 داخل العالم yeah well actually I should have really start my first lesson with this but since every time I say next time but in the end I keep forgetting so this time I said I will do it in the beginning I am sure that I will forget it you know and this is yeah نعم ، هناك 12 ، لا أستطيع أن أتكلم ، أستطيع أن أتكلم 12 ، so maybe I will mention it when I reach the point. So actually most of the material I talked about can be found actually in this reference, especially in the first and the last. And for the classification of finite spaces, it's a paper-wide standard model which is mathematical. There is a physical review version of it, which is not mathematical, where things are summarized. It's called conceptual explanation. I think physical review letters force us to change the title. I think it was called Dress for the beggar. I didn't like this terminology. So today, in the last of my lectures, I will start quickly by summarizing what we have done since it was a break of two weeks. Our basic assumption is that spacetime is the product of a continuous space times, or a semi-direct product, times a fixed finite space. And this should be taken as a basis for unification. In other words, this has to be considered as geometric space. And all geometrical entities built on this space should be used in constructing the action. And since it's a non-computative geometry, it means there is a lot of information actually in both the algebra and well, in the three, algebra, Hilbert space, and Dirac operator. And the Dirac operator has to study the spectrum. And the eigenvalues of the spectrum are the geometric invariance. So we really have applied this idea to the product space of the four-dimensional continuous manifold times a finite space. Now, which finite space we used? We used a finite space. We classified first, actually. We have classified finite spaces subject to what we call weak physical requirements. And we ended up that the space is C plus H, let me say left, plus M3 of C. And I remind you, actually, that originally this was really, due to the neutrino mixing, was really, these three come together in one algebra and this is with the other. And what really breaks the symmetry is the mixing between fermions and their conjugates. And this implies the existence of a Majorana mass to only one particle, which has to be neutral and is no coincidence that this happened to be the right-hand neutrino. What we have shown also is that the basic building blocks, basic building operator is not only the D, but the D that includes fluctuation, which we labeled by DA. And this DA is equal to D, where D is, you know, it could be a direct operator, say, in a previous direct operator that doesn't mix anybody in flat space, plus flat, no commuted space, I would say. Plus some connection supplemented by this term, where we have a, where actually A is a connection and there is a universal formula for it and the AI and the BI are an element of the algebra. So this actually gives all connections and then we have shown that the connection A contains, let me say the following, you know. So let me give you a drawing just to, so let me assume this is H right, H left. And then this here M4 of C. And this we said essentially is broken into same terms of the form lambda lambda bar. And then here, let me say it is caternion Q. And here I have, you know, same lambda. And here I have 3 by 3 meters. So these are the elements of the algebra. It retakes this form. And then what we have seen is that the connection A, in this case, according to the same terminology, would really have this form, which I'm really going to. So for this part, it will have a 0. And then it will have B mu. And here it will have, you know, some B mu with some power and some SU2. Let me call it W. And in the lower part, what it will have, it will have, you know, B, W, and V for quarks. It will have everybody. And here, actually, there's a B because it's broken. Now, what happened, actually, is that along the off-diagonal elements, we really get the Higgs field. And here, what we really get is something that connects, actually, here, the H right with the H left. And this happens to be the Higgs and, you know, H dual, whereas this is defined to be sigma 2H star. In other words, actually, we really have one Higgs field, which would give mass to both the, you know, say, electron and the neutrino and the up and the down quarks. And it's known, actually, that in the standard mode, the Higgs field is not the Higgs field that gives mass to the up and down quarks is the same Higgs field. There are no different Higgs fields. And this, actually, you know. Yeah, but the question is that there is no, here, the two things that are really independent of each other, actually. They are dependent on each other. They could have been independent of each other, but they're really dependent on each other, which shows, actually, that only this is the consequence that only one Higgs field is present. Now, for the Dirac operator, we have seen, actually, that there's here, there's zeroes here, zeroes everywhere, actually. There's only a singlet entry. Sigma is a singlet that gives Majorana mass to right-handed neutrino. So, this is really, actually, our all consequences of the construction. That, I would say, predictions of the construction that a singlet would exist and the singlet is extremely necessary. It plays a fundamental role in giving the Majorana mass to the neutrino, the right-handed neutrino. In addition, so it really can establish the CISO mechanism. In addition. Because in our picture, in the... Yeah, it's this, actually. 2010, so it was published already before Higgs was discovered. And the reason we mentioned that it really has a big consequence on the stability of the Higgs doublet or the Higgs field in the potential, it shows that you really can extend, one can extend the standard model all the way, one extend, one can extend all the way up to very high energies beyond the 10 to the 11 GeV where the Higgs coupling lambda quartic coupling becomes negative. Without this field, it would become negative. But it stays positive because of this field. So that's a very, very important point. In the sense, you know, that many people think that our model is ruled out because of the low Higgs mass, which is well over 25 GeV. And if we had this field, if we had computed the renumberization group equations and so on, we would have found that it was perfect. One can show, actually, from the stability argument that if you don't have the sigma, then there's a prediction that the mass of the Higgs is somewhere between 160 and 180 GeV. And this was the first thing ruled out. This is ruled out. This was ruled out first before the Higgs was discovered at 126. However, once we have this, you know, there is no problem because it makes