ANPA Conference 23 — Session II

0:00 the title of course mentions algebra and a lot of people look at my work and look at the algebra side and say I think that's where I'm coming from but really I'm not coming from algebra I'm using the algebra and forced into using And so, I'm kind of developing the algebra along with the physics, rather than taking algebra and applying it to physics. And I don't want people to think that's the way I work at all, apart from it. I'm more interested in symmetry. And I've always started from this symmetrical picture. And I don't want to say a lot about that, because we could get bogged down at that. But I start with a picture of these parameters, which I believe to be symmetrical, mathematical way. I've already given a talk on this in the past, which people can read up about. And there's also a copy of it on the Lannel.gov slides. I'll give references to those later. Just may I ask a stupid question, what do you mean by a synaptic group? I mean a group, a mathematical group structure. All the four groups. And by divisible, you mean discrete. Divisible and discrete, yes. I don't want to talk about that but I don't think that that answers our leftist question. Oh, that's who was happy. Was she? I don't want to get involved with that because I won't get into what I really want to talk about. I've already talked about that. I want to know what the advice is. I'll talk about that on Monday and Tuesday. Okay, well that's only fine. It is a group symmetry. A group of all the four. Those things form a group. A mathematical group. yeah a group is a symmetry it's a symmetrical concept so that we have what I really want to do is extract something from that which I extract something from that and this is what I'm extracting from that more general concept and it's this that charge and I don't just mean electric charge I mean the sources of all three non-gravitational interactions or what I've always called charge

2:30 And I've always believed those to have a quaternionic basis. And the reason, in crude terms, if you take electrostatic charges, that light charges repel, so you get the opposite sign to light masses which attract. So perhaps one can refer to one as the imaginary and one as real, if you like. So a quaternionic basis for charge, and of course four-vector basis for space-time. However, a quaternion and a four-vector are not completely symmetrical, treats the vector in a particular way. And that treatment of the vector is to make it multivariate. In other words, to have botanical ionic multiplication. And the way you can do that is by making it isomorphic with power matrices. So they are vectors, but they're a bit more than vectors. And if one does that, and people have done this, and this isn't particularly any theory of mine, this is reasonably standard in multivariate algebra. That one can derive spin from that. So spin becomes of using a Pauli matrices-type algebra for vectors. By the way, I'm always going to represent vectors by bold letters and quoted in by bold italic. And that, there, is ordinary I. Now, people like Hestervers and so on have done a lot of work on that aspect, on the spin aspect. so I'm simply there going to write down the rules for multiplication, we don't need to do any of this in detail, it's not important but I'll just show you that it happens and you can see these are power matrices rules if you multiply vector r by j you get ordinary i times k that's power matrices rules of course you can't have ordinary vectors that don't multiply, what's multivariate vectors now what? I mean at first sight, the difference between the left end column and the right end column in some multiplication with I. Yeah, I mean... Essentially, one can write down a vector of two vectors, AB, with a quaternionic type multiplication, A dot B plus I A cross B. And so when you do the units, that's what you get. Well, I'm trying to not get too cluttered up in my own mind with all these equations, but some of them we can just... Yeah, we don't

5:00 what Dirac is looking for when he induces the Pauling matrices is a unitary operator basis so unitary is the first equation that you see if you instead go for whatever Hamilton was doing and pick that negative one then you get the return but yeah it's just an embedded I in the definitions we'll get a symmetry between the full multiplication, the turning product, the vector, the full vector product. So I'm not really interested in this. What I'm interested in is that these represent charge, and mass is the real part. These represent space and time, including spin. if one, because I think these are fundamental, nothing else matters in physics but this. This is all that matters ultimately when you get down to physics. That's what I've always believed. When you start from scratch, dimensions. Where does spin come from? Well, spin... You start from scratch. I start from earlier to get that group. But I don't want to talk about that today. No, don't. But just a quick pointer. I can understand the three dimensions. But spin is something extra. No, but spin isn't if you use a multivariate version of... If you use a multivariate version, let me just write down one thing. Well, maybe you can make it very simple. Well, I can't do it simply. I can't even open this packet. Well, then we'll stop the rotation. It's Swiss. Another IQ. Thank you. What I mean is this. It's probably better to do it on oil. Those are for plastic. Peter, those are for plastic. There's my oils there. Oh, okay. I'm going to use one of these. that's a multivariate vector multiplication and when you have that extra cross product when you do your, say, Schrodinger equation something like that, you've got del squared instead of del squared you have this sense. You don't have del dot del

7:30 you have del dot del dot i del cross del and that gives you the spin turn. But that's fairly standard in people like Hestoners and so on. I'm not claiming any originality for anyone. What really interests me is that these represent charge and mass these represent space and time and that's fundamental in my work. Now quite independent of that I have developed a theory of using quaternions to represent charges, and therefore to represent quarks. Nothing to do with this direct representation here. But, at some stage, I thought, well, if these two are fundamental, what happens if one puts the two algebras together? The interesting thing is that one gets an algebra that's isomorphic to the Dirac algebra. with a 32 part algebra with a number of that's how the 32 part algebra is made up can I just ask what if it turns out what if one of those fundamentals as you call them turns out to be non-fundamental what happens to your algebra then well it doesn't affect the Dirac algebra I'm just saying I started with that concept pulling this together will get an algebra would still be valid mathematically. It's purely mathematics now. I have a question. When you write cross, before, when you wrote cross... Oh, yes, it's not cross product. You were using vector cross product. Sorry, that is not cross product. Just a second. You really meant outer product, then. Yeah, I did. And here you mean... No, just ordinary multiplication. Sorry about that. And what we'll get... I don't want to bamboozle anybody with all this. Those are the 32 terms thrashed out. for the Dirac algebra. This is very well known in terms of people's art. Yes, I'm doing it in a different way. What I'm doing is saying, I don't care what's known in algebra. I have to use this algebra to sort out my theories. If I put it together, this is what I get. And that's the way I look at it. I don't start from algebra and work through it. Now, wait a minute here. In the fifth line, the third equation, you've got there. I think you'll find they're okay. I've gone through them hundreds of times. But I don't want to talk about them.

10:00 What I want to do is this. That's all I'm interested in. Those are the gamma matrices. You can write them in that way. Of course, it's isomorphic to lots of algebraic ways of doing it. I'm not really concerned with that. I started from a physical viewpoint, and I have this before I do my gamma matrices. So the gamma matrices in terms of quaternion points? Quaternion and multivariate vectors. So the gamma 1 is like a double quaternion algebra, but I'm using vectors for one of the quaternion algebras. The gamma 1, for example, is vector i times quaternion i. The thing is that it makes it remarkably easy to do the correct equation using this. It doesn't need any matrices at all in the ordinary sense. is there somewhere there I can see I've done through all this before there's nothing this isn't new in my world that doesn't make sense I've had this for a long while if I start with e squared minus p squared minus m squared equals zero just standard relativeistic equation for energy and then just factorise it using this using what? using this quaternion vector algebra. Simply factorise it. Remember, P is multivariate, so I can multiply it by P by P and get sense from it. P squared means such. Is there a uniqueness theory? I mean, there's only one way of factorising? There isn't only one way of factorising, but they all work out the same in effect. I think Brian Poggle has into that version of the algebra. I've let him look at that side of things, so I think he's gone through that and found that all the versions look like this anyway, whichever way it's not the only way of doing it of course you could replace all the quaternins in different orders, you could replace vectors, I did previously do a version of swapping around the vectors and the quaternins and so forth there are more than one ways of doing it so it all works out the same effect what we're talking about is the different representations of the same algebra yeah, effectively, yes all the same algebra and all one needs to do is factorise it and then simply exchange E's and P's d by dt's and delts to get the three particle equations.

12:30 And so one can then choose to write it in that format and you've got your Dirac equation. Now, okay, why should you bother doing that? Well, there are lots of reasons why you should bother doing that because it... My apologies. Well, we're really pissed at you. I just mentioned your main thing. talking about your work on the So I'm just up to this stage now. The thing is that colleagues of mine who are particle physicists and quantum mechanists and all that say, can you do this, can you do that, can you do the other? And okay, one by one, I've got everything they ask me to do. And I've found out all these ways of doing it make it incredibly simple. One can do all these things. I've done all of those things with it. And I've got copies of the papers here. I've put stuff like this on the web. you can do it a lot easier and you can do things that you can't normally do I'm sorry I should have said that because we've got a square root of 0 that's effectively that object is a square root of 0 which is what we call a nilpotent it multiplies by itself you get zero and it's because it's nil-potent that it has always power now I've had a big argument with Basil Heimler over this the wave function is normalised to unity I'm sorry the wave function is not normalised to unity you can normalise it just ignore the normalising well if it's nil-potent it means it gives zero multiplying by itself no it is you don't multiply the wave function by itself you multiply the wave function by its complex conjugate multiplying by its complex conjugate not non-zero. No, not according to this. It multiplied by itself. If you multiply by its complex conjugate, you get a scalar. You don't get a non-zero scalar. So that's perfectly okay. I always thought that the term in nil-totent applied to operators anyway, not to vectors in Hilbert's place or whatever. It's a fairly general term. You put the interpretive side of the way it has done or as conjugate of both mean. Yes, you have an operator that is zero,

15:00 but you can take it square root. And the square roots are nil-totent operators. Well, that's fine. That's what I've done. But you have to agree that it's a gross line rule. Corresponding to what? If it's square root zero. Yeah. All right, so it's not a member of a different algebra. That's not true. What I'm interested in is this. So this, in some sense, is self-orthogamous? What I'm interested in this. Thanks to the new vector of zero length. It still has a direction, and its length is actually zero. So that's all for probability. I think there's a conversation going on here that should go on. It probably shouldn't go on the right. I do think it should go on. discuss that afterwards because I want to get on to the main point. One has to accept all this before you get on to the main point. Now this is very significant. I'll use Brian's term anticomputing penta for the Dirac thing. There are five terms in it because we have a vector here. These are the things in front of the I's and P's and E's and P's and M's in the Dirac expression that we use. In fact, I should say here that In this version, I've multiplied the brackets by an extra ordinary I, complex I, just to make it simpler. But that's just a pure scalar factor. So really, these are the things that one has. I started with these eight operators, unit operators, and I've ended up with those five. The thing is that these eight operators can generate the whole of the algebra, 32 parts, multiply them all together, get the whole algebra. And so also can these five. Remind us what we saw on something before. Bold italic means quaternion. Bold means vector. Do you want me to write it down somewhere? No, he already said it. I already said it, but people might forget. I'll write it down here, and then people can look. So bold is vector. Bold italic is quaternion. Square root of minus one. And italic is complex on realness. Make them all the same, but they're going to be more in the series. How can I make them all the same?

