 أعتقد أنه سأبدأ. الآن أقسم my lecture is the title is Geometric Unification. And my methods will be some. So title is Geometric Unification and I will be using non-commutative geometry as my tool to achieve this unification. To start with essentially for these physicists who are familiar with general relativity. It's known that general relativity is the example of a geometric unification or geometric theory where the gravity is described through the geometry. In other words, as is now known, matter will tell space how to curve and then the curvatures would dictate the motion of particles and things like that. So I would be seeking in a way a geometric setting to unify all fundamental interactions. And the quest for unification really started long time ago and indeed started with Einstein in his work with Kelluzha. Where he tried to unify the only two no forces at the time, electromagnetism and gravity within a theory which is geometrical, which is five dimensional theory. One of the dimensions was thought to be small and it was compactified. However, it turned out actually that the electromagnetism that Einstein or the five dimensional theory had is not the electromagnetism, but it's a different what we now call a gravity photon. It doesn't have the correct electric charge. Anyway, anyway, the time was wrong because not all forces were known at the time. So now actually we know more or less the full set of forces. So at least we are in better shape to find or to achieve this goal. So of course many recent attempts for unification and the most visible one is string theory where the idea there is that you replace the point structure of space time with strings. So in that case of course you can think of it as also geometric unification, but not in the same spirit. It's not really based on some rules of geometry and axioms and things like that. So what is the, you know, of course actually another problem we would like to attempt to solve is also quantum gravity. So when I say all fundamental, you identify all fundamental interactions, of course I really mean the electromagnetic force and the weak force, the strong force and the gravitation. The first three, of course the general structure of unification is now known, the grand unified theories. And so this actually in principle one would know how to do. Geometrically of course is really based on a simple geometry which is, you know, what's called vector bundles or fiber bundles. So one can do that, but it's not really complete geometric unification in that respect. But one can, you know, give it geometric clothing as I would say. You know, we can write everything in this language of vector bundles and connections and things like that. This one can do. This actually, the first actually three are well understood because one really can develop for quantum field theory and the quantum field theory. They're all realizable theories which means that you really can do perturbation and is, you know, is well behaved I would say. Quantum gravity, of course at the quantum level nothing, little I would not say nothing I would say, little is known. But as a classical theory is really very well established, it works extremely well. And so the issue that how to unify these in one frame within one framework in natural framework that they simply comes out. Well, actually when I say this is a force, now of course, you know, I always say a force is now of course actually the other aim of course is to explain particle spectrum. So it's not only the Higgs actually. And the particle spectrum that we know now of course is that we have three families of quarks and leptons. And also in addition, we have the Higgs field. And the Higgs field is the field responsible to giving all the fermions and the bosons their mass. So one of course actually in this respect actually even there is no geometry here because here at least you can say vector bundles. But here it's simply there's simply a doublet of scalar fields. H is a doublet. Now these forces, you know, for example, these are really described in group theory in terms of Le groups. And for mathematician, it's a vector connection on the u1 crosses u2 crosses u3, which for, you know, for this, this and that. Now one has, when I say explain the particle spectrum because of course first you have three families is a big question. Unfortunately, I will not be able to say much why it is three. You know, we know that it's more than two, but we don't know why it is three and not more than three. Why you have quarks and leptons, you know, the way these particles have set in representations, which I'm really going to derive in a way. Why they sit in those representations will be able to see. At present everything was developed historically by, you know, every particle when it was discovered, it was placed in a certain multiple. And with time, actually, the picture took shape. I think by 1974 or 75, you know, the picture was more or less complete except of course the top quark was not discovered until, I don't know, much later. But people were convinced that the picture, the three families of quarks and leptons should be there. So again, actually one has to explain all this. If one simply writes it down, it looks like, you know, there is no reason. It looks okay, but there is no geometric reason why things are the way they are. Okay? And it's the purpose, actually, of my lectures is that to show you that we are almost led uniquely to predict, actually, the three forces with the gravity and the representations that they are exactly of this form, we will not be able to predict the three. You know, we have still no way of knowing or finding out why there are only three families in nature. We don't know. And not only that we don't know, I think nobody knows. For some time, you know, in string theory it was thought that they have an explanation and the explanation was always, you know, in terms of calabi-ow spaces as some topological number. However, actually, it was always, you know, the three came out as number of families and anti-families. And, you know, the three came as something like 101 minus 98 equal to 3. So it was not really a three. 