everything stable and this argument of stability, which was used in getting this mass. Not only, actually, in our case, it's not only stability, we have some mass relation which forces this prediction to be in this range. Anyway, so now everything is okay because the sigma is necessary and it's re-necessary because one really has to break C into C plus M3 of C that this is really necessary and the symmetry is broken. So essentially, I really can summarize for you, actually, what are the positive or the predictions of the model. Before going on, in that, so let me repeat, actually, what did we assume and what did we get? What did we get out? What we put in that, this is assumptions assumptions that spacetime is a product of a continuous four-dimensional space or manifold, let me say, times a finite space. So this is an assumption. The second assumption while we were doing the classification we did is that one of the algebra M4C is subject to a symplectic symmetry reducing it to M2 of H. So that actually limits two chtorions. Three, we did assume that D of that day is different than zero and four, we did assume at some point that the neutral algebra U of A for the whole thing is restricted to S2 of A. Actually, this condition somehow implies anomeric cancellation. But as you see, one has to assume one has to put some input and this is the input we put. Okay? One can argue actually why you have to put this input out of their way out mathematically, but it's not easy. We did try and it was not really easy to reduce this set. For the first condition, yes, it's possible. For the first condition it just comes from the fact that in M4C you only look at the automorphism so it wipes out these things. For the first condition. For the whole thing? No, no, for the whole thing. Yeah, for the whole thing, exactly. For the whole thing. And then, okay, then when you reduce, you get this condition. So, yeah. No, this is really understood. Okay. So, maybe this can be weakened, actually. This can be weakened, yeah. Okay, predictions. First, no number of fundamental families per family is 16. The algebra is C plus H plus M3 of C. This comes out, actually, as a consequence of the assumptions. And we obtain the corrector presentations of the families with the 16 with respect to H2 3 plus H2 2 plus 1. This is not trivial, actually, to get exactly especially the ion hypercharges really very difficult to get, but really to get the 3, 2, 1 for each one of them really comes out. The fourth is that we predict the existence of a Higgs doublet, the phenomena of spontaneous symmetry breaking. But that actually really comes comes from as a consequence of having the correct hypercharges. This is a consequence, actually. And, you know, especially with the negative with the negative mass time for the Higgs, which is minus whatever, B squared H squared H. So it comes with that a prediction more or less actually of the top-quark mass of top-quark compatible with the experiment, you know, we'll get it within, say, 4% or something. I would say, actually, why you know, we're not able to pinpoint everything to the last digit. We also predicted the seesaw mechanism to give a very light left-handed nutrients. And I will add, actually, something I mentioned last time at the end of the lecture is that you get the correct Gibbons Hawking term Hawking, yeah. In addition to the to the Einstein-Hilbert action. Which is that, you know, when people talk about the minimal with gravities, they take the sum of the two terms but we do get a cross-term which is, in fact, also predicted by Feynman which is the term in R-H squared. Yeah. So I mean, it's not really the sum of the gravities plus the sum. Conformal couplings of scalars to gravities. So you get terms like Roud G different R-H bar H term also R-Sigma squared terms. You know, these terms are really present, you know. They are there with the conformal value, you mean? Yeah. It's, yeah. This, you know, this is a thing which is like a theorem, actually. That in the spectral action you get, actually, first it's cosmological term which is lambda. Then you get Einstein term let me write it cosmological Einstein Hilbert term which is Roud G-R and also you get, actually, H bar H term and Sigma squared term. Then the next term in this expansion in this asymptotic expansion you get really what I call a conformally invariant action which is Roud G-R and then you get, actually, all the D mu H squared you get F mu nu squared and you get R H bar H and it's not H bar H. No, no, this is a, you know, conformal, yeah. Yeah. C mu nu sigma squared. Yeah. Plus it comes with the with the while tensor squared. And there is, of course, actually another term which is Gauss-Bonnet is also there. So this is reconformant. So what does it mean, actually, if one really assumes that this is an effective action from the Wilsonian point of view, it means your starting point is conformally invariant at least for the for all terms which are have dimensionless coupling. All these terms are known have dimensionless coupling and so, of course, the coefficient is logarithmic and that's why it comes it's exactly actually conformal every single term. So this actually I can say these are the main, you know, there are more subtle points but I will not go through the more subtle points. I can explain, actually, why the thing okay, what are the drawbacks? So maybe actually one has also to say a few things which no, no, only the one one term in the expansion is conformal, yeah. This is the whole block, actually. It's conformal. All quartic let me say order four arey conformal it comes exactly conformal. Arey arey conformal. Yeah, but the meaning of the conformal R H square is only respect to the first one. It's conformal means I change it. Yeah, yeah. No, this actually, of course, these two terms destroy conformal invariance. The cosmological and the they If I change it R here the metric by something proportional to one plus H square, okay? Yeah. I I will change the coefficient of R H square so what is the meaning if I have something on conformal you said the coefficient was one six which is the conformal Yeah. Is it one six? Yeah. But one six means that when I change the metric by a conformal thing I can absorb completely this R H square in the metric but I can make disappear No, well actually okay This No, no I say only one part in this expansion happens to be conformal invariance one can prove actually why this term in the asymptotic heat kernel expansion is conformal actually Let me say something actually. See, last time I say that the given hooking term comes exactly with the right sine coefficient however as you know I also added that if I use not the Dirac operator but Laplacian you get the wrong coefficient in the same story actually suppose that I didn't use d squared in my expression