17:30 I feel like they're all physically different, yeah. So, if we have eight units here, and those can generate the algebra, also, because I saw that penta, and Brian's been looking at this, and I think he said that it's got to be in some kind of penta. Well, I mean, you only need four. Yeah. In both of them, because I and J create K. That's true, but but to get the pentad, you need 5. Right, for the pentad, you need 5, and the pentad is the most general you can have for a nil-potent vector, as you're defining it. If you wrote this all out purely in terms of Clifford algebras, then you really wouldn't need... Well, I've got a reason for not writing it. Yeah, right, but I just want to make sure I'm tracking you. I mean, you don't have to distinguish between vector, quaternion. But I do want to, because I've physically... Right, I just want to make sure that I'm tracking what's going on. I'm not complaining about what I'm doing. Well, you could write it as Clifford Algebra, but then you wouldn't understand the physics that I'm getting at, which is that these are physically different objects. Well, I'm not sure I agree with that statement, but I just want to make sure I am tracking correctly what you're saying. You could write all this out directly in terms of Clifford Algebra, and you would not have to really, in that sense, distinguish. distinguishing by the grade of things. But I've never found any version in Clifford Algebra which can explain so much so easily as this way of doing it. Can I make like a funny comment? I think that what you plan to do here to marry is for instance the approach that the gifts you have with the leptures and then Harald. Now what you're claiming is that the marriage of Harald It does not produce color. Well, it is, but not for the physics. No, it's not the same as physically it's different. These are physically different objects. I've never had any interest in algebras. What I'm interested in is the physics. And these are the algebras that corresponds to the physics. But aren't these just tokens in your theory? No, I'm sorry, I don't want to say physics. They're physical objects. that charging and space are totally different objects and therefore have different physical properties. Different mathematical properties. That's why I asked you at the start, what if one of these turns out to be, what if two of them turn out to be combined? Even if all that was not true,

20:00 this would still be a useful and proper representation of Durant. That's all I'm talking about at present. So it doesn't even matter. If everything I said about foundations was wrong, it still would be a way of doing that kind of... So that makes it all theoretical to me. Not yet. Now I'm coming on to the physics. Okay. So now can I talk about what I think the physical significance, because I think this is an immense physical significance, this fact that one can do this. That those eight and those five are the same in generating things. In other words, we can kind of compress the eight down to five and still get the anti-commuting pentad, the anti-commuting operation. We don't need the 8 in principle to start from. And let me see what that means. How do we actually do it? Well, that's how we actually do it. I say, is there a physical significance? Is this just something that's useful for direct algebra, or is there something physically significant in it? Because I've started from physical motivation. Can I go for physical meaning for this? Well, this is what I start with. This is the representation I started with. Complex for time, vector for space, scalar for mass, and quaternum for charge. Now, kind of remove charge from that, and superimpose it on the other remaining things. What one gets is these things here, not like space, they're not like time, they're not like mass. Charges are conserved quantities, and they're also discrete objects. They're also quantized objects. time in the first instance is neither of those things, for example. What happens when we do that? By charge, you mean just electromagnetic? No, I mean all three. I always mean that. I always mean all three. getting, in the last line, the second line, you have the J occurs again. I've switched them around for a deliberate reason. I could have put IJK there, but I've just it's convenient to me to write it for K-I-J. It doesn't make any difference. Am I going to be confused?

22:30 Maybe you are, but I can't do anything about it. Yes, you can, because you're speaking. The I has also slipped down, but they have different meanings. three here, I've taken these three here and put them under those three, those five. See those five there? Yes. I've taken those three and put them under, I've just chained the order, which doesn't matter, it's arbitrary. They're just labels, K, I, and J, labels, it's arbitrary, which you put where? Okay, you forget about the first I, J, Q, first top line. Yeah, but that's, that's a vector on that. I've not chained, taken, that's still there. I've removed those three. There. And I've put them under there. Now in doing that, I've now got that combination. So the k goes with the i, the quaternion i goes with and the quaternid j goes with the 1. And that's what I've got. But I also claim that I have physically altered these quantities by combining them with the charge quantity. Because I now have a quantized energy which is in the direct state, in the stationary state, is in fact also conserved. which it isn't as a classical quantity. I've also got a quantized momentum and a quantized mass, a rest mass. So now I've given those, by combining those, I've created, and I think this is what the Dirac equation is doing, it's creating those new concepts by combining those old ones. Combining the algebra, it combines the physical concepts as well to create those things. Not just energy, but Dirac energy. The energy in the Dirac state. of Dirac-esque mass. I'm also going to argue that if one looks at it from the other side, if these really do represent charges and not just quaternion operators, which is what I first was using them for, in the Dirac equation I was just using quaternion operators, the fact that I'd use it for charges was a pure coincidence. But I'm also claiming that they are really truly charge operators. and that in doing so what we are doing is also changing the nature of those

25:00 charge operators from being the same to being different in kind we are actually the Durak equation already uses a broken symmetry we had a perfect symmetry between these before we combined with those but now we've broken the symmetry of the charges so I don't think those are I think I've used those and I'll leave those to later. So I've now broken the symmetry between the charges, as well as creating this Dirac state. Because I've increasingly come to the belief that these Ki and J don't just mean Ki and J, they also, the charge concept is actually built into the Dirac equation. So how do you break the symmetry? Well, if I combine K with a pseudo-scaler, combine I with a pseudo-A with a vector, and if I combine J with a scalar, the combined objects are not the same. They're not similar. They're not part of a group. You mean you can't imagine a structure than the previous one? Yes. When I started with K and J, they were exactly the same, they were just quaternions. But when I combined one of them with a scalar, one of them with a pseudo-scalar, and one of them with a vector, I got different... Hard to understand what this means, though. I will show you what it means as we go along. When you say a break in a symmetry now, it seems to me, from your answer to our letter, that they now no longer form a group. They form a group. What does that mean? What do you mean they no longer... I would say these don't form a group. These form a different mathematical structure to them. You want us to know what breaking the symmetry means. What I mean is I've got a symmetrical charge there. I've got Kij. and j, you can't tell the difference between k, i, and j. They are quaternion objects. They're quaternion units. They're the same. k, i, and j. You mean they are invariant under S3? Yeah. They're invariant under S3. Well, these are not. When we combine them, when we combine one with a scalar, one with a pseudoscalar, and one with a vector, they become different. The combinations are different objects. And I'm saying that that broken symmetry is the same broken symmetry as we will get in particle physics. these things are just labels i, j and k could be anywhere around in the first place

27:30 well i, j and k still can be anywhere around but the combined objects are now different physical objects that in itself constitutes a broken symmetry it certainly does, it's the k, i and j the same this is something what we call Well, the commutative algebra is a version of that, but this isn't the commutative algebra. This is still non-commutative. Basically, you're breaking the symmetry of the pen tab. Basically, you're breaking the symmetry of the pen tab by defining a three-vector within the pen tab. If you have a pen tab, A, B, C, D, E, and they're all mutually then there's no way to distinguish them. But if you pick three as being part of a triplet, you've broken the symmetry. You've basically defined a subgroup within the larger group. And that breaking of the symmetry is what he's doing to define the Dirac energy, the Dirac rest mass, and the momentum. Without really understanding what you mean by symmetry in the first place, it's very difficult to know what in it are conservative IJK is all the symmetries you define 3 especially he said O3, the group O3 IJK is O3, symmetry it's no longer O3 if one combines it with some other mathematical object another way to say since you choose from a pentad tree you go from a larger group to a subgroup and this is breaking symmetry you have this group of five you have this group of five you can pick any three to be a triplet and however you pick the three breaks that symmetry of five even though the global structure of five is still there you can choose any three as being special but as soon as you define those three as being special you've broken a symmetry in that all five are now we asked Peter what he meant by he said membership of the group so now it doesn't mean that does it that's the same

30:00 I would see it now I would not see it as membership of the group that was the definition the O3 group is broken there was a group O3 so that's broken now but you said any member symmetry was and then it was symbolic. That's what I'm saying, it's broken now. It's the symmetry of symbolism. Yeah, this is... Sorry, Brian. No, but you tell me, yeah. There's different cemeteries being defined in different ways. Yeah, I'm trying to get at the definitions, but I'm sure it's more than Right, in this case, right here, when he's talking about breaking the symmetry, what he's doing is he's basically choosing an of a pentad by defining a triplet. So it's not breaking the symmetry of the group, but it's removing an arbitrariness of the pentad symmetry. Because basically, if you call it a symmetry, the pentad can be defined, you can choose anything to be A, B, C, D, and E. In any order that you want. And they're all mutually anti-commuting. And so they're order to get this structure you have to somehow define a triplet and that triplet is no longer a quaternion or pseudovector because of the extra term that is part of a pentat so you have a triplet which is related to you know a quaternion or or you know pseudovector but it's the mathematical symmetry hasn't changed but you've fixed an orientation with it It's still the Dirac Algebra the Pentad is still the Dirac Algebra and it's still part of the larger group These three physical objects are not the same as those four physical objects That's what I'm saying now So what did you say to someone who didn't understand the maths what would you tell them has happened there We picked up we said up is this way and as soon as we do that in order to be consistent we've broken the arbitrariness instead of x, y, and z z will be defined as up which is different from x and y

32:30 being horizontal because I can walk horizontally on this floor but I can't walk vertically before I could say x is horizontal, y is horizontal v is horizontal I can pick anything that I want but as soon as I've chosen the direction which is what he does by defining the triplet he's chosen which way is up and as soon as you've done that you've now lost your arbitrary nature of vector space you see I'm not sure that the man on the clap of omnibus would understand I'm sorry but this is physics I can't talk about physics I can't talk about the symmetry it's now 5 plus 1 you clearly need to go on for Yeah, can I just get on with it? What I want to ask you, Peter, is how much more time do you think you need? I really don't know. Can I have at least half an hour? Are people happy to stay that long? Yeah, I think that we should keep the discussion-type questions down. Yeah, I think, can I just get through it? Yes, that's right. Anyone who wants to leave early will just have to sneak around a bit, but I think we have to let this go on again. Right. I've hardly had five minutes to speak. Yes, yes, I realise it. No, no, no. You should speak to me. So can we continue? I'm saying, look, those three things are not mathematically the same as each other, are they? Something that's complex, time to quaternion, isn't the same as something that isn't complex, time to quaternion. Go, Peter, go. Now, what I'm saying now, this is the key point. I'm going now to identify what happened to that particular quaternion with the weak component I'm going to identify what happened to that quaternion with the strong component and I'm going to identify what happened to that one with the electromagnetic component and I'm saying that now is breaking. Let's pick an example, the strong component that's a nice easy one to do the electric and weak are harder But if I can start with one that is relatively easy. Now, some time ago, because I've been using a nilpotent to represent a wave function, or anything, whatever you want to call it, a state vector, when I think about a barium structure, how can one write a barium wave function down? If you've got a common energy, momentum, or angular momentum, mass term for the object,

