unit of angular momentum or something. I don't know. I can see your point. I haven't, you know, sort of invented a name for it. Well, if Plank's constant is that 6.6 times 10 to the minus, Humpty Dumpty, of those funny units. Which I never remember. Is that the word? I'd just like to point out that you and Tony have had 55 minutes so far. And how much longer you want to go on for? As far as I'm concerned, you can just continue. No, well, I have no fixed ideas about it. I suppose perhaps what I'll do is just put this up again and, um, ah, right. You'll see if you look at it that they're all in two bits. I mean, you've got, well, for mu, you've got 3 over 2 alpha, that's the fine structure constant, And there's another bit, which is 1 plus 4 over nu squared, which is the weak interaction member of, it's not exactly that, it's got a fractional part, but that's how I'm covering it in my next, how you get that in the next talk.

55:00 but basically that's just if you take the hydrogen atom as modelled by two point particles you've got principal energy levels and then you've got the fine structure and that is, they all take this form apart from the pile itself I only have one for that and basically it's just Bohr's equations all the way up and down all over again I wanted to revisit a question that came up earlier and elicit an opinion from you and from anyone else in the audience who might have an opinion on this, on this issue of whether mass should be considered to vary with velocity when you're speaking of a gravitational problem. Consider a star, and consider that various processes go on in it, which might change its temperature. Does the mass of the charge change because the temperature of the charge of the star changes? That's a good question. Yes. What I can say about that is that Planck's equations of QED, in the form in which I put them, which is just exactly the same form, except there's, in addition, this extra classical four-vector spin, if you want it and not if you don't want it. it's a covariant under either. With those equations, you can have either have a charge of four-vector varying with velocity, or you can have it constant. You can have either. And it's quite easy to work out two forms of the equations, in which you have one and in which you have the other. So whether you can ever, at the end of the day, say it's don't know because if you get a system in which that's all there is, or what I've described

57:30 to you is a system where that's all there is, and therefore QED, then well, either one would work. What do you think would be the answer from a person who was committed to general relativity theory? What do you think would be the answer to that question from a person who is committed to general relativity theory? Oh, well, any form of mass at all would have to be counted energy in mass. You'd have to counted E. So the answer would be yes. Well, I'm no expert, but yes, as I've said. Does anybody else want to comment on that? I don't know. I think it's a hard one. It's a hard one because you don't know how much the energy is really in the field. Where are you going to start measuring it? I mean, this is why when you put black holes together, the total mass, some of it leaks out in the form of gravitational radiation. And you can't stop that happening. So, some wise acres would say, ah, well, it's the mass of the two stars separately, say, if one star's going down to another. But that isn't quite equal to the total mass of the system, because some of it's tied up in the field. I think you'd have to say it changed. Oh, actually, can I say just one extra thing? I'm a copy of the papers there are some mum which I put down here sure um there's some Seasons there they have the papers on them and they have as well the two draft copies I'm working on at the moment so no guarantees for that but if you want those and pick up one of those any more questions isn't there a conundrum about this you were going to talk about the in relation to this here is that a second talk that's another talk

1:00:00 yes well I asked at the beginning which one you wanted first and it turned out to be the first one which is very reasonable Yes, the second one, I'm doing that. It's on Tuesday. Yeah. Sarah, thank you very much. It's the end of Sarah's talk, Sarah Bell. This is Tony Booth's talk on dynamics of complex waves. Okay, I've got some gear, as yesterday, the bullet point list page, and if I'm going to pass these around, anybody's wanting one, and the sets of equations, the main one is the dynamics of complex waves, of course, but I've got the spin one, which I think I should be mentioning today, so you might as well have the equations for it so that I can refer to it. The only reason for passing them around like this is so that if you can't see them up there, you can locate the same thing in front of you, probably, and we can at least talk with some sense of what's going on. Because I've got them here as well, in the overheads. Let me start with this. It's a shame I've done that on the laser, and it's a bit dark. Still, we've got the copies of it, so you can see roughly what we're doing. Right, so general background introduction at the front here. First of all, the confusion of last night was meant to take you through a bottom-up appreciation that there are some pretty straightforward continuum models that will yield an emergent quantization. By that I mean quantization occurs in the solution space of the continuum model which we talk about. And you don't have to be too outlandish to get that to happen, which I believe is contrary to general belief anyway. It's certainly worth thinking about and see what it is. You don't have to force the non-linearities in and make it all happen. It's just an ordinary Minkowski space and some equations on it. But you have boundary conditions and they... No, we don't. You don't? No, that was self-quantizing. That's the point.