I say f of Laplacian or something then of course this term this part of the expansion will not be conformal it has to do with the Dirac operator Yes, the Dirac operator that's more conformal for the artist than Laplacian No, but you have to change the action of course No, actually I will come I will come to the question of scales actually what does it mean to have this conformal invariance I'm coming to it but that's the reason actually I didn't want to stress so much that I have this Rh bar because you're right you know either the whole action is conformal invariance or is not conformal invariance I would say the whole action is not conformal invariance because when I go even to even higher orders in my expansion this conformal invariance breaks down the part which has no lambda as conformal invariance you know the coefficient is logarithmic there is no scale there okay yeah this I think in this explanation yeah okay well actually not completely you know suppose I use I use Laplacian then I can show you actually you don't get while squared you get something else so it's not only that actually there's some hidden hidden aspect okay you know one really can say also things which are not perfect and the thing which is not perfect actually that here there is a prediction on the unification of the coupling of gauge what is this it is tells me that 5 over 3 g1 squared is equal g2 squared equal g3 squared and if you run the Rg equations and you look at if you define alpha i to be g i squared over 4pi i is 1, 2, 3 you find that you know alpha 1 inverse alpha 2 inverse and alpha 3 inverse they almost meet but not quite not depends actually one of course has to run if you do to 0th order you know just run or first order if you run that you discover they don't but in reality you really have to take loop corrections into account now the loop corrections have been worked out even up to 3 loops actually up to 3 loops things have been worked out and it's known actually that the running does change and especially for the alpha 3 inverse actually it does change is like 16% correction or something so however the problem that nobody has worked out the equations in the presence of the singlet because we know a singlet and the Higgs itself changes the other coupling so and nobody has done actually even to 2 loops they have done only to 1 loop we know to 1 loop things do change but still the thing don't really completely meet we are not really sure actually of the story because we are waiting until someday when somebody will compute the 2 loop order corrections and the guess is that things would really improve actually however to discussion actually whether is there something beyond the standard model or not this is a topic that I will discuss later whether is there any physics beyond the standard model besides the fields that I have written because if you look at this point of view with this classification it seems that we only predict the standard model plus the sigma field which is new later actually there is a dilaton which I have not talked about yet possibility you can add actually an anti-symmetric field which is an action by through torsion terms by assuming the connection actually in the manifold part here in the 4 dimension manifold part you can assume that the connection has a completely anti-symmetric torsion in which the B field would appear only through its field strength this you can do and in this case you will obtain a very you will obtain the usual action actually as a contribution so that's the possibility but apart from this this is it actually then you say okay there is nothing else so this actually is but you know of course there is a question mark we cannot really make a definite statement now they don't meet but you know something may happen because nobody has computed the two loop realization group equations this will come at some point I'm sure the next of course actually many things we cannot explain like why there are three generations the structure of the Yukawa coupling you know this Dirac operated in the finite space has many entries and every entry happens to be a Yukawa coupling for one of the observed fermions so if one can predict actually the values of each of course you have solved the eternal problem but that seems to be really tough the question is what determines the Yukawa couplings actually this we have not answered what determines or the entries let me say what determines the Dirac operator in the finite space completely you know this is a big question but if one can do that you have really a full scale prediction of everything that is observed the number of generations also we have nothing to say about if you want to multiply this by three generations it means the algebra is the Dirac sum now no no no no the algebra doesn't change the hyper space changes okay just you repeat three times the hyper space but it's important to say here if you want that okay the difference with the usual problem for Yukawa coupling is that now there is geometric meaning and this means that if one could make a guess of what is a geometric not coming to new space is that new could make a guess for the Yukawa coupling so this is yeah I think now the question you know the the question of the Yukawa couplings of the furnace has been changed into a geometric question is that what determines the Dirac operator of the finite space we have not really assumed anything about this operator and this is the reason why you know let me make for instance you know a home guess but just to give some idea that if you look at spectrum you will find that there is something which is corresponding to the usual music and which which has no incarnation for ordinary spaces but which for if you think about non-competitive space you will obtain what is called the quantum sphere now the quantum sphere has a spectrum which is powers of q and it's not difficult to see that when we look at the structure of the quark there is a little bit of this geometric structure in the sense that there is a non-competitive progression so what one would need to do is a bright guess of what is the underlying geometric space yeah but I think you know the point is made is that now the question of your quark has been changed to a geometric question what's the nature of the geometric space remains to be seen we need to have a huge basket of geometries non-competitive geometries including quantum spheres and everything but you know according to my information there are some examples but not that many actually are there many examples oh yeah there are plenty of examples the thing that you need to understand is what would make examples to be finite maybe you know there are plenty of infinite examples with infinite spectrum but of course we know that it's a finite space so a finite space you have