35:00 in that that have those three concepts in it. Because if one has two of that are exactly the same, let me just write this down like this. As soon as I multiply that by itself, I get zero, so I can't write a bar in that way. And this was long before, well not long before, but before I was thinking in the terms of now thinking. That is zero, So you can't write it that way. Now, what I reckon is, is that you can write it that way if that P is not P, but one dimension of P. Say PX and PY, or P1 and P2. So you can write it that way if you write it like that. where your p's are actually in separate dimensional domains. Okay? So I can write p1, that is not the same as that, because at any one moment one will only have one of them. So in effect you'll have k plus i p plus ijn times k plus ijn times k plus ijn. which is non-zero. And, of course, you will get six possibilities because you've got plus and minus of P, and you've got three-dimensional things. And those six will, of course, all happen at once. Whereas if you have a free ferment, that's probably more like what it looks like. That is a representation I've thought out for a barrier of wave function before, before this. and if one relates that to more standard ways of looking at it using colour, the colour principle see I've got the z direction there I've got the x direction on this one this one down here, that's on the x direction that's on the y direction, that's on the z direction and one can arbitrarily just fix those to six anti-symmetric and six symmetric colour representations so this representation looks like a good match to what we get in standard quantum mechanical theory.

37:30 Now, this is going to be a bit more difficult. So if you find it difficult so far, this isn't going to be as easy. What do we do in particle physics to represent the strong interaction? Well, what do we do in quantum mechanics to represent P? We replace P with... dy dx or equivalent in particle physics depending on what particular force we've got we have to change that to the covariant derivative so if it's dz of course if it's electromagnetic force multiplied by ephi or ephi or ev or whatever It's the vector potential, isn't it, on that bit? Yeah. If it's electromagnetic, there's the F. So if it's strong or weak, there's something else on it. So the strong one, the covariant derivative, in standard quantum mechanical theory, is like that. It doesn't really matter too much about the details of that. The important thing is that there is this A term, Now, there's a d by dt bit, which is a scalar, and there's a d by dx, d by dz bit, which is a vector. So I've written down the vector bit here. So if I write down all of the ones for those, I will get that. This is perfectly standard theory. Nothing original about any of that. So you add a bit of a thing to represent a strong interaction for each. but it's a vector for the p's and it's a scalar for the e's now if I now rewrite my wave functions these things putting those covariant derivatives in instead of the e's and the p's I'll get something quite complicated but I'll just do it for one single term again that x is just the only multiplication the derivatives are just representations of the e's and the p's they are, that's absolutely correct that the e's and the p's are not something different. No, they're exactly the same thing. The e's and the p's are just... They're either operators or their eigenvalues after applying the operators.

40:00 Now, this looks complicated, but what I want you to look at is that. That bit there. Which button? You're shaking all over the place. I'll just try to find it. That bit there. because this is the same thing as having your P there rather than and not in that one. In the last two brackets, you see, that term doesn't exist. Times, of course, is ordinary multiplication. It's not vector product. So in those two brackets, that one and that one, these two aren't there. But in that one, it is there. That's the same as having P there. Can I summarise this process in what it seems to be saying to me that by taking elements from the algebra you started with and multiplying through these compound operations you reorder things not only algebraically speaking but that comes in front of that or whatever you also reorder the nature of the beast. Is that what you're doing? You understand the nature of the beast. I think you don't change it, but you get a different understanding. This new pattern that I get by modifying through is in 1-1 with some physical object that I know about. Yeah, I'm going to talk about that now. Because imagine that that A thing there, the vector A bit, actually change from the exposition in the first bracket to the second bracket and to the third bracket. In other words, you're getting that strong component is actually being transferred to the other brackets. If I wrote down all the other possible states, that A thing, the strong interaction thing, wouldn't be in that bracket, it would be in one of the other two brackets. In other words, if one wrote down all the possible states for the strong interaction, one would write down all the ways where that A could be. So, to put it simplistic, it's like you're transferring that package of strong interacting information from one bracket to the next to the next at a constant rate at a non-local you do this because you can because it can be done you say well it's the straightforward consequence nothing's guiding you here you say you've got no precept as to where you're going as you say I'm doing it because I can

42:30 because it falls into that But long ago, I've had this representation of the strong interaction using those tables of an S-charge transferring between three representations of those tables. And what I'm saying is that this is the same manifestation put in a more quantum mechanical way based on Dirac algebra. And if you like, because that is a constant rate of change angular momentum. In other words, it's going with the spin direction. It's going with the XYZ thing of the spin. The strong interaction is going with that. Which is why I'm claiming that the P and the S map onto each other. The P in that Dirac algebra is in fact coming from the strong interaction. The directional aspect, the explicit use of a directional It's the same thing as explicit use of a strong interaction. And if one takes that as a constant... Well, just let me finish this. I wasn't supposed to have any interruptions, was I? If one has a constant rate of change of angular momentum, what's that? That's a constant force. What's a constant force? A linear potential. And what do we get in the strong interaction? A linear potential. If one then puts the linear potential back into the quantum mechanical expression that I use, one can actually derive the full strong interaction potential. And no nonsense about using lattice gauge or any of that. It's automatic. It has to be. We want to go. And we've agreed not to have any more questions. So try to contain yourself until after he's done. We have. Yeah, you have. Everybody has, except you, apparently. Now, if I were to do the electra a week, that's a lot harder, but I can also do that. And I don't think I want to do that, I think people have got to read that privately. What I would like to say is this, that that is now what I'm planning. That that mapping is in fact representing something of that nature. it should be of interest to you because it's the angle of momentum carrying the information

45:00 it's a great interest to me yeah it's alright it's also a great it's a bit as well I would think now I can't I would rather let me just put the tables that I've always used like this just ignore the B and E bit just look at the A, B and C because the thing that's happening here And the S is in a different place on each. And that to me is the same thing as that A changing over in the wave function term. Now I've actually thrashed through the electron weave as well. And it's a lot more difficult. So can I just leave that out and just go to some conclusions? so the constellation properties of W and E charges are determined by the algorithm and the momentum operator just the same way as the S is I could show that in the most simple case it's harder to do for the other two but it can be known it's the combination of W and E that affects P two side options for IE because the algorithm demands complexification that's already built into the algebra it's IE before we start True to speak, you're P, E, and N are in one of the algebers. Yeah, they're in the Dirac algebra. P, E, and N are what I have in the Dirac algebra. Yeah, K, E plus I, I, P plus I, N. But it's really, I should write it properly. Let me write that way. And let me underline that in that way to be a quatern. It should be really done that way. I normally multiply by another complex item to make it simple. But the complexification comes from I, E. the complexification comes from IE and I've discussed this in previous antlers only the positive solution is physically meaningful and we compensate by creating a full weak vacuum that leads to violation of charge conjugation symmetry, sorry I'll have to just summarise this very quickly I can't really thoroughly explain it in the time available and when we get time reversal parity as a consequence of that we get all this kind of thing only one state of electricity

47:30 in principle hence the introduction of mass because of the heat-p mixing and so forth I'll just have to skip through all that because I would have to explain the lecture will be for you if I were to do that so I'm saying the creation of the Dirac state breaks the symmetry between the three interactions it determines the distribution of units of charge in both quarks and leptons but what I say is this that the lepton one is quite easy but if one wants to transfer it to quarks you have to realise that a particle doesn't know whether it's a quark or a lepton except through the strong interaction it can't tell which it is so we can do it that way it's the transfer of the momentum or angle of momentum between the three components of the Bayer-Way function It's identical to the transfer of people to a strong charge. That's what I explained that. I'll have to be very brief. I'll skip on a bit. I'll just say that one can actually put down specific structures for all of them. So go away. It's not going anywhere. Just to confuse you. The symmetry break is here. Just to confuse you. Now, this is the first model I had. If one wanted to represent quarks, you did it using R1, R2, R3, and these were sort of angular momentum vectors, if you like. Random ones. And if you wanted to change that to a lepton state, you made all your R's the same, and you eliminated the one concerned with the direction aspect. I was able then to produce a sorry I can't really explain this thoroughly but you might be interested in that I was able to get it into a single expression finally this is very interesting because one can see how one can get mass from it in principle if you've got the symmetry breaking you start with, that's the same as the Higgs principle. You get sigma z has only one sign, or the whole thing only has one sign. Sigma z is the angle of momentum orientation. I don't see any equation. Is there a people sign in there? Exactly, I don't see any equation.