1:02:30 The self-quantizing occurred by... Oh, this is the diagram. It's by those inner loops. It's like a distributed self-generating boundary condition. through how that happens you're integrating a field that is derived from the charge and it forms a boundary and it causes a convergence of the periphery and i'll only do it for certain control values this thing exists in space right yeah and therefore you have to say what it does as as you move away from the center of it yeah does it does it fade away like the old soldier It decays away, the same as any sort of... Yes, well, if you impose, with most of these wave equations, if you impose those soft conditions... We don't impose it. Things happen. All we impose is that certain phase conditions will exist in the periphery and at the center. The center's got to be a zero of some sort, and at the periphery, it's got to converge so you get finite energy. And that will only happen when the integral is reached, the remodulating integral has reached pi, your multiple of pi. And when you use that, it produces a value for the pi. I'm not surprised. I mean, you're putting nodes in it, so it's bound to. I don't see I'm putting them in. These are just the ordinary solutions of a Klein-Gordon equation. I didn't generate that. I just look at the thing and say, look, it's got these solutions. I didn't put them in any more than you put the zeros in the position of a pendulum when it sweetens, you know. Yes, and so the solutions have three parameters in which, only when you put some sort of boundary condition, not necessarily the one I stated, some sort, either implicitly or explicitly. I'll take it up with you later. These are the conditions. I know about the Klein-Gordon equation, and I'm speaking fact. Yes, but so am I. You've got H equals 1, C equals 1, no, H bar, actually, C equals 1. Omega C, that's the Compton frequency, equals 1, and the other ones, the impedance of free space equals 1. With those all equal to 1, you solve that Klein-Gordon equation with those loops back over them, as shown on that diagram, and it produces, by the ordinary approximation I was using, within 1% of the right fine structure constant. And it produces a quantization. That's the point. Well, then implicit in that are some boundaries. Now, I'll take it up a little later.

1:05:00 I can't do it now, but that is not so. I'm just trying to understand the way you're speaking. You would say that a dynamical system is very quantized if it exhibited quantum behavior, like quantized energy levels. Yes, it's only got certain eigenmo... Or otherwise discrete behavior coming out of it. Yes, it will only sit steady state in certain... It's like the click of a cam top, you know? It'll only go to certain states and be happy about it. Yeah, right. So some things which we wouldn't necessarily in our more academic way of talking call quantum would fall into this like solitons. Absolutely. Indeed, you can hardly do it without... It's a very nice general question to ask. When does some dynamical system exhibit quantum behavior in that? Yes, that's right. And I've just gone for the most stripped-down version I could and traced it through. And lo and behold, if you come up, you see you've got the quadratic effect of the intensity coming out into the excitation of the field, the electromagnetic field. And that multiplies in the coefficients of the Klein-Gordon equation. So you've got a cubic system, which you need for the soliton. And since it's a real ripple integral, it will only solve on pi intervals. And then you estimate the charge delimited by such two zero surfaces, and it comes out to be close to the fine-structured constant. Well, the electric charge. Yeah. And that's all contained in what's unitalist as far as I'm assuming. Just take the one for everything. And that's where we started from yesterday. And that's what I call a bottom-up argument, because what I did was just put in the simplest equations I could get. The hardest one was getting these intensities to produce the electric charge and the current. That required the Hilbert transform, still with unit coefficients all the time. You put anything in there, I just used the simplest form with unit coefficients that would give us a proportionality in the charge between frequency and this charge-energy ratio. So you get the plant relationship out of it. We know we want that, so I go looking for a wave arrangement that would yield that. we can call synthetic. So today's effort now is to say, having seen those equations and gazed at them for some hours, days, and maybe weeks, you then start to say, can you justify