98 will, you know, families and anti-families will be supermassive and the three lightest one would. Anyway, so anyway, now actually, in reality nobody accepts that this is an explanation, actually. So this is actually one of the problems. Many problems I will not be able to solve, but actually many, many things I'm going to, would be able to answer. Why the Higgs Field? We'll be able to answer this question. Why it's a doublet? Why do we have quarks? Why do we have leptons, you know? And all the charge representations, they will come out as I go along. So this actually- So for our sense, you claim to predict that there is only one Higgs? One Higgs, yeah. Okay. So it's tomorrow, the second week, it's covered. I will, I will say something in that, actually. You know, there is, there is an axiom, actually, which will, there is an axiom. And this one of the axioms is that we have linear connections. The connection is linear. If the connection is linear, we get that. If the connection is not linear, then we have another prediction that it's not that actually all this will fit, but we'll have also on top of it, this will be the low energy representation of a higher theory, which is really the Patty Salam, which is SU2 left cross SU2 right cross SU4. And in this theory, of course, the leptons of the quarks are really unified, and the lepton really comes at the fourth color, because the, you know, quarks have three colors, you know, red, yellow, blue, or whatever. So the leptons thought that it's the fourth color. But of course, actually, this is really at much higher energy, and then it's broken. And then, you know, you are going to get more Higgs and things like that. So this is the alternative picture, and it's unique also. You know, yeah, it's either this or that. Okay, but so it's one of the axioms, and for a long time, you know, we questioned whether one should take the axiom or not. And now, you know, of course, the way I see that this is a very good approximation, and probably it will be the way it is. You know, I think it's unlikely that another Higgs will be discovered soon, or another, you know, U1 field. But of course, you cannot throw it out, because this possibility is there. It's a very nice possibility. This Patty-Stella munification is really very nice, and it will enable people to go all the way up. Okay, so actually, this is the ground setting. Now, okay, so suppose that you are a physicist, and you know, you would like to unify all these forces, and you say, I'm going to look for a geometric way of doing this unification. So you try, actually, for a long time, you say, okay, what are the symmetries of all these forces that I talked about? We know actually that general relativity is really based on diffeomorphism invariance. Everything we do, invariance. Which of course is linked to the equivalence principle. And one way to achieve it, of course, is that you make everything a coordinate invariant, invariant under coordinate transformation. So every action that you write must have disinvariance. Okay? Which is pretty strong. Okay, actually, this really, in a way, guarantees that the metric of space-time will have its universal interaction, because without the metric, you would not be able to write any invariant expression, usually except topological. Topological, you can write, but apart from topological interactions, the metric is everywhere, and in a way that would, as a consequence of that, the gravity would interact with everybody, because there is no way out when you write a density that you start by writing root G, which guarantees gravity is everywhere, actually. So it's universal. One way to look at the universality of the gravitational interaction is that the action you really cannot write except by using the metric. Okay. And then, and it's really a semi-direct product because with the symmetries which I have written, SU3 cross SU2 cross SU1, this is the electromagnetism, this is the weak and this is the color. So this is the symmetry that we have. And one would attempt, actually, to find a geometric theory based on disinvariance. And of course, actually, the problem is that it doesn't exist, actually, you know. In principle, you know, one can show that you really cannot do it this way. You cannot simply find a higher symmetry. Without, of course, having extra modes. For example, Kaluza Klein is a way of doing it, but the price you pay is that you are really going because what happened in higher-dimensional theories, of course, you know, suppose that I linked myself to five-dimension, in this case, you get d5x. Everything is five-dimensional. And in this case, you have fields, you have extra fields. And these extra fields, you would like to think of it from the four-dimensional point of view. And then you would say, okay, what do I do with the dependence on the fifth coordinate? So what you do with the dependence of the fifth coordinate is you start to expand and you assume that the fifth coordinate is like a circle and then you take all the Fourier modes and then you have an infinite tower of states. So what people do is that they truncate or chop out all the states. And, you know, you get, you are really throwing out all the Kaluza Klein modes. In strength theory, you keep, actually, the extra states. And so it's not really the same as, you know, taking a geometry, which includes only these things. So, now, my assumption is that I will use non-commutative geometry as my setting, as the setting for unification. And the first question that we ask is that, you know, why should one go non-commutative, you know? Why should one go non-commutative? So, suppose, actually, you know, I call this N, which is the difthymorphism across G. So N, let N be my symmetry with difthymorphism times some internal group. And what I would like, actually, that the difthymorphism of this my space N would be this, or include this, or at least would be that. G is local, you know. G is local, yes. In other words, actually, I would like a space such that difthymorphism of this space would be what we know. And, you know, I will not really go through the mathematics of proving that you really cannot do it except by going through a non-commutative process in which you really involve matrices, essentially. In which you involve matrices. So anyway, let me, you know, actually start with discussing a non-commutative geometry. So, you know, there are many, many ways of trying to lay the groundwork that, okay, we have to take non-commutative. But in reality, see all arguments will be only suggestive. And the question is that if I take non-commutative, does it work, actually? Because this is, you know, the pies and the pudding. So what do we do? How, okay. So let me start the follow section. Let's take, actually, the data that defines a non-commutative space. And this data, let me write it this way. See, this is a spectral triple. And these, actually, decorations which I'll discuss in a minute. You know, the non-commutative geometry in the end, of course, one may ask a question, would you be able to recover what we know? For example, would you be able to recover remain in geometry? And the answer, of course, is yes. It already includes remain in geometry. So it's not that you don't, you get completely off and you have no way to really communicate with what you already know. So this is really, it's a big test, actually. And it's under the reconstruction theorem in which you can show that you can obtain, actually, all what we know or most of what we know about because the project is not complete. Most of what we know about remain in geometry from the formulation of non-commutative geometry. So let me, actually, define where, actually, the gamma. Yeah, I'll define each in a minute, you know. So A, I will, yeah. Yeah, I will, you know. Well, J, actually, for physicists, it will be a charged conjugation. Gamma is chirality, you know, if you are a physicist. This is the analogy. One is, but it's an anti-linear operator, if you would like