50:00 Zero. What is zero? Well, that top line is... It's just a generator for particles. If one does this, I won't say how many E's there are W's there are, how many S's there are you apply that to the state vector to get different other state vectors which represent other particles is that what you're saying? well, no, it's the charge structure how many electrons what do you do with this thing up here? well, what one can do here is, for example if one makes see, these are each units of quantized angle and they're selected randomly if you make them both the same you make them, you see this thing here, because you've got a chronicle delta here, if B and C are the same, then this term will disappear. This is the term that... Sorry, I understand all that. What I don't understand is this. That thing that you write down first, that complex formula there, is an operator, right? It doesn't operate on anything except itself. But what one gets is something like this. hang on, you can't do that that isn't in the mathematics it's got to act on something no, it doesn't act on anything it's not an operator it's just an expression you call it an equation no, no, it's not an equation I agree it's not an equation it's an expression it's a summary of everything that's going on it's an operator well alright isn't it supposed to be a Nelton no, it's not a Nelton I think it's a creation operator. Actually, it is a creation operator. He applies this thing in its various forms to a state which represents one of the particle states and it produces one of the other particle states. You see what I mean? And if this thing's got so many forms, he can do all of them. But can I talk about math? Because that's what interests me. If we want to talk about a film, start off with the concept of a full mapping I've gone into that theory before and we're talking about a preferential treatment of the concept of the spin vector for example the old thing about teaching I was only being left handed and so forth

52:30 so we're talking about preferential left handedness in the grand state if you want to get the opposite handedness as the well-known argument goes need to introduce mass. So if we start with this being one-handed, then anything that changes the status of the sign of that, that reduces that in any way. So if we zero any of these terms, or we make them negative when they should be positive, then we will that necessarily introduces mass, because you are effectively changing the handedness state of this object. and you can see the various ways in which mass can be introduced this one for example is how one can get masses of bosons and mesons states and so forth collective states through this one through this one one gets the difference between the up and the down masses and so forth and the difference between electrons and neutrinos I don't know what this one is now This one actually is the difference of the masses between generations. You can see how those types of mass generation occur. Anything that changes the sign of that will introduce mass. So I'm saying this is a codification of each mechanism in effect. So can you relate the generation of mass to some symmetry breaking? This is the symmetry breaking. this is already one state of this is already one state of the ultra part so you see that's negative and it can't become positive without generating mass if one also automatically assumes the full value, one of these must be that this automatically becomes one sign, if one wants to change the sign, one has to change the sign or zero, something is this in here so changing the sign of anything in here or zeroing it produces an element of the opposite handedness which produces therefore mass because mass is generated as the opposite handedness I wish I could explain it in more detail but we've pushed the time so I'll leave that for the moment

55:00 but if anybody wants to I said it would take a few years read the papers. I can give you also some references that you can read. Have you related your theory to the discrete symmetries? I mean, CPT? Yeah, that's what I'm straight for. But this is what one has to do. CPT, if you just... To actually do a transformation, CP or T, well, multiply this by, say, K on either side, or I on either side, or J on either side. That will give you three transformations. If you multiply all together you get back to where you started. How many more slides do you have? How many more slides do you have? That's it. Is the formal talk over then? About five minutes if you want. Up to 1.30. Can I ask the question that I was asking? I wanted to ask you. I'm trying to find out what's going on here. is it like as if you had say a lot of things you put them in a bag these fundamentals put them in a bag and you shape it by the nature of things some process occurs inside and at the end you get some kind of precipitate which is the which is what you come up with you have certain rules which apply for various reasons Yeah, I want you to take that for granted, you know, one understands you're a good mathematician and you haven't made any mistakes, right? But what you tell, like as I say to the guy on the crap I'm on the bus, you know, Peter takes these fundamentals and he shakes them and he shows that really there's the nature of the thing, well there's a change in the shaking or you bring it out in the shaking what was there yes you can't actually know that immediately you've got to shake it to find out what there is i guess yeah sure i understand but i don't particularly choose to use algebras they just emerge yeah whether it's a clifford algebra or a dirac algebra or something like that someone could say well okay we don't know about that but but it's the process

57:30 the fact that something comes out of something that would be the interesting thing wouldn't it if we start with concepts of a certain kind and you stick them in these concepts are arbitrary well if they are and you shake it the shaking is a refining process which is refining to something which is more economical. Is that it? Yeah. Sorry, a lot of people have got questions. Can I... I'll answer Dan first and I'll answer some. Okay. By deriving somehow the fixed mechanism, you surpass as I see the standard model by this D-Rack equation. Yeah, I'll tell you, this is better than the standard model. Yes, I think so. That's my question. Do you have any hint further, going beyond the sample data? Well, I can certainly make some predictions. I'll tell you one prediction I can make immediately. It's a trivial prediction in a way, but people haven't made it. CP violation was observed in K0, K0 bar splittings. Immediately one can then say it should be observed separately in K0 and K0 bar, which has now been observed. But I say it should also be observed in K plus and K minus. CP violation. It should also be observed in both Einstein condensates. So there are a couple of predictions that certainly nobody has actually specifically stated. Those are certainly predictions that one can make. And they are far from trivial. I mean, they're not trivial. No, no, they're not trivial. No, you're right. Can I answer Tony first? No, I've got two questions. William Kingdom Clipper. Yeah. Can anybody name a book where I can learn about his algebra as opposed to other people? I'm going to answer that I think that the nicest explanation is in this book by Heston called there's a book it was written in around 1989 it's called it's a classical mechanics not a subject that's a mathematical one this one is on it's called classical Classical Mechanics, I think it's called. New Foundations of Classical Mechanics. Foundations of Classical Mechanics.

1:00:00 I think I know about it. There is Hestonese. Yes, Hestonese, all by himself. And the first 40 pages of that are, starts with the Greeks and talks about how the Greeks looked at vectors, actually, and so on. And the disadvantage of that book is that he restricts himself to three spatial dimensions. I think as a strange introduction there's a lot of work on this and there's a website devoted to a sort of lecture on her work which I can dig up as well but thanks for this because I've seen that reference in connection with quite something else a book called The Joy of Classical Celestial Mechanics it's only the first 40 pages that I'm recommending for your purpose Who's the author? My other question is this. When I do my talk, you will find that I generate four Dirac-like objects. That is to say they're anti-computant, they do various things. They don't square to scalars, however, because the mass turns out to be an operator. However, if I multiply through by a fifth object that does square to unity, which I've just realised, I don't alter any of the equations which I get in the end, which are the kinetic equations of special relativity. Sort of like gamma 5. Yeah, it is gamma 5. I don't alter any of those because they're eigenvalue equations. And I multiply through by something that... so if I show you how to get those things from fundamentals you've got the whole thing in mind well that sounds good your separation if we sort of think back to your very first slide where you are separating space time from charge the thing that struck me drawing in parallel with my bit bang paper a couple of years ago If you look at the signature of Clifford Algebras, if you start with your basic vectors, what Brian has called A, B, C, D, and E, if you take their squares as 1,

1:02:30 then when you do quaternions, you get minus 1. You square them, you get minus 1. When you square the 3 vectors, you get minus 1. When you square the 4 vector, you get plus 1 again. and I'm wondering if it could be that no, there's the next one the one that's got the 5 and the 8 on it I'm wondering if perhaps the thing that allows you to as it were, split off the charge space things, which I think is straight polarity those are the 2 and the 3 the 1 and the 4 are the ones they're using for vectors as it were and in that being the case then your top 5 up there are actually the 3 of them the middle 3 are the middle 3 from my Big Bang paper where I'm taking AB with C and BC with A and so on so I was wondering versus 2 and 3 is I guess my question. The 2 and 3 he needs. Right. The 2 and the 3 squared to minus 1. They both give a square of minus 1 which says they're tracking a polarity that you don't have in 1 and the 4. This is a 2 and 3 algebra. This is a 2 and 3 algebra rather than a 1 and 4 algebra. It must be a 1 and 4. And the 1 and the 4 are the space and time which I don't have this charge-like polarity type. Because I multiply by the other 5 I get a 2 and 3 algebra instead of a 1 and 4. I'm not saying the separation that you started with is a 1, 4 versus 2, 3. Yeah, I think that's where people get confused because what I see as being the profound shift is not simply just multiplying by gamma 5 to make it nil-populous. But in fact, what he's doing is taking the standard Dirac equation, which we see as basically kind of a wave function or a state function, and changing that into an algebraic number, the same Dirac algebra, but it's by changing the state wave function to a state number, which is the real profound shift. Because as soon as you do that, you can look at just the structures of the numbers. That's where he's pulling all this stuff out, is out of the internal structure of the number within this pentad, within this Dirac space.

1:05:00 When you say number, are you saying like ij then? That's a number? Yeah, I mean, one way of looking at it is to say that the general number of a Dirac space has 16 terms with constant coefficients. You can treat this kind of as a number algebraically or symbolically. When you're shifting to that, it shifts into a number that has an internal structure. and that internal structure reveals the quark structures and the force you're talking about Clifford algebra terms yes, Clifford algebra numbers but he's what Peter's done is actually denote it in a very physical way so instead of just having the abstract Clifford algebra, which it is and it can be translated purely into abstract Clifford algebra by using the IJK and italic IJK it links it intimately with this physical structure and so it becomes very simplistic to pull out not simplistic but very easy to pull out these terms people give me a problem and I can usually do it the next day the problem I think you run into is that you're already so ingrained into this shift that I can't understand why other people aren't that you can't understand why other people aren't and other people keep asking questions like well where's the and what, you know, why is this a shift? Where is the symmetry breaking? What kind of symmetry breaking is it? That's true, yeah. Because it's a special representation of a general algebra. But it's the special representation that makes it easy. It's like Feynman diagram or something. Peter, do you want to declare a Peter Rowan support group to go to the mantra together? Definitely, that's something I would support. Thank you. Thank you. Well, I am talking not about particles but about black holes and scents, that's the news,

1:07:30 but the thermodynamics also. And I am supposed to sing the phrases of thermodynamics. And I've done a little work on this actually. The first thing I might mention is that it's very surprisingly powerful. For example, do you know that you can prove the theorem of the geometrical and algebraic mean? Do you know there's a theorem? It's a very simple result, that the geometrical mean is bigger than the other one, arithmetic. Peter, could you use the blue one, that green one is not really hard to see. The blue one is on the tray there. Oh, yes. You know that. It's very, very unique. Now, which one is it? I forgot. The arithmetic mean it's bigger than the geometry. And you prove it from thermodynamics. Yes, it is. Do you know that? The geometric mean is... This is right. The numeric mean is bigger than the geometric mean. Because if you make them all the same, you can see... So, will I give you a very quick run over from entropy? To remind, basically to remind you about entropy. So, you have a number of So you have a number of reservoirs, different temperatures, Ti, T1, T2, T3, and so on, and they're all the same heat capacities here, shall we say, and then they're allowed to come to equilibrium, and the condition is that sigma has heat capacity, Ti minus Tf, is equal to naught. here. That is the condition for equilibrium. Tf is the final temperature, and Ti, these are the various temperatures of the n-bodies, from which you find, not surprisingly, that

1:10:00 tf is equal to 1 over n sigma ti. Okay, now you make a second law calculation of the entropy change. The entropy change delta s is in this case heat capacity again times the integral of dt over t because c over t is the integral for each case. And that gives you, well, let me see, I haven't done very well, I ought to have a sum of an idea for all the containers. containers, and this turns out to be, I think you will agree, log Tf over T1 to Tn to the power of 1 over n, isn't it? One, the power of one when it is. Is that roughly right? You can do it in your head, I think. DT over T gives you log of T for each of the containers, and so if some. You're already getting an N now? Or is that going to come from the log? That's the log, that's one upon N. so that's the entropy and of course the entropy increase has to be positive and so from this you see because we have already got this equation of Tf but actually has got to be bigger than the u I mean, it's a very rather simple argument. And TF is that. So it shows you that the arithmetic mean has to be bigger than the geometric mean. And you obtain it with complete accuracy from seven dynamics. Now there are some philosophical problems