1:07:30 this in any way? Can you pull it together? There's two big reasons for that. One is, I think I mentioned them up here, there's two equations in use, and they both show the same constant c. One explicitly is the velocity in the Maxwell equations, and the other one the geometric mean of group and phase velocities in the in the Klein-Gordon equation those I'm just having to put them in with the same velocity one is in the normalized units but the question is if you have a structure in which two independent parts of your definition I'm having to work with the same constant implies that there's something underneath that very strongly implies that something underneath it well I felt that anyway Occam's razor says, you'd better find where it's coming from. So I start playing around, looking for it. The other one is that you just look at the Klein-Gordon and out of idle curiosity say, what about if the dispersion constant, which is one in this, was a function of space? And you say, I tilt it up a bit and I immediately get what seems like a negative gravity. It twists the phase. Function of space, now what do you mean by that? The one in the Klein-Gordon equation is instead just K of X and T. Just x, if you like, if you want to keep a steady state thing. You just make it a function of space-time, and then ask what it does. And it twists the phases, and the twist gives you phase velocity, and that makes acceleration. You say, well, I guarantee jumps off the page at you then. And you say, if there were something in there that were determining that constant, then it could be involved in gravity. What I find is, the higher you make that constant, the more it repels from the hilltops. it's only the dips in that one that constant as it comes down that creates gravity wells it twists the phase the right way so it's as though there's something noisy behind this system that is creating the one in the Klein Gordon and if you reduce it like a shadow in it you'll get gravity into that well now that's good structure interesting structure we go on from there and start trying to of it, that broadly visible, I then have to say, whatever this background thing is, it's got to create a one from consistently, from when you've got ordinary Maxwell equations around as well. So I start looking at the zero D'Alembertian equation. That's how it feels as though we're dealing with more like the scalar equation producing the Maxwell than the Maxwell producing

1:10:00 the scalar. I've seen lots of people trying to make Maxwell equations be the solution for everything. It doesn't feel like that to me. It feels as though the scalar's got the guts in it. It's got the awkwardness, and we can get the limit cases out of that to produce the Maxwell force, perhaps. So I start dreaming around, just do the 0 d'Alembertian, and then, of course, you get this very empty solution space. It just has got orthogonal waves at the speed of light. But you say, we know there's trouble with spin, and we start saying, is there any way that you can get non-orthogonality? Well, if you get singularities in the solution, then you could have interaction anyway. You've just got to get the integral line to go around it. You mean you can get the singularities to interact? You could get the fields of which the singularity, yes. You can get interaction off the... It's really all one field, but you can put as many singularities in it as you like. But you can now talk in a fairly sensible partition basis about the plane waves, which you've got the non-interacting and the zero-centred, or the singularity-centred waves, and you start finding the plane waves interact with the singularity-centred waves, and of course the singularity-centred waves interact like crazy. They really don't like each other. Because that's a model of particles. Well, of course, but we got to it by ordinary extension of the solution space. the plane waves worry me a bit because they don't have any boundary they can go anywhere and fill up they're generated somewhere by God and absorbed somewhere else by the devil but what are they doing for you they are forming the stress which when interacting when you look at the oh yeah I haven't quite got to the end of that In order to support the spin-centered, the singularity-centered, self-delimiting wave modality, you actually have to have the interaction from the noise. There is no solution which is freestanding. You can't get a solution, even with singularities. But if you put an invasive noise in, then you can get a stochastic solution. You get a quivering solution, which all the time, because of the noise bombing through it, actually vibrates around a certain standard set of forms.

