1/4 Spectral Geometric Unification
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Chamseddine, Ali
A brief introduction to noncommutative geometry with emphasis on the essential tools used in physics.
........................ So I think I will start. So the topic of my lectures is the title is Geometric Unification......................... 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 at least physicists who are familiar with general relativity it's known that general relativity is the example of a geometric unification, a 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 will be seeking in a way a geometric setting to unify all fundamental interactions. And the quest for unification really started a long time ago and indeed started with Einstein in his work with Kallouza where he tried to unify the only two known 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 gravity photon, it doesn't have the correct electric charge. Anyway that 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. Okay, so the main of course many attempts, 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 and 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. Alright, 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 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 in the Grand Unified Theories and so this actually is a principle one would know how to do. Geometrically of course is really based on simple geometry which is 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 give it geometric clothing as I would say. 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 force of what's called 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 forces 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 if 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 you can say vector bundles but here is simply there is 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 mathematicians the vector connection on the u1 cross s2 cross s3 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 different representations which I am really going to derive in a way why they sit in those representations we 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 this 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 geometrical reason why things are the way they are. 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. 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 in string theory it was thought that they have an explanation and the explanation was always you know in terms of Calabiyao 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 100 minus 28 equal to 3 so it was not really 3 essentially 98th of you know families and anti-families will be super massive 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 things I'm going to be able to answer why the Higgs field will 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 one Higgs yeah okay I will I will say I will say something in that actually. You know 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 Patti 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 leptons really comes at the fourth color because you know quarks have three colors you know red, yellow, blue or whatever so the leptons thought that it's 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 so it's one of the axioms and for a long time you know we question 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 in a field but of course you cannot throw it out because this possibility is there, it's a very nice possibility this Pati-Stella unification 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 the Fumovism 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 coordinate invariant, invariant under coordinate transformation. So every action that you write must have this invariance. Okay which is pretty strong but in a way actually this in a way guarantees that the metric of space-time will have its universal interaction because you know 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 you see 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 that the action you really cannot write except by using the metric. Okay and then and it's really a semi-direct product with the symmetries which I have written SU3 cross SU2 cross SU1 this is 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 this invariance and of course actually the problem that doesn't exist actually 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 that you are really going because what happened in higher dimensional theories of course you know 5 the suppose that I linked myself to 5 dimension in this case you get d5x everything is 5 dimensional and in this case you have fields you have extra fields and these extra fields you'd like to think of it from the 4 dimensional point of view and then you would say okay what do I do with the dependence on the 5th coordinate so what you do with the dependence of 5th coordinates you start to expand and you assume that the 5th coordinates like a circle and then you take all the 4-year modes and then you have an infinite tower of states so what people do is they truncate or chop out all the states and you know you get you are really throwing out all the Kaluza Klein modes. In string 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 the assumption is that I will use non-committive geometry as my setting. The setting for unification. And the first question that we ask is that you know why should one go non-committive? Why should one go non-committive? So suppose actually you know I call this N which is the difumorphism across G so N let N be my symmetry with difumorphism times some internal group and what I would like actually is that the difumorphism of this my space N would be this or at least would be that. G is local in other words actually I would like a space such that difumorphism 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 non-committive process in which you really involve matrices essentially. In which you involve matrices. So okay so anyway let me you know start with discussing non-committive geometry. So you know there are many ways of trying to lay the groundwork that okay we have to take non-committive but in reality see all arguments will be only suggestive and the question is that if I take non-committive does it work actually because this is you know the pie and the pudding. So what do we do? How okay so let me start this for a section. Let's take actually the data that defines a non-committive space. And this data let me write it this way okay so this is a spectral triple and this actually the curations shall discuss in a minute. You know the non-committive geometry would in the end of course one may ask that question whether what we know for example would you be able to recover remaining geometry. And the answer of course is yes it already includes remaining 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 known that 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 and most of what we know about the remaining geometry from the formulation of non-committive geometry. So let me actually define well actually the gamma yeah I would define each in a minute you know so A yeah I will you know gamma is chirality you know if you are a physicist this is the analogy one is but it's an antilinear operator if you would like to. Okay what's it? An associative algebra with unit one and involution star H is a complex Hilbert space carrying a faithful representation maybe I should presentation by of the algebra but I think I should have started here anyway D is a self adjoint operator on H with the resolvent D minus lambda inverse where lambda not the amount of R and D is compact is compact D is for physicists to be the dirac operator is to generalize the 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 atiyah, Sanger, 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 the 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 spectrum of the dirac operator actually it really has it carries inside most of the geometric information about the space what is compact is D D minus lambda inverse so if I forgot to write it off I changed the line so yeah exactly okay still let me see if I erase or should I maybe just a second let me push it up okay now actually the J is an antilinear operator sorry unitary it's an unitary it has it involves complex conjugation because on H it's a restructure in other words the reality of the another it really takes us from complex geometry to real geometry the the role of the J take us to real geometry essentially so for physicists charge conjugation gamma 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 yes well well for example you know something like this right and also square the evolution is the identity no no the existence there is identity element you know you need the evolution by default you need to get the element itself okay properties properties first J squared is epsilon where epsilon prime elements of minus or plus one for example zero sorry