1:12:30 that you might wish for this. how is it that you can work you can give a method you can give a proof of a theorem of mathematics from physics it shouldn't really be allowed I think the effect shouldn't be allowed and indeed this is not what I'd like to state I would like to state that the theorem can be made as easily accessible not truth as easily accessible process and that It's simply, I don't really mention, it's not really a part of the talk. Just to make it clear that thermodynamics has certain unexpected parts. Are there any questions? I'm going to change here. Not a long question. I'm just curious, could one argue that the mathematical content of your conclusion was actually already built into the integration and the summing and the log and stuff so that you could claim it was a proof. I could claim well, one could claim it as a proof, I think. But the proof is from mathematics. Well, the proof is from physics but using the hood of mathematics in principle. Well, exactly which physical principles to do? I'll use the second law. I'll use the second law. It tells you the entropy is greater than zero in this case. Yes, that's all. So this tells you that the second law of thermodynamics in this case is not contradicted by mathematics. One we're putting. I prefer to say... That's the way I put it. I prefer to say it makes it... In this case, thermodynamics makes the mathematical result very plausible. I can't say prove it, because although it looks like a proof, It can't be regarded as a proof because the whole of mathematics is assumed to be valid, to do this at all. Have you ever run into an instance where the second law of thermodynamics led to a false mathematical result? That's interesting. That would be nice if one could do it. No, it's probably not possible.

1:15:00 You don't usually find the flaws in physical loss by seeing that they leave the flaws in their life. Rather anew. Well, so now I talk about thermodynamics in general, and here I want to mention there properties which the entropy has and here they are. I want to focus on three properties of the entropy which are really rather useful. The first one is called as a super additive. that is to say the entropy of two systems when they are combined tends to be either bigger or equal to the sum of the entropies that's one property and I call it super additivity called briefly denoted by S the second property is that as a function of U goes up in this way, as I've shown. In other words, S prime decreases, the SDU decreases, and D2 SDU squared is actually negative. And this is through cavity. So the entropy, as a function of U, is always bent over like that. And then the entropy is also homogeneous in general. down the bottom there. If you multiply this, double the system, for example, lambda equals 2, you double the system, you double the entropy. Normal. So these are three basic ideas. And now, question for the class is, in terms of these three properties, how many types of entropy can you expect? I mean, allowing each of these properties to be either true or false. Entropy as a function.

1:17:30 Entropy as a function of what? And when you say we want to have formulas, we would expect 2 times 2 times 2, 8 possibilities. Entropies, S, C, and H may either be valid But of course, actually, the number is smaller because some of these properties are actually not possible, not allowed. They're not allowed because they imply So you have S, C, H and the next one I'll call S H C, S not H C, S not H and not C. S H C, S H C, sorry, C and S not H not C. Can you read that? There are three possibilities in the first row, and then there's another row with another three possibilities, that S is not valid, S, H, C, not S, not H, C, not S, H, not C, and not S, not H, not C. There are eight possibilities, as you write, but some of them are not allowed, because they imply the contradictions. Four, five, six, seven, eight, and the ones that are not alive are going to cross out. Not as easy as I thought.

1:20:00 two of them are not allowed because they are are self-contradictory. I don't think we need to worry too much about that. But anyway, there's six left. And now the next point I want to make, there are some examples. In order to study, now the reason for all this, the original reason was that when black holes were first discovered, one found that the entropy had rather strange properties. namely that Black hole could in fact develop some of the strange things. Namely, could, it's not, doesn't, you know. Once you're in here, in this group, let's say where superadditivity is not satisfied, then you come some of the dynamics in a sense. So you see, if you have two black holes and you put them together, it is not, the entropy is not doubled because entropy goes as m, I think, or m squared rather, and so it's not doubled when you double the mass by putting it together, the entropy is not doubled, so And so, one tried to think of examples of simple systems that violate this. And here I have three examples. One is a generalized gas. Now that is almost like the ideal perfect gas, a perfect ideal gas which you all know, pV equals pT, and for which the entropy is actually, the entropy for that case is like the one I've written up there, E as a constant, Boston's constant, the number of particles, N, and then the log of or A, the internal energy, the volume divided by the number of particles.

1:22:30 You see, this entropy for the ideal gas, when you double the gas, the internal energy doubles, the volume doubles, and this, because it's the power of 4, so this thing is actually invariant. But this, and here, doubles also. So when you double the system, you double the energy. So it's behaving in the proper way, as you would expect. But in order to study the variations that can arise, if you don't have the straightforward super-additivity, I put in V to the G, there was a G there, and there was an H in the bottom. so by this way if you now start and it gives you an example of what happens if your entropy doesn't behave in any way can I interrupt for a moment something that just bothers me you know you say an ideal gas and it's you don't say anything about the extent of it I mean let's say yeah okay And let's say that we were talking about the cosmos here, and we imagine it's normally distributed, et cetera. There's a problem that it seems to me that if you say you had this bit, you had this bit, you had this bit, you had this bit, you know, that implicitly you're talking about one instant right throughout when you can talk about the entrapies of all the systems are there. But in actual fact, I mean, when you think about special relativity about light, you know, the finite speed of light, etc. I mean, how do you find this cosmical instant in which you can add all these things in an equal...

1:25:00 I think you're expanding the range of activity for the moment. Perhaps can you leave it till later as a question? Okay, sure. Because it's just a normal finite system at a given time. It's still quite classic. In maths you can do this, I understand. I think we can talk about it perhaps later, maybe we're getting a bit of cosmology later. When, under what physical circumstances are the G and the H different from what we started with? Well, that is artificial. We've made this artificially in order to find out what the energy is. Right. And then we find analogies to the black hole, and thereby we understand that better. Now, in addition, there's another example, which is dilatonic black hole, which is somewhat sophisticated, but the advantage has a very simple entropy expression. Now, we look around to our black hole experts and say, pass down interesting entropy expression sources. we don't want and I hope you won't ask me to derive any of these because there's quite complicated derivations and I'm on an expert but anyway number 2 is 1 and number 3 is yet another and what can we, now the next our job as some of the analysis would be to try and find out what are the implications of these entities and this we will Well, first of all, I'll take the first example, which is perhaps a generalized gas, a generalized classical gas. We have a quick look at the theoretical laboratory, and we've got to say what seems to be the position You get U, here's the pressure for U, the internal energy, and here's the pressure, you work out the UDS, and you find

1:27:30 the SDU is B, M, T over U, and that's one of the T, we know that, and that's the entry, so the entry really comes. you can see you can come to that and then and anyhow in the end you come to this particular you find that It has the expected behavior of not actually doubling and doubling the system because you see there's a V to the G and there's an N to the H down there. And therefore, on doubling the system, you get 2 to the g plus 1 in the numerator of the log and 2 to the h in the denominator of the log and those two will not cancel and so on doubling the system you will in fact get an amendment at the end of the rule which is not simply a doubling but it's something that depends on the log so it's more complicated that's a funny example one. I will now go to another one. Now here's a rather interesting example, I think. And that is the third one I give. And that teaches us quite a few things. This is another black which has a very simple entropy expression, namely, sigma n is a reduced entropy. You write like a constant and you get a dimension of entropy and it goes like the log of... Now, we have a mother entropy and n true. So, m is the total mass n children of mass of little m. You see up here the top.

1:30:00 m minus nm is the mother reduced by the n children. And for that, because of the general formula for the entropy for this case, the entropy will be the reduced entropy divided by k. In other words, is there. m minus nm plus n naught plus the n children which contribute to that. Okay, so no problem, when there are no children, you have sigma n equals more, n equals more, and now you have a look, find the maximum, for the maximum with respect to a little m, you find the entropy, and you get reduced entropy, never mind these, mathematically used perhaps, you get a change. Now we've got a change of entropy in going from the original to the sigma nought being the original, no mother entropy and no children. And we go to the sigma n, which is the mother entropy plus n with one children. and that difference is worked out here with God now what happens that difference could be positive or it can be negative if it's positive then it means that the entropy is biggest if sigma naught is bigger are you with me I think perhaps if sigma naught is bigger what happens? you will the system will try and establish a case when there are no children in other words, if they were children initially they will disappear how can they disappear? only by merging with the mother back into the womb If it's negative, that is to say, if sigma n is bigger than sigma nought, you will, the system should, by increasing the entropy, produce the truth.

1:32:30 In other words, fragmentation. So what we have here is a nice example where entropy tells you about possible fragmentation What are they called? Coalescence. The lapse. Coalescence. Coalescence. Then this is one and so on. What I actually thought is something. Now I'm going to plot this. I'm going to plot this. I'm going to plot sigma one for one shot. So here's a very nice picture. So here's a reminder of the entropy expressions. sigma nought, sigma 1 is for one child, the maximum of that, and so here, for m over n nought equal to 3, the bottom curve, we have 1. That is when the total mass of the black is reasonably small. When it's bigger, we go to the case of the top curve, which is m over m0 is equal to 5, so bigger, bigger case, and you get another. And now we look and see, we put in sigma0 as well. So we've got the sigma, we've got the reduced entropy, and the entropy for 0, 2. Three cases where m over m0 is 3, or 4, or 5. And the top of our curve, which includes the children,

1:35:00 lies below sigma0 in the first case here. In other words, the maximum with one shuttle is less than what you would get if there are no children. So, if there are children, the system will come and become, you go up, you go up from there to there. And in the other case, up here, when you've got M0.5, you will actually, again, this time the increases go the opposite way, in other words, from no children to one child. So it's a case of fragmentation. So we see that in these cases of such enterprises, you can get fragmentation and you can get time. And I have a feeling this is quite interesting. From the point of view of some of the physics sometimes comes across. But of course, these cases are are somewhat remote from simple, classical examples. So, we have complete or pregnant? Let's see. What else we have? Okay. What else can entropy tell us in its different guises? somebody talked about I mentioned it is it a completely unique topic, I mean do we all know exactly the expression of the kitchen, is it always Shannon type there has been a first of all that's obviously not true because we know of at least one the Boltzmann where is it Shannon entropy is the one we are using Boltzmann Boltzmann entropy is a special case

1:37:30 there is a Rennie Rennie I don't know quite about the He's a famous man and he also has an entry. More recently, one of the man that has been about the same night, the man in the reserve, Konstantinos Sachs. Have you heard of him? We are doing that. And he put forward another expression for the answer. I'll tell you. Generalize angles. Okay, now, here we get some. First of all, we have, I define q. q is the sum of all probabilities raised to some power little q. And that big q, it doesn't normally occur in entropy expression. So it's Shannon, the bottom one That's a channel entropy, and Q, big Q doesn't occur, just sigma P, we all know. But the Iranian entropy, or maybe, goes back quite a few years, YSR, R for red, red. Red is like red. So it involves a big Q, which is the sum of the probabilities going so far. Now, Salus produced another one, and I'm just checking on something here.