1:12:30 What do you mean when you say you can't get a solution otherwise? It doesn't solve the basic equation of the zero-down version that we're using. But if you put the noise in and postulate the singularity, then you can get a local solution, because the interaction with the noise, there are certain conditions where the interaction from the noise to the local solution produces scattering that completes the model. Now, there aren't many things that can do that, but there's a certain combination of two overlaid modes that will, as a factor in any solution. There is a philosophical problem here. There's more than one. Yeah, but there's one that you need to look at, and perhaps you have looked at. If you put singularities in, or you admit singularities into any wave equation, then you can't say, or you perhaps can say, but you have to test it, that this is a quantum mechanical problem, that this wave is the state function or something. Oh, no, we wouldn't do that. Hang on a minute. Well, all right. Yeah, if you put a singularity in psi, then psi star psi will in general also have a singularity. There's a line on which is a singularity. Now, okay, that's fine, providing it's normalisable. Well, if it isn't normalisable, then you're up against the people who say, oh, well, you aren't doing conventional quantum. There are plenty of problems around here, but it's all right, that level of it will come together. the singularity is a line singularity I fought with it for quite a while to try and see a point singularity it wouldn't come with very big long expressions trying to make them balance up but it just won't and then suddenly you put a line singularity and it does they're not spherical or cylindrical it's a cylindrical line singularity that you have to have at the centre well cylindrical except that it tables away in the intensity towards the ends but there's a cylindrical axis that's the one that does it you could then find you've got two-dimensional nature near the axis which merges out to three-dimensional now that's a tricky business merging out to that three-dimensional but this this is all when you say it goes out to three-dimensional you're saying that it's more

1:15:00 like a spherical distribution are you more like yes the distance for instance charge becomes if you had a single charge it would become far away from a line of charge at least the electric field does that it's not so easy gravity does that and various things do that so that's what you're saying when you say it moves out, when you move out it gets 3D. It's that type of solution now that last bit I've just given you I only raised it in passing because it's a tricky the important bit is the cylindrical bit the other middle what i would like to know is what and it might be very difficult to find it is what are the sort of classical dynamics of these point particles that are represented by the singularity that's not the approach how do they move yeah i don't even touch that now what you do is talk of a field and you talk of a singular yeah i know you don't touch it What I'm saying is that if you did touch it, you might have more support for what you're doing. That's an interesting thought. But I'm fairly diehard on this because I'm predisposed to continuum analysis and I'm intent on getting a... If I'm going to get any solution, it's going to come from the continuum and it's going to yield to quantization and things that go on. But as you have said, without singularities, it's not interesting. Well, it can't be done, that's all. Well, I mean, there are... Yeah, you can do it with elaborate equations. but not with these simple things. Well, yes, but I mean the Klein-Gordon or just the de Lambertian wave equation, you can put plane waves in it, for example, and it will satisfy, but the plane waves don't mean anything. There's no quantization. That's because there aren't any boundary conditions. Well, you can put it that way, but I say it's because there's no self-integrating quantizing mechanism. That's just a different way of looking. boundary conditions. But the singularities are boundary conditions. They are very queer boundary conditions, but they are, and that's why you get quantization. I do talk about the central boundary, if you like, and the peripheral boundary, but look, let's carry on down this road. Let's just go down this road. Okay, I've given you the background of where we came from, and I've given you these two challenges as to why you'd want a metamodel, because the synthetic model that seems to be so suggestive just needs its bits

1:17:30 So I start then, and to my amazement, I find if I use just a zero-dumperance here, you know, just play with this and start working out what it might mean. Now, use the same methods with this, more so, you might say, as I was using previously. We've just re-expressed this as something that looks like... So, just pull it apart with a couple of differentiations, and then the next thing you do is, this is pure Minkowski, you know, minus one, one, one, one matrix, and then we start splitting this apart by transforming it, we're going to do quite a lot of that, and put a matrix in here. This is just the same thing seen as in the frequency space instead of time space. Nothing very special about all that, but now what we've got to do is... We've talked about the singularities now. I could give you a hint of what the singularities are. Would anybody be interested to know? They're not difficult at all. Okay, let's just sketch them up a bit. From memory, I've put slip this. You've probably got some equations there. Basically, what we're going to do is have a singularity with some sort of, we'll consider in the azimuthal plane, this is the axis, the main axis of some atomic structure, and we're considering of one electron mode, we consider an azimuthal we will also consider a radial function now we've got to have a zero at the middle but we'll be considering only a factor at the moment it's a factor which has to be multiplied into a spinless harmonic spherical harmonic form the spherical harmonic will have first order zero at the origin, it won't get away with anything more than that it will have convergence with finite energy at infinity as well and we'll now play around with a factor that we can get away with putting on that and still meet the finite energy requirements.