to somehow you know cut this quantum sphere or something like this do something like something like this something like this okay um d Latin when I introduced a spectral action I said that the spectral action is trace of f of d squared of a lambda squared and you know we put this by hand this scale however you know the idea of a scale which is insert by hand not very natural and then it's it's more natural actually to say I'm really going to I can replace d squared by e minus phi d squared e minus phi why actually because here if you let phi goes into phi say plus lambda it really gives me that d squared would go into d squared of another squared okay so let me call it this d phi I define it this way you know it's it gives you so it's it's natural to replace the scale and you hope that the field phi is dynamical and then it will get some expectation value you know since the only scarce I just want to make one remark you know it's important this point because imagine that f is just a cutoff function then when you take f of d squared or lambda squared you look at the eigenvalues of v which are smaller than lambda that's the same thing but now we need to do to hear at all what you do if you look at where it's smaller than its condition so it's longer global but it's localized so this is what it means this this where actually in physics now the best thing that if you really want a number the best thing is to put a field and let it get an expectation value and this expectation value would really okay so now the problem there reduces actually to this computation trace of this d squared phi like that the question what's the answer so one does the calculation and one discovers actually the following that this becomes you know that if you have something like g mu g mu okay so what's the leading operator the leading time the leading time would be e minus 2 phi g mu d nu plus it's known actually that this elliptic operator starts with g mu nu d mu d nu now you discover that there's e minus 2 phi so you say okay let me define a new metric which is e minus 2 phi g mu actually this would be equivalent to going to the Einstein frame this gives me so the capital G all this bad notation actually is to write capital G because in generality capital G refers to the Einstein metric but let me the Einstein frame and then one observes the following that in this Einstein frame in which the metric in this guy in which it has been descaled with the G what happened that the spectral action really becomes like this this phi this appears and then the next time will be root G R of G and then for example even for the mass time you'll get actually let me say this is important you'll get h prime bar h with some mu squared or something well actually there's h prime and there's sigma and everything and where we have defined to be E minus phi h and in addition there is a time which is G mu nu D mu phi D mu phi and then we go on and then we get G with C mu nu rho squared which is conformal invariant in other words actually the action in terms of the new frame which has been descaled with the is exactly the same action as without except for one time one and only one time and this time is the kinetic term for the dilaton it gets a kinetic term so this is really a physical field in addition you have everything is rescades for example phi prime will become E minus 3 over 2 phi h prime is E to the minus phi h and so on so in other words the physical fields you observe have been really scared with the dilaton what is now actually in reality you can go and compare this the model you obtain with what was known as the Randall syndrome model you really find exactly the same model and that picture what happens is that you go to a five dimensional theory in this five dimensional theory you have in the fifth direction you assume first of all actually you assume that you have two brains and this would correspond to the x5 is zero this correspond to the x5 is pi and what happens is that some fields get rescade on one copy and not on the other however in our case you know it's similar situation the terms are identical actually except the fact now we know that h prime is E minus phi h perturbedively you know one really has to compute the action to all orders and but it's really clear that at the perturbative level the phi doesn't really get the potential term the higher terms will be derivative terms and this situation is very similar to what happened in string theory in which the field phi doesn't really get the field phi doesn't get the potential and so when assumes that at the non-perturbative level phi gets some expectation value you had the scale you had the scale somewhere I mean the x potential at the scale minus me squared so exponential well actually well actually here you know there is a scale okay lambda squared or it's really I'm blank squared in all this okay you are saying there is no dependence algebraic dependence on phi exponential phi anywhere there is there were the mass squared all the best no this is a kinetic term but the mass they were like the x minus mu squared h squared because h is a scale if they are not exponential minus 2 phi okay let me you are here going to get true g I don't know in the x potential there was a scale yeah okay h is a scale by the scale h squared but if it fits like that phi phi called zero is a solution I think it's really decoupled then except from the kinetic term yeah yeah but it's decoupled I mean if I put constant you are saying this is an exact solution it doesn't couple to physics it doesn't couple to except through derivative terms yeah so when I when I assume that you know at the non-perterrative level it may get the potential this is not not easy to see but this is a possibility but even if it does this is an extra scale of field it's not a dilapton anymore yeah when the dilapton changes something if here it's called a scale it's a scale of field which is on the side yeah you know you will not see but it has a consequence for example that if the field h after a scaling it disappears so if the only term like field the kinetic term it's called it's called it's called it's called it's not conformal it's not conformal because this term destroys conformal in Venice I think the problem you know now it's a question of interpretation because now you can say actually the field h prime if the field h we get an expectation value we have seen that but now we have to look at symmetry breaking with h prime now we forget about h if what you are saying is correct let me forget about h and all that and I have somewhere on the side and then I have h prime so h prime we get the value 1 that's it but before the value the value of h prime was like a mu yeah you are not sure which was which was London yeah this and therefore as you have modified London it's going