1:40:00 I'm glad to explain to you about why it is. Because I'm afraid that I can't see you. I have it. Okay, I can't find it. So there's a Shannon entropy we've seen at the bottom, and there's a Rennie entropy we see at the top. For both of these entropies, y is 0. I'll show you an explanation of what it is. Now if you take the combined system and you have a combined system and you subtract the individual entities the thing that will be zero if the thing is properly if it's the usual case But, in fact, it may not be, in some special cases, and then we have S A, S B, and some coefficient Y. So if Y is 0, then the entropy is the proper, usual, extensive, normally additive entropy. And indeed, you see, for YR, Ren, Y Ren is 0, and Y, Shannon, is also 0. So it behaves in the normal, in the way you describe it. What a problem. Now we have Sansa. That, I haven't mentioned yet, I haven't given you the expression, but that's what

1:42:30 and there's a lot of work that's been done, a vast amount of many papers. Do you know them? Have you seen them? Not all that widely known in this continent of Europe. So for the size entropy-wise, that makes a bit of a difference. So when I was a physicist, I doubted some. So I invented yet another one, because you can invent other things. Now you can go away and invent inventors. All you have to do, all you have to do, I think, really, is to use that key properly again. Q, SU, you see that R for Renmi, S for Shannon, T for Pseus, and the next one is U, S2 over Q, R, S, G, U. So that SU is, and that's the wrong unit. Yes, what is the little q, the small cube? Is that the exponent of the pi? One is the exponent. It's the exponent for the exponent. It's a chosen exponent. It's a real parameter? No, yes. That's a very good question because on this hinders all these applications they've had so people said well let's work out some standard problem for example the black background radiation is black so somebody said oh let's try let's do Shannon use the Shannon and see what we get and my way of putting it is to see how close you get in fact it gets very close to one so close it is almost pointless however now we've made some quite good use of it I'm confused by your statement in it doesn't seem like

1:45:00 for example if all the PIs are zero except close so then you just have p to the q except for 1x big q will be 1 no sorry log q will be 1 because q is 0 log of 1 is 0 so you get so it behaves more or less And so you could invent other in terms of the Q if you wanted to. And you can't, probably I'll invent this bottom one. No, U, SU, that one does it to see what happens. and let me tell you it's not very exciting you just get you just get more or less what you get well you get more or less what you if you apply it to one particular problem with entropy and that is the channel capacity now a colleague of mine and I wrote a paper using the channel capacity. And things really worked out there. In the sense that the channel capacity made perfectly well for the channel's entropy, but not for the Tsar's entropy, and certainly not for the one we invented either. So, it was, I would probably say it's not, the current capacity one would have thought might be constant for all those for all those in fact the HSA is changed so we've got a bit out for us on whether this very mean there are many people working on this and believe that there is

1:47:30 How much time do you think I have? You run history? You study your time. You have nine minutes. You probably have... About five minutes. Five minutes. You have more? No longer? Well, I don't believe I want to talk anymore than I'm allowed to do it. But I will tell you... I will tell you... I'll tell you two things. One is that there are three properties of energy, which class numbers, superadditivity, concavity, and S-cine. And now you can make a Venn diagram. Where So where are all the systems? Here's the van ligand. When it looks very tidy, if you don't have a mind, it gives you the basic, the real normal thermal dynamic system in the middle. SC and H of that. And in our generalized gas case, so we had h and g, you remember we had uv to the g n to the h in the log. And of course, if g plus 1 is equal to h, you're back to the normal case. It's like having g and h is equal to zero. Oh no, g is equal to, yeah, so when g is equal to h, when h is equal to g plus one, you're back to the normal kind of sum of that. And so, in this central bit here, which was the elementary arrow in the beginning lecture, it's just standing in the pathway.

1:50:00 Stand up, I don't see there, but anyhow, there's S-C-H. there I have written g equals h plus 1 because the first example I gave you would go back to normal when g is h plus 1 but the other cases are all are all different and I have put in some things about I mean I put in some key I think it's a detailed event. Next, so I have five minutes, so I'll tell you two more things that are slightly interesting. At least for me. And that deals with negative heat capacities. Negative what capacity? Negative heat capacities. Heat. Heat. Heat. They are kind of negative. The simplest example is a star. A star is radiated heat, giving light, and radioactivity inside it makes it hotter. So, although it appears to be absorbing, Although the temperature goes up, heat is given up, light, energy is given up, not the radiation. Energy out and it's getting hot at the same time, which is not the usual thing. Normally when you give energy out, heat is, the temperature falls down, it goes down. and that is the case so the star, simple star is an example of a negative heat capacity situation there are others and I have worried a bit about the question can we have classical situations where you have a negative heat capacity and very briefly

1:52:30 the answer is yes the way you can do it is very surprising. This is a very simple example. Black body radiation. Under what conditions can black body radiation have a negative heat capacity? That really is something. And here is a modest example. But I mean, you will say it's people, but I don't want to change the variables. But we change the variables. But here, first of all, the black-quality radiation U equals T to the fourth, entropy, pressure, and now I introduce a new variable. Do you want to make my elementary mistake again? So you replace P by a new variable called capital X. X is defined as V times P squared. For black-quality radiation, P squared equals T to the eighth, and so on. And so, you now have a new variable and instead of v, you can introduce x, capital X. Apparently, it's a fairly harmless procedure. So now, the internal energy which had the V in it now is X. I just replaced V, you know by the expression down my head between minus 8 I get the circle for the energy I get the circle for the entropy and for the and there it turns out I get the negative quantity but of course I have changed the variable in a very drastic manner it's something to think about you can negative That is wonderful. In fact, I think my time's up. It was a pleasure to have such a peaceful audience. We'll believe anything. So far, but now it's your chance.

1:55:00 What you say about negative heat capacity, it seems to me that you have this with thermodynamics versus nuclear synthesis right across the board. You know, nuclear synthesis, you have fission. And, you know, like you say, the most dramatic example, that's a star. Yeah. which has internal nucleic processes and it's losing energy according to the second law and it's heating up. I mean, you mentioned that one thing about the star, but I see it right throughout the whole of nature in that sense that nuclear synthesis in general and thermodynamics are somehow opposed. Not opposed, they balance each other out. It's another thing, if I may mention it. As I said earlier, and you said, we'd like to share with the problem. Adding things. You know, a mathematician can say, you've got the license to say, have it you know you can do anything like that in mathematics but you try to translate that into physics or anything like that and you're up against the problem as I say of defining an instant in which everything adds in which you can add unless you're God you're bringing in simultaneity just for the purpose of an observation but I think from many situations in physics you assume and quite rightly that your observations are done with time being the same everywhere you can do it on a small scale my worry is about how big a scale can you extrapolate this to well if you've only got one observer and I don't matter how many things he's observing at once he has a particular time it's his time here and whatever's going on out there is what

1:57:30 he observes now he observes the light coming from it which may have been on its way for a very long time or may have been on its way for a short time it's up to him to have a theory to make sense of it yes I do but you know but actually even that doesn't work Tony because there could be things that are beyond the light cone Yes, but that's part of the theory that makes sense of it. Yeah, okay. So there's certain things that are not in your frame of somatineity. It depends on whether you're saying a mathematical split or a physical split. Mathematically, we're going to classify the universe into two sets. It splits it down the middle. It's a physical split of the universe. I agree you can go it that way. it's just that if you're an empiricist I mean mathematicians are rationalists as empiricists you've got to make it all well I think it's Viv that wants to cut the universe I didn't hear Peter talk about that I can talk about the universe as well if you want to but it's problematic to talk about cutting the universe I think it's taken as the universe of discourse as opposed to I did have one short question for Peter and Honey. Is there any physical justification for these different entopies, or are they just sort of form a manipulation? Well, no, they are drawn from the literature, from Deco-literature. black hole research they have will have occasionally come across this well I would say they will invent these different entities on some physical basis why it's a bit complicated is that it always involves interactions and this makes it the point very difficult sometime I will talk to you thank you Thank you very much, Keith.

2:00:00 First, I'd like to thank Keith for inviting me here to talk to you on some ideas and some work I've been doing over the last, I would say, five years. Right, I will take you through the emotions first, and then I'll go and reach out. Okay, I'll start with an initial pessimism, okay. Okay, it's a new version of Murphy's Law that says that, you know, if there is a 50-50 chance for a theory to be wrong, 9 times out of 10, it turns out to be wrong. Okay, so we don't have to be so pessimistic. I would love to share with you my quantum theory of gravity, but, you know, the margins on this slide that I did is too short, and of course the talk, the duration of the talk is short, so I won't do that today. I will content myself to an equilibrium I will settle for the following thing I will content that a cogent quantum theory of gravity should start from a quantum theorist of space-time structure per se rather than go directly to Einstein's equations for the gravitational field and try to quantize them, whatever that means seems to be theoretically lame and conceptually lame to directly attempt to quantize Einstein's equations, to arrive say, supposedly at the fine structure of the graviton, in the same way that one wouldn't expect that quantizing the Navier-Stokes equations one arrives at the fine structure of the water molecule. Navier-Stokes equation of hydrodynamics. This thesis is very, very nicely stated, elucidated, and also worked out