one in zero dimension and minus one and four c squared is minus one then we define so let a be an element of the algebra a and b then we define the element b as j b star j inverse and this is called the opposite algebra in this case we say b opposite is an element of a opposite the opposite algebra and is defined I mean now let xa be an element of the an element of the helper space you can think of it as a spinner yeah then you know you can act on the element of the helper space like this you can act from the left action from the left then the opposite would act from the right so in this case we write like this okay no it's okay if you act from the equivalent actually to an action from the right okay so if you have something ba 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 the right action the right action and the left action commute with each other yes these are the properties actually yeah I will you know I will talk about it actually so it so happened actually this is you know if think of it this way you want to do charge properties of charge conjugation operator the properties of charge conjugation operator really in Euclidean space say they depend on dimension and their properties where their square is one and not one and you know you can take their action on the cleft forward 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 cohomology orthogonal cohomology 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 in the corresponding dimension they have exactly same properties yeah no this actually is yeah proper this is the proper this square is up and the two actions they do agree the left and the right actions they do not happen oh sorry sorry sorry action one left and right action commute now second this actually let me say property one property two is that dj you know is JD up to a sign that's what it means actually this means j squared is plus or minus one dj commutes or anti commutes with JD as I said all these are minus of minus one plus one and and and then say d gamma all is anti commutes with the d but j gamma is now not section that now I have introduced 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 on any table and for each you can you know label give 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 it's taken to coincide with the classification of what we call Majorana and Weyl spinners Majorana and Weyl spinners in when we are talking about you know Clifford Algebras and charge conjugation matrices it's exactly same classification so the 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 mod 8 it's minus it's 2 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 1 1 1 you call it 0 and then you have 1 1 minus 1 this actually you don't define except for even spaces for you know minus 1 plus 1 minus 1 and so on so you really can make a table and these are called the co-dimension you know I can do minus 1 how many 1 yeah 4 minus 1's is the 1 1 minus 1 okay oops then you have 1 so this actually called the 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 Weyl spinners for example people in string theory would recognize usually the 10 dimension actually 10 actually 10 dimensions really have to look at 8 which is the 0 which is 1 this one you know why because in 10 space dimensions you are going to say to get minus plus plus 9 times right and the two would essentially cancel each other so you are really talking about 8 essentially and if you make the classification you discover that everything repeats itself mod 8 that's the reason actually why Majorana and Weyl spinners would exist in 2 dimension and then 10 dimensions okay and the 2 you know because the plus minus can see each other is really equivalent you know 0 is equivalent to 2 and you know then the next one would be 10 dimension and what is your real name for gamma 5 actually? Yeah it's the chirality, gamma of chirality yeah in 4 dimensions it's called gamma 5 but of course in 10 dimension it does exist it's still called gamma 11 in that case you know so the traditional yeah gamma 5 okay okay so what do I do with that now should I keep using the center or okay okay okay okay okay okay okay okay okay okay okay okay okay okay it's not trivial to show that if you don't assume this then the connection will not be linear, it will have quadratic, only quadratic not higher actually which is really another 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 non-computed geometries for which this condition is not true actually and this is really a quantum sphere, it's not true then actually yeah for what gamma squared is 1, the gamma commutes 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 as well as that you are really going to get left handed spinners or right handed spinners spinners now we do assume that H is in doubt with an A bimodule 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 a joint action of U on H is given by XI U XI U opposite, not actually the way it acts the joint action it acts on both sides from both the right and the left is very important because actually it would really tell me this is for physics you know I didn't recognize it you know took me years to recognize this thing and it has big implications on the structure of the representations of particles in physics, it's not obvious but this will come and this U J U star sorry U J U J star XI U opposite is that so it takes this sorry U star here yeah U XI U star so it takes this form because remember this comes as when it comes here it comes as U star opposite which means that it is J U star J star which is this guy okay yeah J minus Y okay you know J minus one is the J star in this case right yeah is one which means essentially the J star is epsilon J if you like yeah you can call it J N versus the same story yeah because J square is epsilon you know so it's um yeah because you can do by you know taking the complex conjugate of that relation alright now next okay now we define I will not go through all you know that you define inner product inner product in the space that the D X on the XI is natural you know and D would act the the operator would act on an element of the Hilbert space then you can take the inner product to form a scalar and this scalar takes this form and 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 given the mentioned 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 J 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 so this is not really strange because this is the way that happens when you in electrodynamics 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 is that you introduce a connection to make this thing invariant and then you said D mu would go into D mu plus I E mu right this is the usual way this is of course no different from the other one and so what happened is that this D has to replace with DA which involve fluctuations where A is defined to be a one form summation of all possible ways I will show later actually that this A really is a connection on this non-computer space once we define our proper non-computer space would include all gauge connections as well as the Higgs field so the Higgs field in that case you know would be another one of the Higgs connections one of the gauge connections one thing actually I have not included actually is not really included here which takes this actually I called inner automorphisms this type of transformation here that psi goes this one psi okay now the other thing which I didn't talk about is outer automorphism which essentially has to do with the fact that D you know 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 in 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 curvature and things like that by doing this fluctuating so here actually is no different and in reality actually one really has to fluctuate the D so the outer automorphisms to generate actually 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-commitative 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'm telling you that look instead of using 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 things with the metric and how to measure distances by saying look I'm not really going to study that but I'm 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'll show that it's indeed the case so this way you know of course I can define distances in these spaces and things like that but it's really more elaborate because you don't do it that way everything is done by making a cutoff on the eigenvalues so everything is done through eigenvalues of the Dirac operator of the d operator in this case maybe you should not call it Dirac because Dirac would immediately think that it is simply four dimensional or whatever it's it can have both it's like quantum mechanics an operator can have no it can have both even according to my assumption it can have both and you know okay how do I proceed I have to make an assumption actually the assumption I'm going to make and this is relaxed from our knowledge in a way because if I really know