to. Okay. What's A? An associative algebra with unit 1 and involution star. H is a complex Hilbert space carrying a faithful representation presentation pi of the algebra. Make a mistake. I should have started here, actually. Anyway, D is a self-adjourned self-adjourned joint operator on H with the resolvent minus lambda inverse where lambda not an amount of R and D is compact. D is, for physicists to be the Dirac operator, this is generalized Dirac operator. In other words, actually, for people who are familiar with Dirac operators, you know that on-care spaces, you know, for mathematicians, you know, all the ATIA Singer indices and so the Dirac operator carries a lot of information about the topology of the space and in other words, you know, with the Dirac operator it's really equivalent to knowing the metric, you know. What happens that in the construction of Dirac operator, you always take the square root of the metric with the fear-bind and it enters there. So, and the spring connection there, the geometry, the spectrum of the Dirac operator, actually, it really has, it carries inside most of the geometric information about the space. Of D is compact, yeah, of D, you know, I forgot to write it off. I changed the line, so. Yeah, exactly. Okay, still let me see should I erase or should I just a second let me push it up. Okay, now actually the J is an anti-linear operator D, sorry, unitary. It's unitary, it has, it involves complex conjugation because on H it's a real structure. In other words, the reality of the, in other words, it really takes us from complex geometry to real geometry. The, the role of the J takes us to real geometry. So, for physicists, it's charged conjugation. Gamma is a unitary operator on H and it is the chirality. Okay? Now, properties, I have only simply defined things. Now, of course, we have to know how these objects talk to each other. What is evolution? Yes. Of the algebra. Star. Star. Well. What is evolution? Well, for example, you know, something like this, right? And also, square of the evolution is the identity. No, no, the existence, there is an identity element, you know. You need that. Just think about the evolution by default if you apply twice to an element to get the element itself. Okay. Okay. Properties. First, G squared is epsilon where epsilon prime prime elements of minus or plus one. For example, epsilon is zero, sorry, one in zero dimension and minus one and four. G squared is minus one in that case. Then, we define the element in that case. Then, we define. So, let A be an element of the algebra A and B, actually. Then, we define the opposite element B as J B star J inverse. And this is called the opposite algebra in this case we say B opposite as an element of A opposite. The opposite of A opposite. And is defined an element. So, let XA be an element of the an element of the Hilbert space. You can think of it as a spinner. Yeah? Then, you know, you can act on the element of the Hilbert space like this. You can act from the left. Action from the left. Then, A opposite would act from the right. So, in this case, we write like this. Okay. To an action from the right. Okay. So, if you have something B A, it's the same as B opposite. Now, action from the right is equivalent to action of the opposite algebra. So, here, yeah, this is right action. Now, this is actually first action. Action. The right action, and the left action commute with each other. Do you want to correct the action from the left? Yes. These are the properties, actually. But in zero dimension Yeah, I will talk about it, actually. So, it happened, actually, this is, you know, if, think of it this way. If you want to do charge properties of charge conjugation operator, the properties of charge conjugation are really in Euclidean space, say. They depend on dimension. And their properties where their square is one and it's not one and, you know, you can take their action on the Clifford algebra and you discover, actually, that they satisfy all these properties. And this, actually, in the usual metric dimension. However, these operators would work on something called the KO dimension, which is the Komorji orthogonal. And they have the same properties. So usually we define the dimensions, not in the metric sense, but in this abstract sense. And these dimensions do coincide, actually, with the properties of, say, charge conjugation operator of spinners of the orthogonal groups in this way. And the corresponding dimension. They have exactly the same properties. Yeah. No, this actually is, yeah, this is the property, this square is up. Yeah. Okay. And the two actions, they do agree the left and the right action. They commute, they commute. Oh, sorry, sorry, sorry. Action one, left and right action commute. Now, this actually, let me say property one, property two, is that you know, is JD up to a sign. That's what it means, actually. This means J squared is plus or minus one. D, J commutes are anticommutes with JD. As I said, all these are items of minus one plus one. And D gamma all is anticommutes with the D. But J gamma is applicable around G. Now, I'll introduce three signs. And depending on the signs, it means you can classify depending on the signs of epsilon, epsilon prime, epsilon double prime. You can classify them in a table. Okay. And for each, each you can, you know, label, give you a label. Now, the label actually is really done in such a way that it really coincides with those because in principle you can call it anything you like. With the classification of what we call Majorana and Vile spinners. Majorana and Vile spinners in when we are talking about you know, Clifford Algebras and charge conjugation matrices. It's exactly same classification. So, when I say dimension now, it doesn't mean metric dimension. It means dimension that so usually you are going to get because it's, you know, you have three signs. It means you have only eight possibilities, right? Because it's small date. It's minus, it's two cube. So you are going to get eight possibilities depending on the signs of epsilon, epsilon prime, epsilon double prime. So you actually use this you can do epsilon. And then you can start making a table and you know, you can stay one, one, one. You call it zero. And then you have one, one minus one. This actually you don't define except for even spaces for, you know, minus one plus one, minus one, and so on. So you really can make a table and these are called the co-dimension. You know, I can write it down minus one. Yeah, four minus ones and then I have one, one, one, one. And then it skips. And then you have one, then you have one. Anyway, so this actually called this n k-old dimension. And the way it is chosen, as I tell you, you know, it's really chosen to agree with the existence of what we call Majorana and Vile spinners. For example, people in string theory would recognize usually the ten dimension actually. Ten actually, ten dimensions really have to look at eight, which is the zero, which is one, this one. You know, why? Because in ten space dimensions you are going to say to get minus