2:02:30 in Ted Jacobson's paper, Thermodynamics of Space-Time, the Einstein Equation of State, which in a nutshell contends that it would be perhaps wrong to try to apply the quantum mechanical formalism directly to the field equations of Einstein. He gives some thermodynamic arguments having to do with black hole thermodynamics You know, the Beckenstein-Hawking formula for the entropy of a black hole And I think it is a very convincing argument that he gives for not attempting to quantize head-on the Einstein field equations I strongly recommend this paper. I'm sorry, it is physical review letters 1995. I cannot give you the exact page. It's a beautiful paper. No, on to my main paper. I noticed that also as Keith announced. Have you got copies of this paper, like a copy? Not with me. If you give me your address, I can send you a hard copy of it. No, I'm really thinking if you had a copy here, it could be zero. Yes, yes. No, I don't. Sorry. I noticed that Keith, five minutes ago, said the talk was manifold reasons against the manifold. Yes, I added also the epithet, space-time manifold. Okay, manifold reasons against the space-time manifold. All right. I was stuck with an agnostic aphorism of a colleague a friend of mine he says the space time continuum who gave it to us I write in red there who because it is it is I was reading a paper by an eminent physicist that in the beginning in the abstract he says we assume that space time manifold, okay, smooth manifold. And then he performs some calculations and he ends up with the singularities, he dumps in singularities. And he says, and he says, therefore we infer that nature has singularities, okay. It is, the purpose of this reading there is, is that

2:05:00 it is, it is our model of space-time. We should not forget that the assumption of a smooth continuum it is our model it's not nature presumably nature has no similarities it is our theories that are of limited applicability and models that are of limited applicability and validity and if we change perhaps our theories if we change our glasses perhaps we can evade the the singularity the so-called pathologies so let us let us start with doubting the space-time manifold on seven rounds the first the first is the constants the constancy what I call the constancy in topology in general relativity while the metric is subject to dynamics the dynamics being given by the Einstein equation field equations. The topology of space-time is fixed once and forever by the theoretician. I mean, space-time is a topological man to begin with. And, of course, we ascribe extra structures to these. It's a C0 manucle, it's a topological manucle. We also add to it a differential structure, a smooth structure, and then we add a smooth field, a metric field of signature 2, of Lorentzian signature, and What do you mean by smooth structure? Do you mean that there are places where nothing can be if it's not smooth? Smooth structure, I use it in the strictly, I would say, technical sense that the coordinates are infinitely differentiable functions. In other words, the coordinates can be anywhere. Anywhere? The coordinates, a point, the coordinates of an event, yes, of a point. There are no places in the space where it cannot reach. Right. Okay. Constancy, pointedness, of course. The pointedness of this pathological feature of the continuum, of the geometric continuum, comes with the pointedness of the events. I mean, we initially with a Schwarzschild solution of Einstein's equation, so we were facing two singularities, an exterior singularity and an interior singularity.

2:07:30 Can you move the slide up? Indeed. Sure. More, more, more. It's your order there. We'll tie it off. We don't want to do the name. Okay, the point is, of course, we know that a point mass generated the gravitational field, which is the Schwarz's solution to the field equations, and which has an exterior singularity and internal singularity. We thought that the external singularity was pretty much nature's. I mean, in the sense, we thought that we could not evade the exterior singularity, however, with a very clever trick, Finkelstein, back in 1958, introduced a new coordinate system, and showed us that actually the exterior singularity is a unidirectional membrane. He used the new coordinate system called the Finkelstein-Eddington coordinates, that Eddington had used in the past, but not for this reason. But also, it pointed to us that perhaps the interior similarity shows us that the theory is out of its depth. I mean, we need a theory for quantum gravity to be able to calculate the gravitational field right at the point max. I mean, this is reminiscent of the other infinities that play quantum field theories, that we assume that the sources of the fields are a point like, say, in QED, the electron. Right. Okay. All right. The third point that I will raise against the continuum is, of course, the experimental non-pragmatic continuous infinity of events. Arguably, we have no experience of a continuous infinity of events. And also, we know that the infinities of both the quantum field theories of maths and GR come from the assumption that we can, in principle, pack a continuous infinity of events into an infinity decimal space-time volume. So, this is quite ad hoc. In any case, all our experiments that we conduct record a finite number of events

2:10:00 in experiments of finite spatial extension, laboratories of finite spatial extension, and during experiments of finite duration, of course. So, this continuous infinity of events, it seems perhaps a gross idealization. Okay, the fourth thing that I would bring against the manifold is, of course, spatiality or space-likeness of connections. I mean, the Euclidean topology of the manifold is defined by two-way reversible connections. I use here a nice paradigm that David Finkelstein told me some time ago, that it is perhaps this conception of topologies as a theory of space, of spatial connections, is perhaps intimately tied to the fact that Euler could cross the bridges of Königsberg in either way. So I like that. It's a very nice saying. But of course there are people who have seriously considered the possibility that in the quantum deep, in the quantum space-time deep the structure and dynamics of space-time, must be that elusive quantum gravity must be a time-asymmetric theory in Penrose. It must be time asymmetric. Time asymmetric. So... Another comment on the way that's good is that I think you can make a case that graphs are, in fact, much better evocations of space than the abstract version of the biology. They're much more fundamental. Either two points are accessible to one another way or not. I'm going to be talking about partial order graphs, particular partial order graphs, partial orders, that they incognise topological features subsequently. I'm surprised I thought about that as an analogue, because you could go both ways across the bridge, but never in time. You could never go backwards in time across the bridge. that's a strange kind of extrapolation nevertheless Einstein's stereotravity is purely geometrical and you can go backwards and forwards in time oh I've got Einstein that's a terrible perhaps it is that's a good point because again perhaps it's the undirectedness of the medium or the underlying medium is based the manifold that underlies GR

2:12:30 But there is another point here from the point of view of a physicist. If you make measurements of things, probably, you do it by measuring time. And the measurement takes time, so there's no such thing in practice as an infinitesimal division of time. In practice, the slots always hit. It doesn't matter whether you're measuring distance or actual time. and so in a way your case is solved because you can say well in practice there is no such thing as a continuous manifold, it's merely an abstraction that these theorists use and have done since with you from an operational point of view again we do not mean what, we cannot measure the differential in the test of separation between events, but even from an actual pragmatic and physical You know, if you try to measure the distance between two, you cannot measure the distance between two events below Planck length without creating a manifold. The other remark I would make in the Mars Shabbat is that if you take all the particles out of your manifold, from the point of view of physics, there is no manifold. Because it's relationships between them, which are expressed by an evidence, produces the whole concept of manifold. If you take the particle away, you've got nothing to think about. Right. sum up your, summarise the first slide, would it be that you don't allow any points? In some sense, as I hope to show later, I substitute points by something larger, fatter about them. Say, region, theory of region. Yeah, OK, but I just wanted to say, points are out, and that's very reasonable, because physically, we don't have that. That's not the point. Yeah. Sorry. Just to make a point about .. Classical is time asymmetrical. Is it? Yeah. Because if you can, for example, look at the equation of two black holes, that coalesce. And because you can't do a repulsive force, you can't have it repel.

2:15:00 So it's time asymmetrical because of the lack of anti-gravity, as it were. You can't reverse it. I would not like to do time reversal of that. Is there somewhere I can refer to that? Because this is so simple. Is there people? I remember two black holes. We've got two black holes, I remember that. Garry Korovitz has had some papers on that. Exactly. Yeah, I have to. The thing is that that's exactly what they do when they put wormholes to make it mathematically complete, is that they then make it time-symmetrical, so that you can go through a wormhole and back to a wormhole, and then we cut it in hand and say, that's the black hole solution. Right. Very good. Very good. The fifth is, of course, the non-superposed, non-interfering connection defining the manifest topology. I mean, the arrows, if you look in a graph, the arrows do not have an ability to be superposed. There is no linear structure on this, I mean, mathematically speaking. This, I would say, this is the crystal rigidity of the topology. There's no interference, there's no coherence between the connection defining. This would prompt one, of course, to think of the possibility of a quantum topology, whatever that means. I think I will touch a bit of that problem later on. Are you saying that there's no endurance or no duration or development of this manifold, the Einstein manifold? That's right. background structure that, as Einstein questioned it, even shortly after, I hope again to give you some very nice expectations. Even in 1916, one year after the advent of GR, at least certainly in the 1924 paper, Uber Deneter, on the ether, he identified the space-time manifold on which he erects the GR. He identifies it with an abstract notion of an ether. He says, well, also I managed to get away from the ether you know with my special theory and I find it back again and it is it is beautiful it is beautiful because this shows

2:17:30 what a great physicist he was you see the manifold that served him so well in GR he was ever ready to abandon especially with the advent of plant mechanics and the molecular picture of nature I think that's an indication of a great theorist anyway he was happy yeah well metaphysics seems happy you know the manifold in that he says it is a substance that is acted upon but it does not act back I would say that it doesn't act back because it isn't really there the matter away there is nothing to talk about think about it we take away the manifold where does differential what is what is left what are these differential equations it's just a set of rules that you can express algebraically or you can think about these pictures if you want to it's a set of rules that's all convenient though I don't know absolutely Right. Okay. Of course, the sixth that I would raise against the manifold, of course, the conductivity of the algebras of the coordinates led in the manifolds. These are, this shows some classicality there in the starts. They're all, how do we actually determine the logs of an event? We send radar signals, okay? They binge, okay? So, presumably, it's always a quantum, always a quantum sees a quantum there. But yet, we haven't got effective theories for such localizations. events all our localizations are classical okay seventh the seventh is the globalness of the classical I mean topology as we consider classical in mathematics is a theory of the shape of space shape you know has to do with holes handles global features because because I I think especially with gravity the relies very heavily on the notion of locality. All the variables, all the variables in general theory of relativity are local variables. For instance, the metric is a local variable.

2:20:00 Locality, okay, at a continuous level, locality can be very nicely encoded into the statement that all laws of nature are differential equations. That is to say, effects connect infinitesimally separated events. But we can say a lot of things about manifolds and we can say a lot of things about manifolds. Sure, sure, sure, sure. The global things of the manifold, this shows, perhaps this shows a bit of... I mean, to my mind, this is a personal criticism of the manifold. I think global features of space, whatever that means, and time, whatever that means, shows a bit of the arrogance of the theoretician. Yanis, excuse me, dude, if you are standing in between this Oh, I'm sorry, I'm here. And for us here, it's soft. I'm sorry, I'm sorry. are you then saying well maybe you like something that only global profit, only local profit yes, I would like to build a theory only local even locality the original theory was entirely local, it's clever people like Penrose who have considered the topology of the whole business, but again you can only consider that by example you have to say well the matter distributions curve in certain ways, and this produces a policy, and da-da-da-da-da-da-da. It's all very clever stuff, but the original theory was entirely local, wasn't it? They're all differential equations. I'm glad you brought this up, because I think we have another example. I mean, we have now with these global methods applied to GR. There are also heavily realized singularity theorems of course, but this brings to life that we have learned that space-time is not a Euclidean manifold in the large. And then we said, OK, it is a locally Euclidean. It is a locally Euclidean, so a C0, a topological manifold.