some non-competitive 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 the space I don't know what space should I take which is really non-competitive so what do 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 space time is defined to be a non-competitive space non-competitive geometric space and is given by a h d j gamma such that that it is really the tensed up 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 the pair has suggested that cross d2 and where gamma 5 with d1 anticomutes now it is true actually that this looks like a trivial product space however it is 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 envelope 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 am going to be very conservative and I am 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 the differentiable function in this case and a2 is the Hilbert space of square integrable spinors and dm actually or d1 in this case I am going to call gamma mu dmu 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 dmu okay I would be able to generate this for you how do I generate this for you well you know you apply a general first of all in this gamma mu of course it's not invariant on general transmission 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 transform under the orthogonal group you discover actually that you are really going to make it covariant 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 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 you know this is in principle is part of the remaining geometry or the continuous aspect of this non-competitive 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 is this a2 h2 well I'm going to call it you know a finite h finite j finite gamma is some finite space you said a2 with the a2 spinors on the same I said 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 long ago 97 or something and for a long time this was taken ad hocly you know the space was simply taken to be 2 by 2 matrices and 1 by 1 matrices and things like that and in the end we 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 I think they put so much grease probably so that it can fly by itself okay 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 that we take this because it works that's not and this was one of the criticism actually against this program say okay you know the answer you try to fit the answer 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 in mathematics there's only some little cases which come out and will be difficult to exclude but for you know most practical purposes the analysis gives us you know almost unique answer if yeah but we look at the exceptions we look at the exceptions 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 would show you that sometimes you have to impose some symmetry if you don't impose then you are going to get some nonsense but the question is that of course is not convincing that you say simply I take it out so it's there will only be something definitely okay however you know 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 now the question is how to go to Minkowski or in other words how does time emerge is really a deep question and is really very similar to 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 is really a thorny issue and you know it's still but why because D is Hermitian as a real spectrum you know but if you consider suppose you have any you could have a continuous spectrum so the problem is a continuous spectrum it's the unboundedness not the continuous is okay but unboundedness is the problem because with plus minus signatures you can go to the negative and you would like things as you see exactly when you do path integrals in 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 yeah D is not the problem actually you know the problem with the discourse is usually you are going to discover that the spectrum D is the squared the deciding spectrum and then things will become unbounded if you take negative yeah so the action in other words the act you need you need the action yeah you need compact compact what do you mean Euclidean Euclidean is a signal yeah I think Euclidean 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 is 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 you have this psi element of H then you can take g psi and then the problem is g psi is an independent spinner or not if you have no relation then it's 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 doubling problem now to solve the fermion doubling problem it would mean the following that 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 let's think now in Minkowski for a moment and the four-dimensional part is really equivalent to 2 as we know now you know why it's 2 because the plus minus of the signature remember I told you that if you have signature say plus minus minus or minus plus plus plus minus plus essentially we cancel each other and then you are talking about effectively 2 and so obviously if you 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, 8 is 0, 8 right so this really forces so to start with actually solve the fermed up problem we are really forced to take finite space to be of k sorry k o by 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 nonlinear sigma model in two dimension and the next one is 10 and that's where the heterotix string would exist and you know so this is for a long time and you know it's it takes different the same you know mathematics is exactly the same you know the wording is different so now actually it's exactly 4, 4 we know 4 is the 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 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 if I want to impose on the full space then I know the k o dimension of the full space should be 8 because then in this case according to my table still up you know 1, 1, 1 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 and I will talk about it later actually what happens in that it's more interesting actually the way it solves itself in Euclidean signature is different than the way it solves itself in Minkowski signature there actually you determine you don't get terms you get the Phafian instead and you know so it's different mechanism I will say to solve the same problem I will come to it so now you have 4 Euclidean dimensions you want to add 4 Euclidean yes 0, 0, 0, 8 yes so you don't get 6 because in the signature it's not this person yeah exactly so I know now actually yeah the way it solves itself is different as I said you know because the way it solves itself is that you take actually the 6 and this is 4 because we know 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 integrals you don't really get determined of the D you are going to get the Phafian or the D and the Phafian 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 cut you cut the degrees by half and it solves the problem in this very nice way anyway but think of it actually just this way that we really need a k0 dimension 6 for the finite dimension space so now this is the problem is how to there is something very strange in the discrete space this is the one particle I mean for the continuous space in the space yeah why don't we want to continuous geometry based on many particles the fact to say oh there are spin-offs and things like that you say you know something different I don't have anything to say about that but this is not convincing at all we are talking about physics so we are talking about the physics but like suddenly to say fermions and spin-offs it's not natural I mean it's a big action you mean to take spinners as your basis that's what you are objecting that to take spin half as your what about the whole states coming in but physics is not 100% 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 is this you can think of it as an answer to the question you know if you look at Wheeler book you always thought that the spin half is the building block okay and indeed actually for example out of spin half you can build spin 0 spin 1, spin 2 you can build you know out of principle you can think of the spinner as the building block but of course you know you may argue that why should I take this as a basis of geometry you know this is well taken but I think if you take the other point it becomes too complicated I would say this is a becomes too complicated so my answer is that you would have to perform the second quantization out of that this is the first step yeah then actually the problem is that okay the proper way of doing it 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 much more complicated but of course it doesn't rule it out you know I cannot say it complication is 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 of finite now commutative spaces of k well actually in okay k o dimension 6 yeah well actually the part of the classification will go without this dimension 6 but dimension 6 you know immediately will cut things out but before we say I will take that