plus plus nine times, right? And the two would essentially cancel each other. So we are really talking about eight, essentially. And if you make the classification, you discover ten dimensions. Okay? And the two, you know, because the plus minus cancel each other is really equivalent, you know, zero is equivalent to two, and you know, then the next one would be ten dimensions. And yeah, it's the chirality, gamma of chirality. Yeah, in four dimensions it's called gamma five, but of course in ten dimensions it does exist, it's still called gamma 11 in that case, it happens here. Okay, so what do I do with that now? Should I keep using the center or axiom two, which actually in the last lecture I'm going to throw to see what happens if I don't assume dA B opposite zero. What does mean actually? I will show later, it's not really clear actually that okay, it's not trivial to show that if you have only quadratic not higher actually which is really a phenomenon here because we don't know it actually in any other field that for all A B element of A it's called first order condition. However actually there are no non-commuted geometries for which this condition is not true actually and this is a quantum sphere the gamma commute with elements of the algebra for all A element of A and therefore gamma is the chirality operator which implies the Hilbert space would split into H plus and H minus which would have actually you know implications for physics because it will ask that you are really going to get left handed spinners now we do assume that H is endowed with an A bi-module structure such that this I did XI B is A B opposite this actually tells me exactly where to put the opposite element when I transpose it okay A has a well defined unitary group where U is an element of A and U U star is equal to U star U as well now finally coming to an end it's not just definitions because it will finish at some time the natural adjoint action of U on H is given by XI go U XI U opposite not actually the way it acts the adjoint action it acts on both sides from both the right and the left is very important because actually it would re-tell me this is for physics you know I didn't recognize it you know it took me years to recognize this thing A3 has big implications on the structure of the representations of particles in physics it's not obvious but it will come and this yeah U J U star sorry U J U J star XI because remember the opposite is that so it takes this sorry U star here U XI U star so it takes this form because remember this comes as U star when it comes here it comes as U star opposite right which means that it is J U star star J star which is this guy okay yeah J minus Y okay you know J minus 1 is the J star in this case right yeah is 1 which means essentially that J star is epsilon J if you like yeah you can call it J inverse J is epsilon you know so yeah because you can do by you know taking the complex conjugate of that relation alright now next now we define you see I will not go through all you know that you define in a product in a product in the space that the D acts on the XI that's natural you know D would act the occupator of the help of space then you can take the inner product to form a scalar and this scalar takes this form okay now what one would like of course is that this be invariant under the above transformation the unitary transformation about the transformation which is here we have written it this one be invariant under under the given the mention transformation now and however it's not actually if you do then what happen you know this would really go into you discover that this product you are really going to get things like U J U J star into D of U J U J star star and then actually you discover that this is there's nothing invariant and it really moves along U D U star plus epsilon prime J U D U star of course it's obvious why because the D is going to act on all these guys you know it's a differential operator it will act and obviously this is not invariant because this is the way that happens when you you know in the electrodynamics it's known that you really have what we call phase invariant right psi and if you let this be a function of X you discover that this is not really invariant anymore and the solution of that is very simple what you do that you introduce a connection to make this thing invariant and then mu plus iE mu right this is the usual way this of course knows different from the other one and so what happened is that this D has to replace with DA which involve fluctuation where A is defined to be a one form summation of all possibilities I will show later actually that this A is a connection on this non-competitive space once we define our proper non-competitive space would include all gauge connections as well as the Higgs field so the Higgs field in that case would be another one of the Higgs connection one of the gauge connections one thing actually I have not included actually it's not really included here which takes this actually I called inner automorphism this type of transformation is here that xi goes this one xi okay now the other thing which I didn't talk about is outer automorphism which essentially has to do with the fact the D and that in a way I will try to explain later that this way even if I start from really a trivial flat geometry let me say I will be able to discover all the most general geometry from fluctuating around the flat geometry okay this is not strange because after all you know for people who are aware in particle physics they always like to obtain generativity as taking a flat space and then introducing some metric fluctuating around it and of course you will be able to discover all the the curved space the curvature and things like that by doing this this fluctuating so here actually is no different and in reality actually one really has to fluctuate the D with the outer automorphisms to generate to generate actually the curved the curved metric out even of a flat metric so in other ways actually in principle even if one knows very little you only know some flat space you will be able to generate the most general curved space in the non-committive context okay so what do I do so essentially now I claim that even with this little mathematics that I introduced I am really somehow I have enough information to start and attack the physics problem how should I do that now you know here the thing is that what I am telling you that look instead of using you know the metric starting with ds squared equal j mu nu dx mu dx mu we are really replacing this with some spectral data so I only replace you know things with the metric and how to measure distances by saying look I am not really going to study that but I am going to study that I define my space through some data and then I look at the spectrum of the Dirac operator the spectrum of the Dirac operator in principle has all the geometric information that I need you know this looks abstract but we will show that it is indeed the case so this way you know of course I can define distances in these spaces