2:22:30 Okay, okay, okay. Take it from me, I mean, this is... Excuse me! You don't have to give up global topological properties, because they can be a consequence of discrete structures anyway. Yeah. In fact, what he's interested in is the way certain local interactions could rise to the global structure, I think that was what was in the back of my mind as well. Would you still say it the same way? Sorry. Can you put that in there? Yeah, we can't read the Bible. I'm sorry. Let's get on. I mean, this was just this, you know, glimpses of inside. Can we just see the . Of course. It's the rest of the slide. Pardon me. I mean, even the notion of locality, do we expect locality to survive in the quantum deep? I mean, locality isn't locality, the concept of locality intimately connected with the space, an ambient space. the country that don't I don't but of course we have we must content I mean we have to start from somewhere and let's see let's let's let's try to be a bit Spartan laconic in our assumptions okay and okay this so these are the in my doubt. Instead I have to suggest something instead. To all of this I have something to say. Suggest instead. First of all, the constancy has to come up again. Again? Like that? As I reach down, we can suggest instead. That's not the problem, it's the machine itself in the way. It's fine. You should go closer. If I take care of it. Anybody wants to read 5 in the way? Alright, yes. We can suggest instead of the costacy, of course the alternative will be a dynamic.

2:25:00 It's too big. Yeah. That's what I'm saying. Bring it closer. There we are. Straighten it out. That's a small enough room to see. That shows by the way. Even more. No, that's excellent. Pardon my letters. No, no, it's warm. We probably spent hours together to get it, Mark. Better to read it. It's okay. What the hell? Yeah, we're good. Instead of constancy, of course, we can see a scheme, a theoretical scheme, where the topology is treated as a And if, say, in the quantum context, maybe as an observable, as a property, say, of space-time that is, in principle, observable. Because, again, we are motivated for that alternative by the fact that only geometry is an observable in classical GI. But topology is given once in forever. Constancy, we can look for something by the dynamic conceptions of topology. Now, of course, the pointedness, the pathological, the pointedness. We can substitute the offending points by regions of water. I mean, in algebraic geometry, I think this is a burden of water. what they do is the offending point, what's called they smear a space of directions over the offending point. Is that something like if you think about that in practice are you really thinking about uncertainty kind of a thing? That's where the smear comes from? Yes, the smear in such I think that conception that is a quantum observance, a topology instead of like Wheeler's phone, that it participates into quantum superposition, the topology. The connections defined in the topology superpose with each other, okay?

2:27:30 There is some coherence between them. So topology is subject to dynamical fluctuations, okay? That, in some sense, fuzzies. It's an idea. A theory of regions. I mean, even Tarski had that theory of regions. But these regions are determined by quantum uncertainty, that's where the Smearing comes from. Yes, I will show you an algebraic model of such a region theory that we can interpret the algebraic structures as a Planck structure. Of course, yeah? As far as I understand, I think it is not a case. The locates in the cosmoid are as far as I understand. Still classical. Still classical, but they are pointless. Is this your word, pointlessness? I think the whole theory should be called pointlessness. It's like the theory of relativity, you know? It has such a weird name. The theory of relativity is like the theory of pointlessness. In some sense, this is quite important. Because if you think about it, if you want to, if you wish to view generally I think there's a gauge theory where the gauge group is the diffeomorphism group, then indeed the diffeomorphism group is of course the symmetry group of the labels, of the infinitely smooth functions labeled in these points. What do we mean by point? Presumably, we mean the coordinates of that point. The diphthomorphism group, which is the symmetry group of the GR, it evades that. So at the classical level, it is, in some sense, pointless, the theorem of course. I don't mean to make a technical point. It's just a great word, this pointless. Pointless. Yes, it evokes some Eastern . No, no, I didn't mean that as a dog, I mean, I think it's selflessness, pointlessness. On the panel of pointlessness, can I tell you what, how do you always decide something, maybe a point, it could be anything, it could be another smear, is or is not in a certain place, or that's not

2:30:00 in other words I'm trying to introduce no I don't think it is meant that this sphere point has a background but they make up the space there's no background the idea is that they constitute the space otherwise it would make the observations the only observations and when you make a measurement of the distance you decide whether your answer is between here and here on the loo as it were or between there and there on the clock and that smears it for you the thing is that all our theories we tend to think of these things as exact because otherwise life gets very complicated but we need to work our theories back once we've got them and say well what do we actually get when we do measurements. And I think what quantum theory, it may be an incomplete theory, but it does help us to get back to that kind of thing. It does give us rules. It says, if you start with these abstractions, I will smear them out for you. Okay. The third one is, of course, the continuous screen. Let's substitute it by something finite. I prefer the epithets finite, reticular, over, you know, to over discrete, because discrete, I put it in inverted comma discrete there, because for me, at least from the topological point of view, discrete has a very particular misleading, I mean, you know, a discrete the structure I'm going to be talking about they are not discrete in the topological sense they are not completely disconnected sets, they have quite rich topology I associate it with them, but it's so discrete, perhaps at the at the cardinal level however, the word discrete is used quite a lot for what you mean I mean, we use it a lot for what we mean But is it correct as well, because discreteness always reminds you to unconnectedness, like the structure of the natural numbers.

2:32:30 You know, we bothered you a lot with the thing of the connectivity structure, which brings this shear between continuum and discrete in this community. Discreet is easier to say than fine writing. Of course. I think it makes a good part, too, to make a difference between this. In any case, from a topological point of view, I wouldn't like to use it, because maybe for some people it can be disillating, but it's not . All right. We continue temporality, of course, speciality, when we said that a theory perhaps should be based on one-way connections, you know, like as Lewis suggested, we have like a graph, we have a set with arrows, and you can study certain topologies. The graph defines, actually the connectivity of the graph defines them. of course. This is rather than, of course, the spatial or space-like conception of topology, as we said, because in any case, even from a relativistic point of view, we do not have actual experience of typhoons. Okay, okay. This should suffice, I mean, to base one's theory on causality. We do not have material signals traveling faster than light in space-like directions. Are you trying to use the directed graph? Theory of directed graphs? You haven't mentioned it, but take it. It is partial, I'm sure. The incidence of over-partial, but we'll see. Okay, the of the manipulse connections of chaos, of course, as with the form conception, we should perhaps, from the graphs, that should motivate the incidence algebras, the root algebras. And also, this is talking about the Gervan duality a bit. And of course, the non-commutativity, the algebras, co-ordinatizing whatever that is that we suggest instead of, as a quantum replacement of the classical space, and the algebras of those points should be non-abelian. Should be what? Well, to be

2:35:00 in line with some basic ideas for quantum mechanics. If one can do it commutatively, for instance, there is a famous magician, mathematician, Bill O'Veere, claims, I haven't seen the paper, who claims that, for instance, the well-popular and advertised non-commutative geometry, he claims that the aphorism is the following. Everything that one can do non-commutatively, he can do commutatively, entirely commutatively, in a certain larger, quite abstract space, . OK, so again, the thesis has to be careful of cliches. we have seen that there is underlying non-commutativity of quantum mechanics but of course one should not forget especially the physicists that non-commutativity is not a characteristic it is not an idiom of the quantum I mean you know that boosts in special relativity do not commute and of course also once you should treat with a pinch of salt the finiteness, radicularity of course we have a classical system that exemplifies a discrete an innate discreteness. A discreteness is not a defining characteristic of the quantum, of course take an experiment roll and dice it's a classical system and the experiment is the phase basis, discrete. Okay locality local causality, this is as Since you objected, Keith, and you convinced me, for now, you didn't convince me, actually, don't think that you would work. Pandering to the audience. I had a question on your point before. You're talking about processes, not objects. now I think that's a little dodgy because that's for me I translate it to meaning there was no hierarchical structure process then you know I see there's no objects there you cannot sort of

2:37:30 make anything bigger, because you can't have any objects. But you can't build structure in a mirror. I understand this. This perhaps, I said, processes over objects, I think. Yes, the hierarchy will say the hierarchy of structures. Don't you think that's a characteristic of . Except like objects? Why can't we think of a hierarchical process? Well, if time goes in one direction. But then you see if we have a higher level of process, quote, unquote, then it is cheating with the invariance of the lower level of the object, basically. But the objects are out of the process. True, true. Process coming out of the object. That's how we do it. Perhaps you're reading more into what I mean in an object. I haven't had any Tonyan concept of an object, which is just conservative long numbers. Right, right. There is 15 minutes in there. I hope I'll just speed up a bit. Let me speed up a bit. Very good. Dan, at the end. For you. Some philosophical strategies, OK, the continuously. This is a load. I mean, in philosophy, we have the Christophites, Parmenides, Heraclitus, and the atomists. They were four discrete, four discrete. I guess the ideatics and, of course, .. Not Parmenides. Not Parmenides. Oh, Parmenides, yes. No, no, no, no, no, no, no. He is the godfather of the Eleavits. Look, look, look, how many? Come on, come on, come on. I mean, we have Zeno. I don't want you to do this. And for Zeno, Zeno lies somewhere in the middle with his paradoxes. Of course, we have the Riemann's Dilemma, the famous in his habilitation, of course, that he defined also the discrete manifold. He said, well, a discrete manifold in, as opposed manifold, the pro, is that it carries its own metric. The metric doesn't have to be prescribed from the outside. I mean, merely the metric is provided just by counting. But also, the dilemma was the following, that a continuous manifold has continuous symmetries. Has continuous symmetries. And he thought back then, he said, nature has continuous symmetries.

2:40:00 therefore the manifold therefore space and time must be continuous he was very much influenced by some ideas of sophistry of course in the second in the second form of the dilemma but of course we respect that this is now that we have even quantum theories for symmetries, we have quantum groups this dilemma starts I think going towards the, for the discrete battle, we'll see. Of course, the quantum dilemma, the quantum dilemma is well known to you, is nature that deep down, the field or particle, right? How about space-time itself, originally, as I said? Should it be subject to some sort of quantization? I don't know how much that is, you know, reticularization is quantization. Tongue-in-cheek, more adequately, what reasons? More adequately. What do, what reasons altogether do we have for assuming space time as an a priori concept of construction in the quantities? It was a discussion that I had suggested. Perhaps we can talk about this. Now, I'll do something more quick about this. Okay. Okay. My first encounter was with a beautiful paper by Ray Sorkin. I hope that he would be here with us today. Is he coming today or? I believe he's coming today. Right. Sorkin on a T91 paper in the International Journal of Theoretical Physics.