really only two possibilities interestingly there are only two set two kind of spaces that you can have regardless of the 6 okay finite you know what are they are they are the algebra of matrices are they caterions or what are they are they octonions you see this is a question finite dimension this does make set of matrices they are the okay physics wise it's a family of the occupants you know it's you can get okay the question that what is the set of all possibilities and we discover actually that this matrix will be nothing but the coupling of quarks and the leptos okay so the derivative is one matrix from the yes yeah there is yeah okay it will turn out to be but before I don't know a mission matrix yes yeah but you know this is the answer so you know what these rules are yes we have these rules yeah 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 we must anyway they are really okay yeah only two possibilities because if I take a C full algebra it can be a very large dimension 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 m n 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 is z where here is z plus z okay and for this actually I only use only use the same n for most factors no no different you know this is for arbitrary arbitrary here they are the same yeah here is 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 possibilities two possible algebras that are there sorry the distance of the gamma and the j I'm not used the tensor j of course 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 actually we assume that the x that now of course actually you may tell me look maybe everything is an axiom the j squared everything is an axiom but at least okay the only thing which is not used in this condition the daa b 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 allent to improving theorems to to crack it down because and was really very surprising that you know because a prior we don't know it could have obtained million possibilities and however we only obtain these the proof is based on the following actually I cannot I not because that would take me one 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 simply using things abstractly the only thing actually which is used that the existence of an idon potent element is equal to E, E equal to AE so you assume that this this projection operator does exist and once you do that then you can show that you can have at most either only one or only E or E1 and E2 such that you know so in this case E1, E2 would be 0 and E1 squared is E1, E2 squared is E2 and this would be E squared equal to E so I you know it means that you 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 would talk to the J and you'll discover that you know this is the only proof is not that complicated actually part of it, part of it is really difficult but part of it which is this part that you can have only E1 and E2 is not difficult you know maybe next time I would go through it but let me let me so in this case actually in this case the J would act on the X like X star one and the other actually this case if you have J of XY it would be Y star X star so here because you have two J so the way it's it acts on it that takes complex conjugation or her mission conjugate and now yeah dagger for physicist no okay now what do we do after the question 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 okay okay okay now now at this point actually at this point so this is actually the complex for the algebra let me see okay so anyway in this case actually the algebra is the same 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 a lysometry of 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 an operator i acts on elements 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 where n is 2a in this case and h is the space of caternions in other words actually it really restricts the form of the n by n matrices to be formed on the caternions they become a symplectic in a way 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 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 is yeah I think i alpha i probably is alpha in this case but i would involve also complex conjugation i squared sorry is minus 1 not plus 1 you know so think of it that you have a matrix like you know think of it that you have a matrix with 0 1 minus 1 0 and then each matrix like you call it a b c d s and 0 1 minus 1 0 is equal that will give you some conditions on this but I think there must be some complex conjugation in this one so this it acts with a complex conjugation in this case a bar I forgot the exact action but something like that and that 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's really it really becomes the space of caternions so this actually is something that and in a way actually now the space really becomes the algebra becomes m a of h plus m n 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 yes n is 2a so this is yeah you can see a and 2a right now we have also the existence of a chirality operator gamma and it has to act you know it may act here or it may act here now usually we take it to act on the first and if it acts on the first this m a 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 what's the easiest possibility m2 of c plus h 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 non-chlorion so you cannot act with chirality on it if you have to act on this guy if that on this guy then it splits m2c into c plus c 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 it's 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 it doesn't work okay so then the simplest case where you can impose the chirality condition is this a equal to 2 a equal to 2 what's in this space is m2 of h okay yeah I should finish in one minute if we act with you know we have to put remember we have this gamma a is a gamma or something you know it's sorry gamma psi equal okay it also commutes gamma a is a gamma we have to put this chirality in and how do you put the chirality in then if you impose chirality on this this splits into h left plus h right plus m4 of c is the first possibility you know now now without doing anything else actually now I really can tell you my fermions how many fermions I have in my space because the number of fermions the dimension of the Hilbert space is in this case is n squared yeah this is 2n squared but it's cut by chirality it's you know half of 2n squared is 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 dimension of 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 a particle spectrum is concerned because tell you what to do this actually is nu right plus e right actually usually you know of course you say it's a doublet so it's a nu right e right nu left e left left down left yeah then I have upright down right it's everything because this is one plus color so we are going to get nu right e right nu 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 come they give you something extremely complicated you know they are talking about SU6 or SU8 gauge groups and we know they don't work so what the claim is the following if you classify algebras finite dimensional spaces the first non-trivial possibility you obtain is that of what would turn out to be the standard model remember I have not re-import the first order condition because when I import the first order condition it would really give me that this one this would follow you know this also this comes on yeah this is you know 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 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 not only this actually from this picture they look at leptons and quarks because of this you know already the 4 here the leptons looks like a fourth color you know it's really unified with color in this respect and if you really would insist on this symmetry then you can have you know all this p minus 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 there are many 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 leptons is a fourth color for quarks this is the idea of pati salam actually it's called the pati salam unification and it was done almost the same time as georgia you know georgia 1974 salam almost assigned me this problem you know he told me to choose between this and supersymmetry but then I chose supersymmetry pati salam pati salam pati is an Indian physicist you know 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 with this connections that I have the indicated I'm able to derive the full connection of the model and how the Higgs field comes out naturally we have formulas very specific thing 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 so it's a...
