 Okay, so I'll try to do my best to put it on the level. So I will speak, it's a privilege to be the last speaker, so I'll try to take much time to make it short. The idea is to present the generalization of Dirac's equation, incorporating colors. There's a special, the quarks are endowed not only with half integral spin, but apparently they have another variable, which takes on not two but three values and they are exclusive. So, we'll try to include this colors as three valued variable into the equation so we'll need to generalize the idea of half integral spinners, they'll have much more components. And then you will find the spinorial representation of the Z three graded Lawrence algebra, because everything will become Z three graded Z three graded means that there is instead of Z two is based on two. There is one generator who is minus one for example in the square is one. Here, the generator is cubic root of unity. But it's a different one, and finally only the cube is equal to one. So you have three grades. Now just to tell you that what I will be exposing here is based on the common work with yes you look at skip from Poland from Brussels University. And there are publications which are on archive. If anybody is interested, you can look at them. So now let's just see, this is just illustration to tell you what it is about it's about quirks which are in a way, not really hypothetical because the experimentally they give signs of existence, but they are in not like electrons or protons or other particles they cannot be observed free. Obviously they are inside the new clans. They are as small as electrons, if not smaller. So you see for example here, the new clone is about 1000 times smaller than any atom. Now the nuclei are in the nuclear even 10 times smaller, but quirks or electrons are even 1000 times smaller than nuclear than protons or neutrons. But what is also strange that quirks cannot propagate freely, they can propagate freely inside the proton, but we cannot extract them. So there is something very strange. They cannot, if they were just obeying Dirac's equation like any fermions, like electrons, there would be no reason not to observe them freely so so probably there's something different in them. This is the confinement mystery. This is the deep in elastic scattering. I mean when physicists have very energetic electrons, they penetrate inside a new clone inside a proton or inside neutron, and then they scatter and the scattering image proves that there are some very small point like particles inside, but they cannot be extracted. They do not. They are free only inside but not outside. So they cannot be directly observed. And this is what I said so I passed to another slide. Okay. This is just to remind that there are three different kinds of elementary forces. One is electromagnetic force that we know well strong forces quirks. They are the strong forces and they carry this new new degree of freedom which is called color. There are three different possible states, and there's, there are weak interactions so quarks interact with everything they interact electromagnetically weak and strong. And electrons or other leptons like mu mesons, they do not see strong interaction, they do not have colors, they see only weak and electromagnetic interactions. Here's the image. And this is, I would like to underline because the present knowledge, we have not all types of quarks, we have six different types of quarks. And this type, there are three families, and in two different states, of course, so the, the family that is really well known is up and down this quarks constitute neutrons and protons, and this, these are the most common ones. There's a strange and strange and charm and top and bottom these quarks are so heavy that they are observed only in very, very energetic collisions, and usually we don't see them. Only this ones, but there are three different families, and three, and each family has two work states. What is also interesting that we have this three colors. Quantum chromodynamics quarks are considered like ordinary fermions, but endowed with, sorry. There's something I'm not. So, they are endowed these colors and this is how they compose particles that are observable, the proton, the neutron, these are hyperons, and so and only different colors cannot coexist. Exactly like half integral spin of electron in, in any atom if you have two electrons, this, beside the spin they have the same energy the same magnetic number the same angular momentum. So they cannot have the same spin, they must have two opposite spins, but here, if you have quarks inside the proton or neutron, whatever you choose, they must have different colors this is the new variable is like spin but it takes on three colors and not to spin remember to describe this Z two symmetry that there are only two states of spin, which cannot coexist or is this plus or minus. Here to describe three states, of course the natural thing is Z three grading and not Z two grading. So let's go on to the expert to Dirac equation, how it was, it could have been discovered by Pauli it was not but this shows you how the new degree of freedom can impose some new symmetry. So after the discovery of spin of the electron, Pauli understood that one Schrodinger equation for one power for one wave function is not enough to describe this two different states. That's why he proposed to describe the dichotomous spin variable by introducing two components function to functions, which are called now they're called Pauli spinners. Of course on this two component functions, you