and things like that but it is really more elaborate because you don't do it that way everything is done by making a cutoff on the Eigen values so everything is done through Eigen values of the Dirac operator of the d operator in this case maybe we should not call it Dirac because Dirac would immediately think that it is simply four dimensional or whatever it can have both it can have you know it's like in quantum mechanics an operator can have no it can have both still even according to my assumption it can have both and okay how do I proceed I have to make an assumption actually the assumption I'm going to make and this is really relaxed from our knowledge in a way because if I really know some non-competive space in which I can give you the data in principle I can compute everything out of it and I should be able to derive all the physics but I don't know this space I don't know what space should I take which is really non-competitive so what we do we make an approximation what is this approximation and anyway this approximation one expects to hold up to extremely small distance this approximation is that the space that we live in or let me say spacetime is defined to be a non-competive space non-competive geometric space and is given by A H D J gamma such that that it is really the tensor product of two spaces one continuous which is the four-dimensional we know and the other discrete which we have to find out okay so in this case I can say A equal A1 tensor A2 H is H1 tensor H2 J is J1 tensor J2 gamma is gamma 1 tensor gamma 2 however the D is not trivial it is D1 cross 1 say plus gamma 5 let me assume you know we call gamma 5 as Pierre has suggested that cross D2 and we are gamma 5 with D1 anti-commute now it's true actually but this looks like a trivial product space however it's not trivial and the reason with that we are going to discover that all the physics somehow is re-dictated by the spectrum of D squared essentially the D squared has the geometric invariant and if you compute D squared you have a mixing you know of course D squared in this case is D1 squared plus D2 squared but nonetheless you know the D squared knows about both in this case so this is the okay what's A1 let me say okay say I'm going to be very conservative and I'm really going to take A1 to be the usual what's equivalent to the data that corresponds to a remaining manifold what's the data corresponding to a remaining manifold the algebra of functions and you know differentiable function in this case and A2 is the Hilbert space of square integrable spinors and Dm actually or D1 in this case I'm going to call gamma mu D mu plus omega mu now of course actually here you may start to object to tell me look how do you know that you are really going to get that as I said even if I don't know I only know that I have flat space that I have gamma mu D mu okay I would be able to generate this for you how do I generate this for you well you know you apply a general cone first of all in this gamma mu of course this is not invariant on general cone transformation you have to make it invariant and then you write it like this yeah and then of course actually when you act on the spinors psi because the spinors you know they they would they would transform under the orthogonal group you will discover actually that you are really going to to make it invariant you have to put a connection which is the spin connection you put the spin connection and then you can you know I can do the calculation if you are interested in which I show you the real condition on the most general D that you can construct which is really consistent with the with the property of the hermiticity of the operator D then I can show you that you can allow spaces with special torsion which would correspond you know to what's known as the B field in string theory it's still consistent you can go you can push it a little bit further but not more anyway there are certain conditions on the torsion that you can do so this calculation can be done but the important thing that is the the the continuous aspect of this no computer space so but now we are really faced with this question I said that this space is a product of you know continuous times discrete what are these A2, A2 well I'm going to call it you know A finite H finite J finite is some finite space you said A2 if you have two spinners on the same and four A1 sorry I don't know I said sort of H1 I'm talking only about you know one now two is discrete now the question that we really now have to ask is what is this finite space for a long time historically you know so this work I have started to do with Alan you know in the longer goal 97 or something and for a long time this was taken ad hoc you know the space was simply taken to be two by two matrices and one by one matrices and things like that and in the end we yeah corresponding to you know knowing somehow it was constructed phenologically looking at the answer and see what would fit geometrically so it's like you know you are given to solve a problem and you are given some data and you try to fit it and you find some answer okay and the answer was found and in the end actually the algebra was found for example to be this finite was found to be like C plus H plus M3 of C and this was done after you know some trial and error and you know it was the answer this I think they put so much grease probably so that it can fly by itself okay or so now actually after many attempts we decided to do a classification of all possible finite spaces you know what are the possibilities to get a good answer because it's not really it should not be God given you know that we take this because it works that's not and this was one of the actually against this program say okay you know the answer you try to fit the answer and so on so what we try to do with that we classify all finite spaces which are consistent with the axioms that I have written before okay now I'm going to show you that you know almost uniquely because you know a problem in mathematics there's always some little cases which come out and will be difficult to exclude you know most practical purposes the analysis gives us you know almost unique answer if yeah but we look we look at the we look at the exceptions we look at the exception the problem is that the exceptions are not interesting but the problem how to rule them out from you know higher principles is very difficult you know because you know I will show you that you know sometimes you have to impose some symmetry then you are going to get some nonsense but the question is that of course is not convincing that you say simply I I take it out so it's there will only be something definitely okay however you know before before moving on I will point to two things that we are considering first of all since we are using spectrum of Dirac operators we are really forced to take Euclidean spaces and the issue of how to go to Minkowski or in other words how does time emerge is really a deep question and is really very similar to is exactly the same actually the question facing