... ... ... ... ... ... ... ...  So I think I will start.  So the topic of my lectures is  the title is Geometric Unification. ... ... ... ... ... ... ... ...  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 at least physicists  who are familiar with general relativity  it's known that general relativity is the  example of a geometric unification, a 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 will be seeking in a way a geometric  setting to unify all fundamental interactions.  And the quest for unification really started  a long time ago and indeed started with Einstein  in his work with Kallouza  where he tried to unify the only  two known 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  gravity photon, it doesn't have the correct electric charge. Anyway  that 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.  Okay, so the main  of course many attempts, 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  and 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.  Alright, 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  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  in the Grand Unified Theories and  so this actually  is a principle one would know how to do.  Geometrically of course is really based on simple geometry which is  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 give it geometric  clothing as I would say. 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 force of what's called 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 forces 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 if 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  you can say vector bundles but here is simply there is 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 mathematicians the vector connection on the  u1 cross s2 cross s3 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  different representations which I am really going to derive in a way  why they sit in those representations  we 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 this 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  geometrical reason why things are the way they are.  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. 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  in string theory it was thought that they have an explanation  and the explanation was always you know in terms of  Calabiyao 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  100 minus 28 equal to 3 so it was not really 3  essentially 98th of you know families and  anti-families will be super massive 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 things I'm going to be able to answer  why the Higgs field will 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  one Higgs yeah okay  I will I will say  I will say something  in that actually. You know 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 Patti 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  leptons really comes at the fourth color because you know quarks have three colors  you know red, yellow, blue or whatever so the leptons thought that it's  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 so it's one of the axioms and for a long time  you know we question 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  in a field but of course you cannot throw it out because this  possibility is there, it's a very nice possibility this Pati-Stella  unification 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 the Fumovism 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 coordinate  invariant, invariant under  coordinate transformation. So every action that you write must have  this invariance. Okay  which is pretty strong but in a way actually this in a way guarantees  that the metric  of space-time will have its universal  interaction because you know 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 you see 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  that the action you really cannot write except by using the metric.  Okay and then  and it's really a semi-direct product  with the symmetries which I have written SU3 cross SU2 cross  SU1 this is 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 this invariance and of course actually the problem that  doesn't exist actually 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  that you are really going because what happened in  higher dimensional theories of course you know 5 the suppose that I  linked myself to 5 dimension in this case you get d5x  everything is 5 dimensional and in this case you have fields  you have extra fields and these extra fields  you'd like to think of it from the 4 dimensional point of view  and then you would say okay what do I do with the  dependence on the 5th coordinate so what you do with the dependence of 5th coordinates you start to expand  and you assume that the 5th coordinates like a circle  and then you take all the 4-year modes and then you have an infinite tower of states  so what people do is they truncate or chop out all the states  and you know you get you are really throwing out  all the Kaluza Klein modes. In string 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  the assumption is that I will use non-committive geometry  as my setting. The setting  for unification.  And the first question that we ask is that you know why should  one go non-committive? Why should one go non-committive?  So  suppose actually you know I call this N  which is the difumorphism across G so N let N  be my symmetry with difumorphism  times  some internal group and  what I would like actually is that the difumorphism  of this my space N would be this  or at least would be that.  G is local  in other words actually I would like a space such that difumorphism 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  non-committive process in which you really involve matrices  essentially.  In which you involve matrices. So  okay so anyway let me you know  start with  discussing non-committive geometry. So you know there are many  ways of trying to lay the groundwork that okay  we have to take non-committive but in reality  see all arguments will be only suggestive  and the question is that if I take non-committive does it work actually  because this is you know the pie and the pudding.  So what do we do?  How  okay so let me start this for a section.  Let's take actually  the data  that defines a non-committive  space. And this data  let me write it this way  okay so  this is a spectral triple  and this actually the curations  shall discuss in a minute.  You know the non-committive  geometry would in the end of course one may ask that question  whether what we know for example would you be able to recover remaining geometry.  And the answer of course is yes it already includes remaining 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 known that 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  and most of what we know about the remaining geometry  from the formulation of non-committive geometry.  So let me actually define  well actually the gamma yeah I would define each in a minute you know  so  A  yeah I will you know  gamma is chirality you know if you are a physicist this is the  analogy one is but it's an antilinear  operator if you would like to. Okay what's it?  An associative algebra  with unit one  and involution star  H  is a complex  Hilbert space  carrying  a faithful representation  maybe I should  presentation by  of the algebra  but I think I should have started here  anyway D is a self  adjoint  operator  on H  with the  resolvent  D minus lambda inverse  where lambda  not the amount of R  and D is compact  is compact  D is for physicists to be the dirac operator is to generalize the  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 atiyah, Sanger, 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 the  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 spectrum of the dirac operator  actually it really has it carries inside  most of the geometric information about the space  what is compact is D  D minus lambda inverse  so if I forgot to write it off I changed the line  so yeah exactly  okay still  let me see if I erase or should I  maybe  just a second let me push it up  okay  now actually the J is an antilinear  operator  sorry unitary  it's an unitary it has it involves complex  conjugation because on H  it's a restructure  in other words the  reality of the another it really takes us from complex  geometry to real geometry the the role of the J  take us to real geometry essentially  so for physicists charge conjugation  gamma  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  yes  well  well for example you know something like this right  and also  square the evolution is the identity  no no the existence there is identity element you know you need the  evolution by default  you need to get the element itself  okay  properties  properties  first J squared is epsilon  where epsilon prime  elements of minus or plus one  for example  zero sorry one in zero dimension  and minus one and four  c squared is minus one  then we define  so let a be an element of the algebra a and b  then we define  the element b as j b star  j inverse and this is called the opposite algebra in this case we say  b opposite is an element of a opposite  the opposite algebra  and