must have hermitian matrices that act upon because all matrix or all quantum operators acting on states should be hermitian in order to have real expectation values. Very well probably this are the three, this is the basis of three traceless hermitian matrices. They must be traceless because if you want to exponentiate if you have the algebra of such things. You have unitary representation. And there is another hermitian, the fourth hermitian matrix two by two which is just a unit matrix, but it is not traceless so the exponent will not have the term equal to one. The three Pauli matrices span the three dimensional the algebra, which is algebra of rotations. And they also spent the Clifford algebra of three dimensional Euclidean space. So now how Pauli proposed first, he wrote the simplest shreddinger like equation in shreddinger equation you have to the energy is replaced by minus IH time derivative and momentum minus age gradient. So the simplest shreddinger like equation acting on this two component wave function would be will take energy just proportional to unit matrix mass is proportional to unit matrix, and then momentum has it is a vector. But it has to act on two by on two component column. So, we multiplied scholarly by Sigma matrices and this is another two by two operator, which is hermitian. So fantastic we have a linear equation looks like shreddinger equation for these two components, you know, fine, but unfortunately, it is not Lorentz invariant. It does not obey the Lorentz invariance because if we square such an equation. It, it becomes diagonal, but then we have the relationship you see there will be a square, there will be momentum squared multiplied by c square. The mass will give us squared fantastic, but there will be a double product. And this double product destroys the Lorentz covariance because relativistic invariant is like this. This is the square of pseudo scholar this is pseudo scholar product of a four vector, which is called for momentum energy and momentum in relativity you have for momentum. And the square is constant this is the mass square, but this equation, although it's very simple does not obey the relativity requirement, this is not relativistic invariant. In order to, you see, when you have something that is different of squares, the natural thing is to think well, you can, you can produce it like a product of difference by a sum. If E plus P minus multiplied by E minus P, then you have a square minus p square, but how to do it. Well, in order to do it. Introduce another another policy spinner and mix them up you see if we have psi plus which is a policy spinner with two, a column of two way functions. The momentum acts on a minus and then E on minus has to, we have to have minus sign here, then by iteration will get will get rid of this double product you see. There will be no double product anymore if you put it on the other side. And as a matter of fact. And this is what happens if we iterated. Now both psi plus and psi minus will obey the Klein Gordon equation, which is, of course, it is Lawrence invariant. And of course this two equations can be written in a more concise form. We introduced gamma zero, which is Sigma three tensor tensor with unit matrix. This is gamma dirac gamma zero, and gamma K, the three remaining space components are obtained by matching with i Sigma two. Why we must put I here, because this matrix is her mission. This should be anti her mission, because the squares should span the Minkowski matrix so the square of this will be one, because Sigma three is one, and the Sigma K square gives one, but this I will give minus one. So we'll have the proper signature of Minkowski space like this. So we have the, so we have now created the Clifford algebra related with Minkowski metric tensor. The Sigma matrices created spent the Clifford algebra of Euclidean three dimensional space. This gamma matrices of Dirac, they spend Clifford algebra of Minkowski space. And of course they commentators give the generators of the Lawrence algebra. Now, as you, you certainly notice that the price to pay for it was the introduction of minus mass of negative mass, or of negative energy depends how you see it but so this was the, the problem. He certainly was scared of but Dirac accepted it and of course he predicted the positrons, the electron is of mass positive mass but positron can be regarded upon as an electron with negative energy, or negative mass. And it's just the same. And they have been discovered of course. So now, relativistic invariance. The spinners, Pauli spinners that compose the Dirac spinner, they under Lawrence transformations they transform differently, because they're two different representations of SL to see group, which is covering group of the Lawrence group. And everything becomes. Sorry, everything becomes Lawrence invariant. Now this couple there are Pauli equations, they can be written like this and they are interpreted this is called Dirac's equation. You have this Dirac spinner is a four component because psi plus and psi minus are Pauli spinners which are two components. So now let us see, you see how Z two symmetry acts on these equations on these states, because if you change spin, if spin changes sign and momentum changes sign. This is the same. The equation remains the same. And if mass changes sign, but psi plus goes to psi minus psi minus psi plus. Again, it is invariant you get the same equation. So you have Z two cross Z two group one Z two group is this is describing the half integral spin spin up or spin down the two exclusive