to be h prime should be a square term the square term cannot be corrected it's only the conformal piece which is in right but not this term yes so this term cannot be the same yeah thank you yeah this term cannot be the same because the volume risqué that the force power this term risqué as a second power so there is an exponential in front of this this collapse okay yeah sure when you see this term is not correct because the volume risqué but the force power is not not from this term it cannot be correct this term yeah I have to check just a second yeah one needs to see the ratio because there is an h4 no h4 is correct this term this term has to to be on the scale on the square sure no it's clear you don't have to look this term is clearly changed you can look at the previous thing if when phi is constant it just changes what was on the square before on the square yeah on the square there is a radian on the square somewhere yeah everywhere there was a on the square because the exponential probably if this would cancel through g no no you cannot do no I'm Michelle that's it what's g it's a determinant so it scales as a force power and that has a signal okay so I think you know the important point is that if this is h then the mass has to be h' which is e to the minus 5h so essentially if you can think that h can get an expectation value of the order of the Planck and the phi is of order 40 then the h' would have a لذلك يجب أن يكون هناك أشياء كيومة مع السنة أيضاً يجب أن يكون هناك أشياء كيومة فاية ديلة تون which is in the game and this somehow we discovered that the only other particles other fields which are present beyond the standard model is the sigma, the phi and maybe you can have a torsion term through a beam you knew that only appears through its field strength if the phi doesn't come in front of the d-laton and are you sure that the phi doesn't come in front of fmu square no of course because everything is conformally environment the d-laton comes in front of it it's not the d-laton of strength theory it's different alright one more thing maybe I should mention is that if the question of parity violation which is written as a theta problem in QCD see we have written this spectral action trace of f of d squared over lambda squared we can also write actually another trace in which we insert the gamma remember actually this the geometric data includes the gamma the chirality operator and in reality you can compute this guy and for trivial models of course if you do it naively you obtain terms like this mu and mu rho sigma r mu and mu AB r rho sigma AB you get such terms also terms like AB mu rho sigma AB mu rho sigma with the trace on the internal space this is known for QCD it gives you what's called the theta parameter this will be present you can compute and you get so this is the Euler terms which destroy parity because it has only one epsilon I think in generativity what they call it a barrel in mercy or something terms it has some name in generativity now so if we do the calculation for the spectral action here for this what do we really get this as I said is a huge trace this is an arbitrary function but the trace is 384 by 384 matrix and in reality you re-obtain this term but the term have a coefficient zero you know 3 comes to 24 minus 24 so respect the reaction this term is really absent this parity violating term and in addition for QCD I don't know let me call it G mu nu or G I don't know to call it G mu nu mu rho sigma 4 SQ3 is also absent is also absent it's not absent for everybody for example you get a term like G mu nu rho sigma B mu nu mu mu mu nu mu mu nu mu mu mu nu mu mu mu mu mu mu mu mu mu mu but this is a total derivative doesn't count however there is a term which did bother me actually at some point and this has to do exactly the same at the theta term of QCD but for the weak interaction but then I went and I discovered that people actually this term is observed actually and it is taken into account ترى؟ ليس أنه ليست مرحبا. إنه مرحبا، وإنه هناك. فإنه مرحبا في الوقت لأسيو 2، لأنه ليست مرحبا لأسيو 2، لأنه يجب أن يتبقى مرحبا. ولكن إذا كنت أردت أن أقوم بشكل كبير، يجب أن يكون لديه أكثر من 2 مرحبا. حسناً، أصدقائياً، أنا مرحباً. أخبرني هذا، وأخبرني ذلك. هذا لدينا جميل. هذا ليست مرحبا، هذا المرحبا، هذا المرحبا، هذا المرحبا. هذا مرحباً بهذا الندخ. if you compute you will find that in reality, it's an Euler term, but it's absent. And similarly ... It's absent because of some un-American relation. It's a numeric section. Everything is rigged so that ... There are circles on the left. Exactly, I think this is the good answer. This is the good answer. It's left and right symmetrically. Make sure that this term is not there actually. حسناً، ولكن الأمر، دائماً، بالنسبة له3 أنه بإمكانه أكثر من ثيتا وه3 أنه 0، لكنه لا يجب أن يقوم بأكدداد إنه أيضاً بإمكانه المفتاح في الممتاح، حسناً one can show that actually, at one loop you have to make an arrangement to cancel, but if you manage to arrange it to cancel at one loop it will cancel to all orders. However there is a condition which we don't know what it means here, the determinant of the up and the down coupling should be real. If you can arrange that one can show them that the theta would be absent to all loop orders. This is the condition that it is eliminated at one loop order, but if you arrange that it will be eliminated at all loop orders actually. So again this is a question which is related to the structure of the rack operator of the finite space, why should this condition should be there? But at least actually at the three level is not there and there is no fine tuning it is natural and it is automatic. So I think you know that's one of the positive things from the next. Any questions before I move on? Yeah I just want to say one thing about this, you know when you say the connection not to be symmetry when you have these anti-symmetric terms because I mean of course it's very important that when characterized it's abstract to the rack operator. Yeah you know usually you have to make sure that Psi deep Psi well there is a J but okay you know. Let me say deep Psi Psi. I cannot do it this way. It has to be hermitian. And this hermiticity it means you really have to integrate by parts. Integrating by parts there is certain you know there is in generativity it means it's a condition on the spin connection. And this condition on the spin connection it's really I think you can call it like this. Omega mu mu B is equal to zero. You know it means the trace of the connection should be equal to zero. And this of course is satisfied by connections where the torsion is completely anti-symmetric. What I would say is not this. You know what I would say is that if you want to characterize algebraically the direct operator on the manifold okay. The condition that you obtain very naturally