people who do path integrals in quantum gravity because there or you do path integrals you are forced to take you know Euclidean spaces otherwise things don't make sense and the question how to go back and what's the criteria for an issue and you know it's still but that's why because D is the Hermitian as a real spectrum yeah but you know but if you consider suppose you have any because I have a continuous spectrum so I have a continuous spectrum the problem is a continuous spectrum it's the unboundedness actually it's the unboundedness it's not the continuous continuous is okay but unboundedness is the problem because with plus minus signatures and you would like things as you see exactly when you do path integrals for gravity or even for gauge series you know you would like things to be positive and this you can only achieve in Euclidean space but D has both plus and minus second value yeah D is not the problem actually you know the problem is that since usually you are going to discover that the spectrum is really the deciding spectrum and then things would become unbounded if you take negative yeah so the action in other words you need you need the action yeah you need compact how about Euclidean Euclidean any compact what it's called what do you mean Euclidean yeah Euclidean is Euclidean signature yeah I think Euclidean signature is enough I think in our cases is enough okay so this actually is one of the problem the other problem we will face is that okay as a consequence of that of course then the question of fermions which we are really going to get would become problematic because it's known actually that especially this problem faced lattice gauge theorist that then immediately we get what we call doubling of fermions and the reason of the following you know I can tell you suppose that you know yeah this psi element of H and the problem that is g psi is an independent spinner or not you know if you have no relation then is not related to psi then it will be a new spinner and then you know physics wise you are going to find that every particle has a mirror and this of course is unacceptable anyway so essentially we really have to require that we have to solve the fermion this product space that I talked about must have something like k all dimension 0 why because then and only then I can impose for example both what we call the chiral condition and the Majorana condition simultaneously if you don't do that then you are going to double your fermions which is a catastrophe physics wise now however we know actually that at least the four dimensional part the four dimensional part you know as already mentioned you know let's think now in minkowski for a moment the four dimensional part is really equivalent to two as we know now you know why it's two because the plus minus of the signature remember I told you that if you have signature say plus minus minus or minus minus plus essentially we cancel each other and then you are talking about effectively two and so obviously if you I only take space time by itself you can never put both conditions simultaneously you are really forced to tensor it with another space whose k all dimension should be 6 because 2 plus 6 is 8 is 0 8 right so this really forces so to start with actually solve the problem we are really forced to take finite space to be of k sorry k o dimension 6 in other words actually the total space should really have dimension 10 like 6 plus 4 is 10 now this should not come as any surprise for people who are familiar with supersymmetry because supersymmetry you know for the first dimension where you can employ both conditions simultaneously is 2 and this is where you have this non-linear sigma model in two dimension and the next one is 10 and that's where the heterotix staring with the existent you know so this is for a long time it it looks different but the same you know mathematics is exactly the same you know the wording is different so now actually it's 4 4 4 we know okay 4 is the continuous 4 continuous but I want the product to be 10 I want the product to be 0 the product should be 0 of the product this should be 0 yes of the product of the full space because then I can employ this condition of course I cannot impose on part on the other that makes sense I have to impose on the full space because then I have to impose on the full space if I want to impose on the full space then I know that the all dimension of the full space should be 8 because then in this case according to my table still up you know 111 is really dimension 0 or 8 whatever okay now we know that dimension 4 now of course now actually you tell me you are cheating because you are saying you are talking about Euclidean and suddenly you jump to Minkowski it looks different actually in Euclidean actually the way it solves itself in Euclidean signature is different than the way it solves itself in Minkowski signature there actually determinants you don't get determinants you get the Phaffian instead and you know so it's it's different different mechanism I will say to solve the same problem I will come to so now you have 4 Euclidean dimensions you want to add yeah 4 Euclidean when you do 8 yes so you don't get 6 because it will be the signature it's not it's first yeah exactly so now actually yeah the way it solves itself is different as I said because the way it solves itself is that you take actually this 6 and this is 4 because we know that when we continue analytically this would be the answer now how does it solve itself I will talk about it later that when you do path integral you don't really get determined of the D you are going to get the Phaffian or the D and the Phaffian of the D it means you don't integrate over the fermions and conjugate you can only integrate on the fermions you know so you you cut the degrees by half and it solves the problem in this very nice way anyway but think of it actually just in this way that we really need a K0 dimension 6 for the finite dimensional space so no this is the now this is the problem is how to by the way there is something very strange in the discrete space because this is a one particle and it presented here I mean for the continuous space this is the one particle in the space yeah why don't we want to continuous geometry based on many particles the fact to say oh there is pinor the thing like that you say innocently it was I didn't say that but this is this is not convincing at all yeah we have to get more physics okay it was based yeah but but like suddenly to say fermions the spin houses it's not it's not natural I mean it's a big action you mean to take take spinors as your basis that's what you are objecting that take spin half as your as your one particle state coming in but physics is not a problem okay well actually okay see the question that I know exactly what you intend but the question how to formulate geometry in terms of other particles will become more and more complicated no the issue is whether is spin half is a building block or not actually this you can think of it as an answer the question if you look is