1:20:00 The most important part is to understand the function of this angle theta here that the wave function takes around this circle. What we need is the ability to get a half order solution function and you do that but first of all you spoil this game by putting a square root function there reciprocal square root as the radial modulator that spoils the that gives you a bias all the time in the in the balance of the terms that you're going to be working with now all right so it's a half order infinity but it's at the point of a first order zero so we get only a half order zero left in the solutions that we will get at the end. I believe this is right anyway because when you look at functions they do tend to show that cusp of the centre which has got that square root shape but this is producing that anyway having put this radial biotin the phase around here now has to compensate that to be harmonic by a half turn now you can't have a half turn and get back do something else at the same time so we can think of this as putting together two square root solutions and the square root solution gives you this looks like with unity and you get this sort of it's a clamshell which, because the square root gets bigger and bigger and smaller and smaller and smaller and further away that's the real part of the square root of the x plus j y this is x here, this is y here we take square root x plus j y, it's a clamshell and normally it's imaginary part, it's the same thing looking the other way I can probably just do it and just figure like that and they should cross on this line here so you've got these two clamshells which point lots of directions and if you just add them together it's a square root function but if you consider a function where one of these has actually got the opposite direction of twist this is a pretty nasty thing now

1:22:30 it's not just an ordinary square root function in fact it's two disjoint parts then you start to get something which is realisable, would sit on this, and will still satisfy this thing here. It's that sort of concoction you have to put together. Now, there's about a dozen ways of doing this. That's one way of doing it. Another way is to... Let's just... We can consider on this circle that two functions can be problem. One is constant all the way around it, And you put cos omega t for this function. So the whole thing is doing a circular complex number. Yeah, that's right, cos. It's just doing oscillation. And then we put another term with it, which is tilted, and this one is j sine theta sine omega t. If you put those two together, you've got one oscillating so, and the other one, with a complex argument around it, is moving in the sine phase. And you'll find if you put these two terms together, you can make it so that the magnitude is the same everywhere, but there's a half twist in both of them. So now we can get it to solve against this. This is another way of looking at the same construction. In fact, there are other ways of constructing. You can just algebraically transform this thing to any of a good half dozen. So it's got a singularity in the middle. The singularity is actually a zero. And in conventional speak, what we would say is, well, there's a particle in the middle there, and that's the source of attraction. That would be, yes. Pulling another particle around. intent upon describing things in particles, yes you would, but this is a continuum based thing from which particle is now emerging. Well that's right, I want it to emerge. So the other particle, the one that's going around in the circle, that's going to have to produce, well, this is the electron, we're talking about the electron. Yes, okay, but you're not talking about one electron, No, this could be, they're all superposed on the same axis for a given atom. This is the image in the middle of the atom. Yes, so that's the centre of the atom which you presume is charged.

1:25:00 How many charges there are? No, not at all. Charge comes from the square intensity of this thing times its frequency, as I was saying yesterday. This is the trouble, your language isn't tying up with any thing conventional. I don't know what you mean. No, indeed, but I did say yesterday, but I'll have to remind you, because I apparently covered too much practice today. Charge is derived by the square in the magnitude and multiplying by the frequency. Energy is obtained by square in the amplitude. This is a complex square, you know, and multiplying by the square of the frequency. And the ratio of those two is Planck's constant. That's proportionality. Well, I agree about the energy business. I don't agree about the charge. It's not a matter of agreeing now. It's a matter of going through this model. We can model like this. I'm not just asserting a model and calculating what it produces. Okay. So now, this is a type of function. That's one way of thinking, a fairly easy way. You can see it's got this curious constant intensity, yet the ability to have a half turn in it. Now, that is simply a factor. It's a twist factor. and you can in fact, yeah, that's right, it's a twist back to you what you do is apply that to a spinless wave, in fact it's better if I divide those by two to make it more apparent you then take some spinless wave and multiply this into it and the spinless wave has got first order zero at the middle which means you end up with this half order pole producing a half order zero in the middle and that's the most important thing, you can solve centre and show that the modes are like in that it's satisfying that that dictates all of this. Once you've satisfied that centre all you've got to be left with is enough convergence in the far field that you get finite energy and as it happens we push the charge nearer the centre so we get better convergence at infinity with this You've got to be able to normalise the various things that you're interested in that actually come out to fix values Okay, but you can substitute in all this. They might be quantized values. I mean, they might not be one value. There's no quantized values at this level. The solutions of these will only be... The point is, this is only the spin factor.