states of an electron. The other Z two symmetry that has been produced, because we wanted to make it Lawrence invariant. The other symmetry is called charge conjugation. It is a symmetry between between particles and antiparticles. So now let us see how the same thing can be done with colors and with these three. So if we want to describe not only half integral spin, but also new variable that takes on three values. So what we, of course, what we could do what is done currently in quantum chromodynamics is that we consider just three Dirac particles, satisfying Dirac equation. We attribute colors to them. And, but now they have to interact by a potential in order to understand why they cannot propagate. There is a special potential that is created very strange one, instead of decreasing with distance, it increasingly it increases linearly with distance. So the, the farther you go, the more the forces that push you together, grow. That is why but this is it this theory works, it gives good predictions. But there is another possibility that you want to propose, which is to attribute colors not to Dirac spinners, but to give the colors to Pauli spinners. And then you'll have five plus is a Pauli spinner. We will call it red. This one kite plus is blue, and this one is cycle is this green, but remember that all particles, even if they are Dirac particles they have to have partners which are anti particles so we must have also particles that are anti colors. So there are six other functions, we'll call them five minus five minus one this is a Pauli spinner but corresponding to anti color, anti color of red is called cyan. Cm, I'll take anti color of kind who was, which was blue is yellow and anti color of green of psi is called magenta. These three colors by the way, you know that the former colors. Red, blue and green. These are the colors of pixels you see on your screen or on your TV. Because they're additive these colors add up when you look at them, they add up red plus green will give you the impression of yellow. But you probably know, you probably observed that these three colors, see and yellow and magenta, they are used, not in TV but they're using your printers. Because they are, you subtract them, the white pages white, and then when you put something, you subtract one of the real colors and then you get the anti colors. For example, if you subtract Cm, you'll get red from out of white. Okay, so now how we'll do, we will follow the same, the same logic that produced Dirac equation out of Pauli equations. But now we'll have to incorporate, not only Z2 cross Z2 but also Z3. So we'll have one Z2 for half integral spin, spin up spin down. One Z2 for the fact that there are particles and anti particles. And finally, the Z3 symmetry, which describes the fact that we have three different colors. All in all, the wave function now will have 12 components. Three times two times two is 12. The Dirac particle had four components, the Dirac spinor, now we'll have 12 components. So this is what I just pronounced, let us see what kind of equation can we will follow the same logic as in Pauli from Dirac. And remember that when we passed from particle to anti particle from psi plus to psi minus the mass parameter mass was changed to minus mass. Now we have not only minus but we have also, we have also the generators of, we will call it J. J is just the cubic root of unity it is e to power two pi i over three. When we pass to another color, we have to multiply mass by J. And if pass even more than J square and only after a third step we'll have, we'll come back. So this is the, now this is the generalization of Dirac equation, but which takes into account not only particle anti particle symmetry, but also the color symmetry. We start with phi plus red Pauli spinner. The mass is the same, the masses positive okay, but you have to go to the next color and to anti particle. So the momentum arcs on chi minus. Now we apply energy to chi minus. We have to change sign because chi minus is an anti particle, but also we change color. So we have to employ also the generator of Z three. So here we have mass multiplied by minus J. But then you pass to another color and two particle. And so you see you have to. We must do six such steps in order to come back to five plus from pi five plus to chi minus from chi minus to psi plus from psi plus to phi minus phi minus chi plus and so on so so you see, and we exhausted all six possibilities It is Z two cross Z three and Z two cross Z three, the simple product of Z two by Z three is the six. So these are all all six order roots of unity. JJ square and one are third roots of unity but if you multiply by minus one you get six roots of unity. Okay, so this is the system will have to investigate. And I remind that this this five plus five minus this, this are Pauli spinners. So each of them has two components so these big things are 12 component. Now this is just to remind you what is what are the coefficients with mass. And then we can write down the whole thing with six by six matrices. In fact, they are 12 by 12 right because they act on a 12 column vector on the column of 12 complex functions. But here we behind each of these items is a two by two unit matrix. So this of course is 12 dimension. And this is also 10 dimensional because each of this small matrices you see Sigma P is a two by two matrix, but it is better to see it as a six by six block matrices. So now it's easy to see what is this matrices are of course can be obtained by as a tensor. This is just reminders that they are two by two matrices