thought that the spin half in the building block okay and indeed actually for example out of spin half you can build the spin zero spin one spin two you can build out of so in principle you can think of the spinner at the building block but of course you know you may argue that why should I take this as as a basis of geometry you know this is well taken but I think if you take is too complicated so my answer is that you without to perform the second quantization after that this is a first step yeah then actually the problem is that okay the proper way of doing it then actually then you have actually to consider enlarging your geometric space to consider operators that act on you know higher higher representations yeah and then of course then it becomes not an argument that it's not there okay alright so now actually we are at this juncture and at this juncture we have to now say what are the possible spaces that would agree with all the actions so what do we do now okay classification finite now commutative spaces of k well actually okay dimension 6 if you want yeah well actually the part of that classification will go without this dimension 6 but to dimension 6 you know immediately will cut will cut things out but before we say I will take there really only two possibilities interestingly there are only two set two kind of spaces that you can have regardless of the 6 okay so finite dimensions so here it means finite algebra are they octonions do you see this is the question finite dimension finite dimension and the dirac operator from the finite dimension this does set of matrices they are the okay physics wise one dirac operator it's a family of dirac operators you know you can get okay the question is what is the set of all possibilities and we'll discover actually that this matrix will be nothing but the yukawa coupling of quark and the leptos okay so the dirac operator is one matrix yeah yeah there is yeah okay it will turn out to be but before I don't know yes yeah but you know this is the answer but of course in this case of course you tell me what's the algebra in this case okay and you know okay I can tell you the result and then you go on maybe some people will not be here next week and they say okay we missed anyway they are really okay yeah but this is not allowed actually it's not allowed the only two possible algebras will turn out to be m n of c where n is k squared actually or or a mn of c well I should say the complexified algebra the complexified algebra can only be this of this one only two cases so in this case the center of the algebra z well here is z plus z okay and for this actually I only use only use the same n for most factors no no no this is for arbitrary here here they are the same here the same but up and down you know so we only use a zero order condition that a we didn't use the first order condition only this that left and right commute left action right action commute is all what you need to show that these are the only two possibility two possible algebras that are there we don't use the existence of a gamma energy sorry the existence of a gamma energy are not used here that's our j yeah of course it's used the j is used look but this is an axiom look the rest are properties let me say rest are properties which you know but this is an that actually we assume that the x that now of course actually you may tell me look maybe everything is an axiom the j square everything is an axiom but at least okay the only thing which is not used in this condition the DAB opposite is zero is not used for in this classification but later actually I'm going to use it to cut it down into into that answer which I told you about it comes in and you know it really has physical implication on the neutrino masses which of course is immense interest for physics now how is the proof done actually you know the proof is really involved you know it took all the expertise of Alan to improving theorems to to crack it down because and was really very surprising that you you know because a prior we don't know we could have obtained million of possibilities and however we only obtain these the proof is based on the following actually I cannot because that would take me an hour simply to to go through the proof but I don't maybe I will do it next time in which I I will give the steps of the proof and I will give you know some tensorial notation for everything I said today I write in terms of matrices so to become more down to earth because up to now I have been you know simply using things abstractly the only thing actually which is used is that the existence of an item important element E squared equal to E E equal to A so you assume that this element the operator does exist and once you do that then you can show that you can have at most either only one only E or E1 and E2 such that you know so in this case E1 E2 would be zero and E1 squared is E1 E2 squared is E2 and this would be E squared equal to E so you know it means that one can show that you can have so the way it's done that you assume you have many and then using all the properties of the J and how you cut all the projection operators how they act on each other how they they would they would talk to the J and you'll discover that you know this is the only the proof is not that complicated actually part of it is really difficult but part of it which is this part that you can have only E1 and E2 is is not difficult you know maybe next time I will go through it but let me let me so in this case actually in this case the J it acts on the X like X star one in the other actually this case if you have J of X Y it would be Y star X star so here because you have two algebra so the way it's it acts on it makes complex conjugation or Hermitian conjugate and now yeah that got for physicists yeah so now now what do we do after then essentially here here actually here the first case is inconsistent with k o dimension 6 and this is how you rule out this thing it's inconsistent with the o dimension 6 you can take it it would give you you know mirror fermions and things like that so this is really thrown out and you are left with this guy now now at this point actually at this point so this is actually the complexity of the algebra let me see okay so anyway in this case actually the algebra is the same as the complexity of the algebra so the question what do we do with this case and here we were forced to make an assumption that one of the algebras one of the algebras on one on one of the algebras there exists an isometry i such that i squared is minus 1 and it acts on the first one and it really limits or restricts well actually here no it's like think of it like i it acts an operator i acts on an element say of m and c like this let me call it alpha i I think let me see is like alpha transpose or something like that maybe alpha star I forgot actually the exact condition but it really limits this m and c to become m a of h which where n is 2a in this case and h is the space of caternios in other words actually it really restricts the form of the n by n matrices to