from the concept of geometry is that you still have a little bit of figure and it corresponds to this figure. So I mean this this figure might be interesting to investigate. Of course we don't really look at it. Yeah the action it is most irrelevant to dark matter and you know Cp violation and things like that you know. Okay next. The question that is there anything beyond the standard model. In other words you know it's now known that experimentalists are looking for new physics mostly in the form of supersymmetry. But in the form of anything actually everything is open for possibilities. And it's a relevant question that anything beyond there anything any physics beyond the standard model. We have seen that in our case these are the only possibilities and they have you know and consequences as something at 10 to the 11 GeV with the sigma and with the neutrino masses and things like that. But the question is there anything observable at low energy. In other words you know if tomorrow something at 7 they announce that they discovered in the data news sign signals. What would it correspond to. So the question for us is that is there any room for generalization. Have we been exhaustive in our analysis. Well there is a good point obviously in our analysis which is the breaking from the chart. So obviously we remember that at one point we said there is a non-trivial mixing and you know the sigma does exist and this broke into C plus H left plus F3 of C. And this we you know we did it by assuming that this condition holds. Now I mentioned before that this condition does imply that the connection is linear. What does it mean actually? It means if I define DA which is D plus A plus epsilon JA J inverse then you know that under so for example if you let psi goes I don't know you opposite then DA would go well at least in this case it would be UDA you start. Where U is an element of A and U is unitary. Provided you know this transformation is okay provided this condition is satisfied. Do you have psi on both sides? No. You have to be careful. It's not UDA you start. It's UJUJ start. The U. The line above. It's not UDA you start. It's UJU start. Here. No no psi is UPSI you start. And the action on the right is given by J. UPSI you start okay. Which is UPSI okay here we have to take actually it comes to the right as J. UJ. UJ times psi. It's UJUJ times UJUJ times UPSI. The J makes a bible. So this actually let me note as U U hat and UPSI where we define U hat to be J U star J. I don't know if you write it like this put D UJ. So this is quite an important point because what happens is that because of the J the hyperspace is a bible you will. And the fermions are in the adjunct. This is very very important. Moreover when you write DA it's not UDA you start. It's U U hat times DA times U U star J. Now we can see actually that in this case actually in reality if we want to define the A one has to define the A as something like A A hat. What you have written in the back is not correct. It's U U hat. DA goes to U U hat. Capital U. Do you know here? No no. I mean it's not the U in the other part that you are there. It's U U hat. The rest of the operations you have to use. I didn't try to have the A transplant okay. I don't know about DA. Yeah but DA is not U D U star. It's U U hat D U star. U U hat. So essentially if I define U to be U U then DA would go into U DA U star. Well actually remember I was talking about the case where I didn't assume this condition. This is the thing that if you assume this condition reverse to the other because all the hats would drop out. No? They don't drop out. No? They don't drop out. They are out of U term. They don't drop out at all. You see. I thought actually that we have A. One second. A one here. But you know I thought that A one if you assume this condition would go into U A one U star. But plus the other term yeah. Yeah okay. So it's 3D. Similarly for the other terms. Okay. So in other words actually in reality one has to introduce such transformations and see what is the invariant operator or what's the invariant quantity you can produce. And in this respect. So I want to discover actually that the A has to be written as something like A I A G hat into D B I B G hat. You know there is a tensor operator between them but I will not write tensor operators obviously. So here we can make without any loss in generality. The summation of A I B I is 1. Why? Because you can always increase your set by 1 in which whatever you have you add one more element and so that this is satisfied. And here summation I and J. Now remember actually A I B G hat is equal to zero. This was the zero order condition or order zero condition. Which implies that left and right actions commute with each other. Now if we try to compute this object we discover the following that we have A I A G hat and then I'm going to get D B I B G hat plus B I summation on I and J. Let me look at the last term first. So the last, the second term can be written as A I summation I on J A J D B G hat. And A I because this the B I and the A G hat is zero it goes through, it commutes. Okay? And of course actually this term because A I B I is 1 this term has become A G hat D B G hat. So obviously I obtain my next term which is this way. Okay? Now what about the first term? The first term this one I have to commute actually this guy with this guy. So obviously I get the term which is A J hat D B I into B J hat plus another term which is A I D B I A J B J. But this is one summation on the J and this of course is A one now we call it one but we are left with this term. Now this term remember we did assume the order one condition which tells me that the A had commutes with the D B and this term drops out. Now suppose that I don't assume this. What does it mean? It means that the A connection will have a new term and this new term can be written as you know because this commutes with this in reality I can take it in but this is nothing but it can be written as it can be written as commutator of A one with sorry A G it can be written as A G hat A one will be G hat. Note that it depends on bi-module terms in other words it depends on the elements in the A and the elements in the opposite and it's mixed term that's why it's really quadratic in the dependence it depends on A and A D B but from the other side. So these are quadratic terms but once we admit actually that this term is there we introduce this connection then in reality you can have more general connections than the one we considered and these connections would have certain properties. So this is the most general connection you can consider without assuming the order one condition. I think it's very important to say two things because I mean at the conceptual level it's extremely important so the first thing that we discover