1:27:30 To get quantization, you must solve the ripple factor, which is not part of the spin factor. It's in the other stuff from yesterday. And it's that which converges to infinity. That will only quantize. But you can't have any stable, persistent matter singularities at the centre. This is the spin singularity, and you'll find if you work out the current in it, and the energy in it, it's got actually twice the energy it should have. I've got the right way around. It's got the gyromagnetic ratio of 2. Because what you do is multiply this function onto the same frequency omega t over 2, which means now, one of the two complex terms goes to 0, and the other one goes to omega. Okay, that's the two frequencies that are present in any mode that has got this spin form in it, and now you find that produces all the charge, but oddly, the current, we think of current as something moving, but in fact, if you look at the original equation, it's only a tangential derivative, it all divides out and in fact the static term produces a current effect as well as this so you get a result that comes out, anyway you have to multiply it through and you'll see you can do it just with a circle and you'll find that it comes out with the effect being twice what it would be for a single that all hangs together rather nicely and there's a note there the paper called topology of a spin mode or something. We can go through that. It's in its fairly early days. But you can see enough, I think, to see that that type of function can exist. In fact, no other persistent function can exist that doesn't have that. There's one or two questions left where there might be some special functions, but I haven't been able to... Well, if somebody said to me, this wave represents a spin of a particle, it doesn't matter which particle you're talking about, Then I would want to say, okay, we'll integrate over the space, we will integrate so as to calculate the expectation of some kind of operator which represents an angular momentum. It should come out to H over 2 or something like that.

1:30:00 I think we'll get that all right. I haven't looked that way, but all I've done is calculated the ratio of the energy to the current moment. And that comes out of the two. What I'm telling you is that unless you do things like that to sort of shore it up, you can't communicate with people who work. What makes you think I haven't done it? When I present the paper yesterday, nobody followed it. When I explain it like this, you tell me you can't receive it. Now you've got to go one way or the other. No, I haven't. I still don't understand some of the things you're doing, but this is better than yesterday. I didn't understand it at all. Let's carry on. I only wanted to make a slight diversion into this to show you the type of function. It's a two-component function. This is in its real-life form, this is in its factor form. If you use those functions and apply them to the spinless functions that represent pure spatial wave functions, you can get the electrons to have the right spin characteristics. There's a little doubt about that. This isn't beyond the wit of man, though it's a little bit difficult thing. It's very nice to work out what goes on in the centre. I'll do it in that paper and give you some idea of how you establish what can be done in the middle of that. Right, let's set that aside now and go back on to the original quest. the nature of the singularities we have to allow in. Right, now we're back on the quest of talking about the zero-d'Alembertian. the correlations produce the required dispersive structure? It's only if we put in, it's a matter we need to scatter plane waves to radio waves for that to happen. This prospect of coupling, this is the reasoning going through, this prospect of couplings if singularities are present. The spin structure from electrodynamics that we've already worked out, I've just been describing, fits this. But it still will only work if there's a stochastic field present. This is only a solution if you have the singularities and if you invade it with noise. It's got to be in a heavy noise field all the time. That is the mainspring of the system. Now, what do you think that noise field represents? It's just a thing I think up at the moment. We can talk about that later, but I've got nothing more to say than that that's a necessity of the model. and when you consider how simple the model is it's worth tracking it it seems to me I can tell you what energy density is it's the annihilation energy of 77 kilograms per cubic centimeter it's intense

1:32:30 that's what it was when I worked it out a year ago I hope I can get it wrong but anyway it's very intense that's the only thing we are skaters what they call skater beagles on the surface of a quiet pond with a lot of thermal energy in it little ripples just going around on the surface of it, where the acoustic fluctuation is. So you're talking about the vacuum energy, which is huge. Yes, it's huge. This is absolutely huge. It needs to be to get that constant up. Now we've got it, we can start working out how we analyze it. So we say, now we know we've got some singularity, set them aside, and start to work out how you scatter. So we take Vena correlation analysis and apply it to the force space. That gives us, now if you start bringing us some equations up, and you can see things. travelogue first. We autocorrelate the matrix autocorrelation gives a 16 matrix.