be formed on the caternios they become a symplectic in a way something like that now of course actually this is why do we have to impose this condition we don't really know actually you know if you don't impose this condition you really have to deal with this case is one of the cases we really could not rule out okay but if we take it it gives us something physically not correct actually and you know it's anomalous and things like that so you know this is it's an operator that acts on the elements of the algebra in this case yeah maybe i alpha i yeah I think i alpha i probably is alpha in this case yeah maybe i i would involve also complex conjugation i squared sorry is minus 1 not plus 1 you know some think of it you know think of it that you have a matrix with 0 1 minus 1 0 yeah and then each matrix like you call it a b c d yes and 0 1 minus 1 0 is equal that will give you some conditions on these but but I think there must be some complex conjugation in this in this one okay so this it acts with the complex conjugation in this case a bar I forgot actually the exact action but something like that it limits actually the n is remember is like 2 alpha is even and it limits the structure that it is a sub subgroup instead of you know for example instead of s u4 you may get s u2 cross s u2 or something like that but in this case it really it really becomes in the space of of caternios okay so this actually is something that now the space really becomes the algebra becomes ma of h plus mn of c where n is even but we have to remember that we have a chirality operator gamma where n is yeah n is 2a yeah exactly yeah yes n is 2 so this is yeah you can see a and 2a now we have also the existence of a chirality operator gamma and it has to act you know it may act here it may act here now usually we take it to act on the first and if it acts on the first this ma of h should split into two okay anyway what are really the first possibility okay let's say even and then you say okay let's start by considering the easiest possibility m2 of c plus h do you agree that happens a is 1 now if you really consider this model but here of course actually here I cannot operate with chirality because if I put chirality if I have to put it here then it means you know because this actually is just a chterion so you cannot act with chirality on it if you have to act on this guy if you have to act on this guy and it means you know your particle spectrum anyway is too little and it really would correspond to the you know originally to the u1 cross s2 model the old Weinberg model essentially if you do physics of this you get the old Weinberg model okay no quarks just leptons of course is anomalous and everything but this is the easiest but of course we rule it out because we cannot impose on it the chirality okay so then the simplest case where you can impose the chirality condition is this A equal to 2 A equal to 2 what is this space is M2 of H plus M4 of C okay yeah it should finish in 1 minute if we act with you know we have to remember we have this gamma A sorry gamma okay it's also gamma we have to put this chirality and how do you put the chirality then if you impose chirality on this this splits into H left plus H right it's the first possibility you know now now without doing anything else actually now I really can tell you my fermions how many fermions this is the dimension of the Hilbert space is in this case is re n squared yeah this re is 2n squared but it's cut by chirality it's you know half of 2n squared is re n squared where n is 4 you get 16 so first of all we can predict actually the number of fundamental fermions in this space which are the elements of the Hilbert space it's 16 if I stick to this finite dimensional space in addition they are they follow the representation as follows you know let me call it A dot A I okay this is 1, 2, 3, 4 and this is 1, 2 which is the right this is the right handed and this is left handed yeah so we are almost there actually as far as the particle spectrum is concerned because I tell you what to do this actually is new right plus E right actually usually and of course you say it's a doublet so it's a new right E right new left E left up left down left yeah then I have upright down right it's everything because this is 1 plus color so we are going to get new right E right oh sorry yeah a new left E left upright down right up left down left and then actually the other but at least you have already predicted that you are really going to get 16 fermions in your algebra and this is like the first possibility now other possibilities we can we can go on but then they really become they give you something extremely complicated you know you are talking about SU6 or SU8 gauge groups you know they don't work so what the claim in the following if you classify algebra finite dimensional spaces the first non-trivial possibility you obtain is that of what would turn out to be the standard model and remember actually I have not re-imported the first order condition because when I import the first order condition it would really give me that this one this would follow this also this comes on yeah this is I already warned you I already warned you right from the beginning that we really have no explanation for the number of families that it is 3 3 but on the other hand actually there is nobody else in the world that has been able to give to have a clue about why this number is 3 it's not really clear yeah yeah not only this actually they look like leptons and quarks because of this you know already the 4 here the lepton looks like a 4th color you know it's it's really unified with color in this respect you know and if you really would insist on this symmetry then you can have you know all this b-l and you know you can get really a very nice model but really at grandification but if you are interested in the physics at present scales this is the model to analyze and you know many more indications that will come along the way but I think this the fact that out of a simple classification I did not prove for you the theorem but one can you know I can go through it and it's a miracle that you know something really arbitrary can go and tell you look here it is you know this is yeah this is the idea of Patti Salam actually it's called the Patti Salam unification and it was done almost the same time as Georgia you know Georgia 1974 see Salam almost assigned me this problem you know he told me to choose between this and supersymmetry but then I chose supersymmetry you know what I said Patti Salam Patti is an Indian physicist so next week I will you know maybe I will go through the proof and then you know not only this I'm going to the way my plan is as follows I plan to and these connections that I have indicated I will be able to derive the full connection of the model and how the Higgs field comes out naturally we have formulas very specific thing you know it comes uniquely there is no and then what to do with that after you obtain the connection how to really get the dynamics out of the model you know so it's okay