is that when you drop this order one condition which was making things very simple you find out that the inner fluctuation of the vindictive of this added connection they form a seminal loop and how did we find that we found that because you know these quadratic terms so you can get very scared and then you can say okay imagine that I do an inner fluctuation I get this new Dirac operator now imagine I start from this new Dirac operator and I do again an inner fluctuation I should get a quadratic term because I got this quadratic term no now our cancellations you still get a quadratic term and so what does it mean? it means that in fact underline this whole story that is a semigroup and this semigroup only depends on the algebra it's an extension of the gauge group and it governs all the inner fluctuations so this is a big change with respect to the ordinary Dirac picture that it means that the inner fluctuations are in fact the action of the semigroup and when you perturb by an element of the semigroup it composes the element of the semigroup so this is what I started to say that these inner fluctuations that we considered with the element Q capital and that they form a semigroup and what does it mean that if you fluctuate with respect to the new generated inner fluctuations you fluctuate again it still gives you terms of the same form so inner fluctuations are still again our fluctuations which have the semigroup structure and when you describe the semigroup so the semigroup is simply you take elements of the algebra which satisfies sigma of ai di is equal to 1 and you compose them left and right multiplication they still satisfy the same form we got rid of the semigroup of perturbations so this is the property opposite summation on J which is an element of A across an opposite and such that A J B J it's one as I said it's not really strong condition because you can always make it and such that A J B J opposite is equal it's the star of itself it's B J star A J opposite I'm not writing tensor products they are tensor products actually here so the definition of the semigroup anyway now I'm not going to because the computation are quite important show spacetime point of view this new term in the connection what does it look like I'm going to I'm going to present it now it's only if the dirac operator was not first order but for instance the reason why we are also there as it falls in spacetime is that we knew that the dirac operator is dressed by the quantum corrections so this means that the true dirac operator will not be order one because it will have a form factor and then it does not satisfy the order one condition this perturbation will be necessary for this case even for the ordinary case because of the quantum corrections so let me try to answer to you a question what happens actually if we allow for this case practically how those things would look like so now no more breaking because there is no order one condition to break this this has to come dynamically in other words the breaking must be attained dynamically and you simply say now let me compute the dirac operator and you discover that how the dirac operator looks like first of all you have h right h left it would imply that your gauge groups is su2 right and su2 left and in the bottom you have su4 color actually so diagonal elements in this will be right left and color so obviously you know the gauge group in this case is su2 right cross su2 left the dimension of the Hilbert space is still 16 so the prediction doesn't change but now you are 16 is written as 2 right plus 2 left and a 4 so 4 times 4 is 16 and this is the way it's decomposed and of course this is known as the patty salam model in which it has lepton comes out as a fourth color now what about actually how the dirac operator looks we know we have learned actually that whatever happens in the off-side in the elements would be hex fields so you are really going to get new hex fields the new hex fields would look like as follows you know let me give you an example so this would be let me say sigma a dot b sigma a dot b as far as a dot b so it's 2 right 2 left so you really get this type of new hex fields and along the off diagonal elements you are going to get I don't know sigma a dot i b j so you really get higher representations with respect to the sq2 right cross sq2 left cross sq4 so you get new hex fields so there is no more one bubblet so you have which is necessary because in the end you need more hex fields to break this symmetry into the lower symmetries now you know there are technical details because if you really start from direct operators that satisfy the order one condition because of the semi group structure you would obtain actually a new phenomena where the hex fields and this are composite of hex fields that appear here so you don't really get independent fields you get fields that depend on only a very small set of hex fields and you get another one which is delta a dot i this is two right two left and this is two right and four and yeah so now actually you may ask what is this old sigma so if you look the question which configuration would generate for me the standard model as in its set and it is the following yeah to this one so a dot i if we take it to be if we take this index to be one one of them it means okay this is the expectation value and if you take delta i to be one it means the lepton color would get a valve and through sigma then that and similarly actually this guy you take it phi a dot b you take it to be delta a dot one epsilon bc like that then you can show everything reduces to standard model so you can obtain the standard model as one of the vacuum expectations of the full theory but of course actually there are many vacuum the question which vacuum to consider depends of course I can say the following that the hex structure is completely fixed it's not that I unlike gauge theories in gauge theories actually what killed grand unified theories like su5 is the following in grand unified theories first of all the hex sector is completely arbitrary first you have the proton decay problem yes I mean once you have a hex in some representation you have two or three scalars invariance so it's not completely arbitrary well look I give you how arbitrary you know let me take say the s or 10 model then what are the possibilities you can have 45 you can have 210 you can have 120 you can have 126 and all these actually you can put in and if you look at the potential you will discover that you know it can have in principle 20 times or so all with arbitrary coefficients so group theory cannot help you in this respect because there are many possibilities once you start taking product representations then many things are possible and in reality I was given an example of the s or 10 a grand unified