is defined  I mean now let  xa be an element of the  an element of the helper space you can think of it as a spinner  yeah then you know you can act on the element of the helper space  like this you can act from the left  action  from the left then  the opposite would act from the right  so in this case  we write like this  okay  no it's okay if you act from the equivalent  actually to an action from the right okay so if you have something ba  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  the right action  the right action and the left action  commute with each other  yes  these are the properties actually  yeah I will you know I will talk about it actually so it so happened actually this is  you know if think of it this way  you want to do charge properties of charge conjugation operator  the properties of charge conjugation operator really  in Euclidean space say they depend on dimension  and their properties where their square is one and not one  and you know you can take their action on the  cleft forward 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 cohomology orthogonal  cohomology 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 in the corresponding dimension  they have exactly same properties  yeah no this actually is yeah proper this is the proper  this square is up  and the two actions they do agree the left and the right  actions they do not happen oh sorry sorry sorry  action one left and right action commute  now  second this actually let me say  property one property two  is that  dj  you know is  JD up to a sign that's what it means actually  this means j squared is plus or minus one dj  commutes or anti commutes with JD as I said all these are  minus of minus one plus one and  and  and then say  d gamma all is anti commutes with the d  but  j gamma is  now not section that now I have introduced  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 on any table  and for each you can you know label give 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 it's taken  to coincide with the classification  of what we call Majorana and Weyl spinners  Majorana and Weyl spinners  in  when we are talking about you know Clifford Algebras and charge conjugation  matrices it's exactly same classification so the  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 mod 8 it's minus it's 2 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 1  1 1 you call it 0  and then you have 1 1 minus 1  this actually you don't define except for even spaces for  you know minus 1  plus 1  minus 1 and so on  so you really can make a table and  these are called the co-dimension you know I can  do minus 1  how many  1  yeah 4 minus 1's is the  1 1 minus 1  okay  oops then you have 1  so this actually called the  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 Weyl spinners  for example people in string theory would recognize  usually  the 10 dimension actually 10 actually 10 dimensions really have to  look at 8 which is the 0 which is 1 this one  you know why because in 10 space dimensions you are going to  say to get minus plus plus 9 times right  and the two would essentially cancel each other so you are really talking about  8 essentially and if you make the classification you  discover that everything repeats itself mod 8 that's the reason actually  why Majorana and Weyl spinners would exist in 2 dimension and then 10 dimensions  okay and the 2 you know because the plus minus can see each other  is really equivalent you know 0 is equivalent to  2 and you know then the next one would be 10 dimension  and  what is your real name for gamma 5 actually? Yeah it's the chirality, gamma of chirality  yeah in 4 dimensions it's called gamma 5 but of course in 10 dimension it does exist  it's still called gamma 11 in that case you know so  the traditional yeah gamma 5 okay  okay so what do I do with that now  should I  keep using the center  or  okay  okay  okay  okay  okay  okay  okay  okay  okay  okay  okay  okay  okay  okay it's not trivial to show that if you don't  assume this then the connection will not be linear, it will have quadratic, only quadratic  not higher actually which is really another 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  non-computed geometries for which this condition is not true actually  and this is really a quantum sphere, it's not true  then actually yeah for what gamma  squared is 1, the gamma commutes 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 as well as that  you are really going to get left handed spinners or right handed spinners  spinners  now we do assume that  H is in doubt  with an  A bimodule 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  a joint action  of U  on H  is given by  XI  U XI  U opposite, not actually the way it acts  the joint action it acts on both sides  from both the right and the left is very important because actually it would  really tell me this is for physics you know I didn't  recognize it you know took me years to recognize this thing  and it has big implications on the structure of the representations of particles  in physics, it's not obvious but this will come  and this  U J  U star  sorry U J U  J star XI  U opposite is that  so it takes this  sorry U star here  yeah U XI U star so it takes this form  because remember this comes as  when it comes here it comes as U star opposite  which means that it is J U star  J star which is this guy  okay  yeah J minus Y okay you know J minus one is the  J star in this case right  yeah is one which means essentially  the J star is epsilon J if you like  yeah  you can call it J N versus the same story  yeah because J square is epsilon you know so it's  um  yeah  because you can do by you know taking the complex  conjugate of that relation  alright now next  okay  now we define  I will not go through all you know that you define inner product  inner product in the space that the D X on the  XI is natural you know and D would act the the  operator would act on an element of the Hilbert space then you can take the inner product  to form a scalar and this scalar takes this form  and 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  given the mentioned 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 J 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 so this  is not really strange because this is the way  that happens when you  in electrodynamics  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 is that you introduce a connection to make this thing invariant  and then you said D mu would go into D mu plus  I E mu right this is the usual way this is of course  no different from the other one and  so what happened is that this D has to replace  with DA which involve fluctuations  where A  is defined to be a one form  summation of all possible ways  I will show later actually that this A really  is a connection on this non-computer space once we  define our proper non-computer space would include  all gauge connections as well as the Higgs field  so the Higgs field in that case you know would be another  one of the Higgs connections one of the gauge connections  one thing actually I have not included actually is not really included here  which takes this actually I called inner automorphisms  this type of  transformation here that psi goes  this one  psi  okay now the other thing which I didn't  talk about is outer automorphism which essentially  has to do with the fact that D you know 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  in 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  curvature and things like that by doing this  fluctuating so here actually is no different and in reality  actually one really has to fluctuate the D  so the outer automorphisms to generate  actually 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-commitative 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'm telling you that look instead of using  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 things with the metric and how to measure distances  by saying look I'm not really going to study that but I'm 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'll show that it's indeed the case  so this way  you know of course I can define distances in these  spaces and things like that but it's really more elaborate because  you don't do it that way everything is done by making a cutoff  on the eigenvalues so everything is done through eigenvalues  of the Dirac operator of the d operator in this case maybe you should not  call it Dirac because Dirac would immediately think that it is simply  four dimensional or whatever it's  it can have both  it's like quantum mechanics an operator can have  no it can have both  even according to my assumption it can have both  and you know okay how do I proceed  I have to make an assumption actually the assumption I'm going to make and this is  relaxed from our knowledge in a way because if I really  know some non-competitive 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 the space I don't know  what space should I take which is really non-competitive  so what do 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 space time  is defined to be a non-competitive space non-competitive geometric space  and is given by  a h d  j gamma such that  that it is really the tensed up 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 the pair has suggested that cross d2 and where gamma  5 with d1 anticomutes  now it is true actually that this looks like a  trivial product space however  it is 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 envelope 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 am going to be very conservative  and I am 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 the differentiable function in this case  and a2 is  the Hilbert space of square integrable spinors  and  dm actually or d1 in this case  I am going to call gamma mu dmu 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 dmu  okay I would be able to generate this for you  how do I generate this for you well you know you apply a general  first of all in this gamma mu of course  it's not invariant on general transmission 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  transform under the orthogonal group  you discover actually that  you are really going to make it covariant  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 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  you know this is in principle is part of the  remaining geometry or the continuous aspect  of this non-competitive 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 is this a2 h2 well I'm going to call it  you know a finite  h finite j finite gamma  is some finite space  you said a2 with the a2 spinors on the same  I said 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 long ago  97 or something and for a long time this  was taken ad hocly you know the space was simply taken to be  2 by 2 matrices and 1 by 1 matrices  and things like that and in the end  we  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  I think they  put so much grease probably so that it can fly by  itself okay 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 that we take this because it works  that's not and this was one of the criticism actually against this program  say okay you know the answer you try to fit the answer  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  in mathematics there's only some little cases which come out and  will be difficult to exclude but for you know most practical  purposes the analysis gives us you know  almost unique answer if  yeah but we look at the  exceptions we look at the exceptions 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 would show you that  sometimes you have to impose some symmetry if you don't impose  then you are going to get some nonsense but the question is that of course  is not convincing that you say simply I take it  out so it's there will only be something definitely  okay however you know  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  now the  question is how to go  to Minkowski  or in other words how does time emerge  is really a deep question and is really very similar to  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 is really a thorny issue  and you know it's still  but why because D is Hermitian  as a real spectrum you know but if you consider suppose you have any  you could have a continuous spectrum so  the problem is a continuous spectrum  it's the unboundedness not the continuous is okay but unboundedness  is the problem because with plus minus signatures you can go to the negative  and you would like things as you see exactly when you do path integrals in  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  yeah D is not the problem actually you know the problem with the discourse  is usually you are going to discover that the spectrum D is the squared  the deciding spectrum and  then things will become unbounded if you take negative  yeah so the action in other words the act you need  you need the action yeah you need compact  compact what do you mean Euclidean  Euclidean is a signal  yeah I think Euclidean 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  is 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 you have this psi element of H then you can take  g psi and then the problem is g psi is an independent spinner or not  if you have no relation then  it's 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  doubling problem now to solve the fermion doubling problem it would mean  the following that 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  let's think now in Minkowski for a moment and the four-dimensional part is really  equivalent to 2 as we know now you know why it's 2 because  the plus minus of the signature  remember I told you that if you have signature say  plus minus minus or minus plus plus plus  minus plus essentially we cancel each other and then you are  talking about effectively 2 and so obviously if you  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, 8 is  0, 8 right so this really  forces so to start with actually solve the fermed up problem  we are really forced to take  finite space  to be of k  sorry k o by 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 nonlinear sigma model  in two dimension and the next one is 10  and that's where the heterotix string would exist and you know so this is  for a long time and you know it's it takes different  the same you know mathematics is exactly the same you know the wording is different  so now actually it's  exactly  4, 4 we know  4 is the 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 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 if I want to impose on the full space then I know  the k o dimension of the full space should be 8 because then in this case  according to my table still up you know 1, 1, 1 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 and I will talk about it later actually what happens in that  it's more interesting actually the way it solves itself in Euclidean  signature is different than the way it solves itself in  Minkowski signature there actually you determine  you don't get terms you get the Phafian instead and you know so it's  different mechanism I will say  to solve the same problem I will come to it  so now you have 4 Euclidean dimensions you want to add  4 Euclidean yes 0, 0, 0, 8 yes so you don't get  6 because in the signature it's not this person  yeah exactly so I know now actually yeah  the way it solves itself is different as I said you know because the way it solves itself  is that you take actually the 6 and this is 4 because we know  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 integrals you don't really get determined of the D  you are going to get the Phafian or the D and the Phafian 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 cut  you cut the degrees by half and it solves the problem in this very nice way  anyway but think of it actually just  this way that we really need a k0 dimension  6 for the finite dimension space  so now this is the problem is how to  there is something very strange in the discrete space  this is the one particle I mean for the continuous space  in the space yeah why don't we want to  continuous geometry based on many particles  the fact to say oh there are spin-offs and things like that you say  you know something different I don't have anything to say about that  but this is not convincing at all  we are talking about physics so we are talking about the physics  but like suddenly to say fermions and spin-offs  it's not natural I mean it's a big  action you mean to take spinners as your basis that's what you are  objecting that to take spin half as your  what about the whole states coming in  but physics is not 100%  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 is this you can think of it as  an answer to the question you know if you look at Wheeler book you always thought that the  spin half is the building block okay and indeed actually for example  out of spin half you can build spin 0  spin 1, spin 2 you can build you know out of  principle you can think of the spinner as the building block but of course you know you may argue  that why should I take this as a basis of geometry you know this is well  taken but I think if you take the other point  it becomes too complicated I would say this is a becomes too complicated  so my answer is that you would have to perform the second quantization  out of that this is the first step  yeah then actually the problem is that okay the proper way of doing it 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 much more complicated  but of course it doesn't rule it out you know I cannot say it  complication is 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  of finite  now commutative spaces  of k  well actually in okay  k o dimension 6  yeah well actually the part of the classification will go without this  dimension 6 but