group why actually in the end things really become very complicated for grand unified theories because one can say okay why don't you consider s or 10 because an s or 10 is like us the fermions all live in the 16 dimensional representation however there it's not the only representation allowed you can have actually many other representations like as I say the 126 is a spinner so you can have the 126 for us it is the only representation allowed 16 is the only thing which you can have in addition when you compute the spectral action all the Higgs interaction are fixed you compute you get an answer and this is the end of the story there is no room for maneuver the nature of the Higgs fields it's more actually we really find two cases let me say the two cases one case in which you start from a d which does not satisfy order one condition on arbitrary you know the thing that before you fluctuate you allow a d which is which looks random then in this case you are really going to generate Higgs fields which are listed which you can list actually I can write them it's what 2 right 2 left and 15 and you know so I write them yeah I written them actually 2 right 2 left 1 plus 15 3 right 1 left 10 1 right 1 left 6 and so on so this actually you can have or if you start from a d 1 condition because of the semi-group structure you will go there and you will stay there you cannot get out and in that case the Higgs fields that you can have is only these two guys that I have written here actually which happens to be 2 right 2 left and this happens to be 2 right 4 these are the only Higgs fields that you can have and then this potential does become simplified the problem actually in this case of course the connection becomes quadratic in these fields quadratic in these fields then you have to understand the phenomena whether this semi-group structure would help you with the realization scheme or not this we have not investigated we don't know the answer it may help or it may not help but the worst possible case that you can have all of this now all of this can have this breaking mechanism and in this breaking mechanism if you ask me what would happen if this was the case then of course you are really going to generate this SU2 SU2 right for example and SU2 right has extra gauge fields these extra gauge fields of course it depends which scale you are going to do the breaking if you do the breaking at the intermediate scale or you do the scaling at the unification scale you are really going to get different physics but it's too early to tell actually there could be consequences in which one really can go beyond the standard model and this thing would have consequences but this thing has to be studied seriously and we have not studied this thing seriously because nothing has been observed yet we are not going to go to jump and say okay because as we say the possibilities are many are plenty the model is fixed the model is fixed so in principle you know with enough expertise one can analyze this model one thing about the proton decay problem is no one actually that the SU2 the party cell model SU2 right cross SU2 left cross SU4 does not have the proton decay problem proton does not decay in this with this symmetry so you avoid it and in reality actually you have really killed the SU5 model and the SO10 Bosonic model not a supersymmetric one supersymmetric one is not that the unification scale in supersymmetry is raised to 10 to 16 GV and there are some cancellations so there is proton decay which is invisible you cannot see it you need at least 4 orders of magnitude in order to become observable so supersymmetric unification models are still okay but here we don't have actually the problem so this problem although it is Bosonic but it has problem of proton decay it has the advantages of unification it can give you physics beyond the standard model but the physics beyond the standard model is somewhere you know at unification scale which is something like 10 to 16 GV and it could be even higher actually the symmetry depending on where the coupling constants meet and whether you want to say the gravity because there are 3 orders of magnitude it depends on some numbers that come in which unifies gravity with other three interactions and it is possible that gravity does unify all the gravity is at the plank it does unify with the same 16 because we have relations with numbers coming in and it's easy to explain a thousand with all the numbers coming in so the conclusion is the following is that if we allow the most general possibility in which we drop one of the conditions on the non-commutative geometric properties which is what we call the order one condition order one condition means connections are not linear they could be they could have quadratic dependence then you can go to the Patti but again this is unique you really can do nothing else apart from that another issue which I did not discuss that what happened if you go and consider you know we stopped at the first case where the dimension of the Hilbert space is 16 but we know that the next dimension of the Hilbert space this was 4 squared the next dimension will be 6 squared or 8 squared depending actually whether we want to consider let me see you have to take m6 of c and maybe you have to take m3 of h or something but if we insist that this to be even again this would really take us into m4 of h plus m8 of c this it means you really have 64 fermions it means you'll have a new fermions it's too big it's too big except actually some people are saying ok maybe but I'm not really going to really something very speculative remember we always write we took the product of space time times so there is a spinner index and then you say ok maybe this m4 of h has also the Lorentz group inside you know one can say such things but one has to prove it and that it's a realistic picture some people did assume that like Lizzie and company but it's a suspect I would say it's a suspect they have not shown that it works so there are possibilities but it really would need too many new fermions and until they find something I don't think it's worth venturing into something which I said I said it in words but I didn't say that we have this relation this would never change that the summation of the square of the weighted I would say m squared you know in this case e plus m squared u plus whatever but there is weight actually with colors you know this will always be there it's independent of what you do it has to do with the scaling of the hex field once you scale the hex field to have canonical kinetic energy then you are stuck with this relation so you know obviously it's not really possible to go way beyond what we have and it seems that the picture is extremely tight you know my time is up I will stop here if you have any questions