dimension 6 you know immediately will cut  things out but before we say I will take that really only two  possibilities interestingly there are only two set two kind of spaces  that you can have regardless of the 6 okay  finite  you know what are they are they are the algebra of matrices are they caterions or what are they  are they octonions you see this is a question  finite dimension  this does make set of matrices they are the okay  physics wise it's a family of the  occupants you know it's you can get okay the question that what is the set of all  possibilities and we discover actually that this matrix will be nothing but the  coupling of quarks and the leptos okay  so the derivative is one matrix from the  yes yeah there is yeah okay it will turn out to be but before I don't  know a mission matrix yes yeah but you know this is the answer  so you know what these rules are yes we have these rules yeah 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  we must anyway they are really okay yeah only  two possibilities because if I take a C full algebra it can be a very large  dimension 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 m n 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  is z where here is z plus z  okay  and for this actually I only use  only use  the same n for most factors no no different you know this is for  arbitrary arbitrary here they are the same yeah  here is 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 possibilities two possible algebras  that are there  sorry the distance of the gamma and the j I'm not used  the tensor j of course 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  actually we assume that the x that now of course actually you may  tell me look maybe everything is an axiom the j squared everything is an axiom  but at least okay the only thing which is not  used in this condition the daa b 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 allent to improving theorems to  to crack it down because and was really very surprising  that you know because a prior we don't know it could have obtained million  possibilities and however we only obtain these  the proof  is based on the following  actually I cannot I not because that would take me one 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 simply using things abstractly  the only thing actually which is used  that the existence of an idon potent element is  equal to E, E equal to AE so you assume that this  this projection operator does exist and once you  do that then you can show  that you can have at most  either only one  or only E or E1  and E2  such that you know  so in this case E1, E2 would be 0  and E1 squared is E1, E2 squared is E2  and this would be E squared equal to E so I you know it means that  you 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 would talk to the J  and you'll discover that you know this is the only  proof is not that complicated actually part of it, part of it is really  difficult but part of it which is this part that you can have  only E1 and E2 is  not difficult you know maybe next time I would go through it  but let me  let me so in this case actually in this case  the J would act on the X like X star  one and the other actually this case if you have J of XY  it would be Y star X star so here because you have two  J so the way it's it acts on it that takes complex conjugation  or her mission conjugate and now  yeah  dagger for physicist  no  okay  now what do we do after  the question 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  okay  okay  okay  now now at this point actually  at this point so this is actually the complex  for the algebra let me see  okay so anyway in this case actually the algebra is the same  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 a lysometry  of 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  an operator i acts on elements 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  where n is 2a in this case  and h is the space of caternions  in other words actually it really restricts the form of the n by n matrices  to be formed on the caternions they become a symplectic in a way  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  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 is yeah  I think i alpha i probably is alpha in this case  but i would involve also complex conjugation  i squared sorry is minus 1 not plus 1  you know so think of it that you have a matrix like  you know think of it that you have a matrix with 0 1  minus 1 0 and then each matrix like  you call it a b c d s and 0 1 minus 1 0 is equal  that will give you some conditions on this but  I think there must be some complex conjugation in this one  so this it acts with a complex conjugation in this case  a bar  I forgot the exact action but something like that and that 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's really  it really becomes the space of caternions  so this actually is something that and  in a way actually now the space really becomes  the algebra becomes m a of h plus m n 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  yes n is 2a so this is yeah you can see  a and 2a right now  we have also the existence of a chirality  operator gamma and it has to act you know it may act here  or it may act here now usually  we take it to act on the first  and if it acts on the first this m a  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 what's the easiest possibility  m2 of c  plus h 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  non-chlorion so you cannot act with chirality on it if you have to act on this guy  if that on this guy then it splits m2c into c plus c  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  it's 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  it doesn't work okay so then the simplest case  where you can impose the chirality condition  is this a equal to 2  a equal to 2 what's in this space  is m2 of h  okay yeah I should finish in one minute  if we act with you know  we have to put remember we have this gamma a is a gamma or something  you know it's sorry gamma psi equal  okay it also commutes gamma a is a gamma  we have to put this chirality in and how do you put the chirality in then  if you impose chirality on this this splits into h left  plus h right  plus m4 of c  is the first possibility  you know now  now without doing anything else actually now I really can tell you  my fermions how many fermions I have in my space  because the number of fermions the dimension of the Hilbert space  is in this case is n squared  yeah this is 2n squared but it's cut by chirality it's you know half of 2n  squared is 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  dimension of 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 a particle spectrum is concerned because  tell you what to do this actually is  nu right plus e right actually usually you know of course you say it's a doublet  so it's a nu right e right nu left e left  left down left  yeah then I have upright down right it's everything  because this is one plus color so we are going  to get nu right e right  nu 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  come they give you something extremely complicated you know  they are talking about SU6 or SU8 gauge groups and we know they don't  work so what the claim is the following if you  classify algebras  finite dimensional spaces  the first non-trivial possibility you  obtain is that of what would turn out to be the standard model remember  I have not re-import the first order condition because when I import the first order condition  it would really give me that this one this would follow  you know this also this comes on  yeah this is you know 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  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 not only this actually  from this picture they look at leptons and quarks because of this you know already  the 4 here the leptons looks like a fourth  color you know it's really unified with color in this respect  and if you really would insist on this symmetry then you can  have you know all this p minus 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 there are many 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  leptons is a fourth color for quarks  this is the idea of pati salam actually it's called the pati salam unification  and it was done almost the same time as georgia  you know georgia 1974  salam almost assigned me this problem you know he told me to choose between this and  supersymmetry but then I chose supersymmetry  pati salam pati salam pati is an Indian  physicist you know  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 with this connections that I have  the indicated  I'm able to derive the full connection of  the model and how the Higgs  field comes out naturally we have formulas very specific thing  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  so it's a...
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