behind. And now we can. So yeah, this is another important feature that in order to diagonalize it. Remember that the Dirac equations, once you square it, it gave you the proper clean clean Gordon equation. Here, it is not possible because we have this entanglement of six different way functions with three colors three anti colors. So in order to get rid of all mixed of all combined double products, we have to go to six to six power. And it is very interesting because the six power gives you something that looks exactly like. The Lorentz invariant you remember is squared m squared p squared. If you put squares here. This was the client Gordon equation. So here it is looks like but it is six order. This is not Lorentz invariant. But if you write it if one writes it in this manner, which looks like a Lorentz invariant but it's not of course, then you see that it can be decomposed that it is a product. This is a product of three different factors. They look like Lorentz invariant look this one is a Lorentz invariant. This is a square minus p square. This is fantastic. It's like, if you write that this is m squared. Very good. This is a Lorentz invariant quantity, but it is multiplying by two other quantities which are complex conjugate. They look like Lorentz invariant, but they are not because they have this to possible roots cubic roots of union, but they are conjugate and the whole this gives you the real expression. So the idea is that probably behind this there's a Z three graded Lawrence group one is a zero grade. This is grade one and grade two, and all three. If they are then it becomes Lorentz invariant. Now, how much time do I have. You are until 10% for about 20 minutes. Fine. So now let us write all this in terms of we see you remember there were these two matrices one was for mass and another was for momentum. So if you want, if we introduce this to trace this matrices three by three, we call it be and you call it q three. Then the this 12 by 12 mass matrix can be written like this be tensor with Sigma three, because there's one and minus one, and the unit matrix two by two. The momentum was q three, it was like this. There was Sigma one because they were off diagonal and there was this little two by two momentum operators with Pauli matrices. So now it's interesting that these two matrices. This is this is traceless and this is traceless of course, if you take the enveloping algebra. They generate the lead algebra, which is now this is the equation how it looks like now with this tensor products. This is the unit matrix. This is the matrix that you saw, which is one minus one minus one j minus j and j square minus j squared. And this is also this off diagonal matrix. So now, in order to make it again, like Dirac equation will put this on the left hand side, and the mass on the right hand side, and there is still something that is not very pleasant because the mass is not a unit to make it proportional to 12 by 12 unit operator but this is simple we have to multiply everything from the left by conjugate matrices be dagger and Sigma three then you'll have one here. This is the unit matrix here and this will be this is what we get. Now it looks like exactly like Dirac equation because this is can be called gamma zero. This can be called gamma. I, and this is just mass operator. Fantastic. So this is like standard your operator the only difference is that only six power is proportional to 12 so this is the diagonalization of the system because now each of this components satisfy the same equation. But unfortunately this equation is not Lorentz invariant, but we'll show that it is invariant under a generalization of Lorentz group, which is the Z three graded Lorentz group. You see this is exactly a Dirac equation but the problem is. Of course, one can say there are many different choices. The problem is why we choose this one. It depends because we choose one of the generators which was J. We could have chosen J square, then there will be different matrices would appear. And we will have a different representation of the same color Dirac equation. Now the question is how much, how many such direct equations are possible, because eight different, the eight different generators. You see this six matrices traces matrices traces her mission matrices. We will expand the space of, but this is not complete you have still two other traces matrices which are diagonal, but we will, we shall give them grades. Sorry, I'll come back a little bit. So these three, you see they have the same shape as matrices, they will be given grade one. And their permission conjugates will be given grade two and grade zero will be. Yeah, grade zero will be two diagonal matrices but trace less one will be called be another be dagger permission. Now this is they span a very interesting ternary algebra, I will not. You see these combinations. The skew, these, these three commutators are zero and the anti commutator. They have three different permutations, and they are all proportional to one, but this is, this is the tensor one j j square. So this is, this is called ternary Clifford algebra. And of course you have the same for complex for Hermitian conjugates with Hermitian conjugate of this spinorial metric or. These are the two matrices that were not numerated so we have eight different generators, which generate SU three algebra. This is a base basis of SU three algebra, and this basis was already started by Victor cuts in 25 years ago. We have this symmetry SU three, you see this is interesting because we started with Z three, we produce an equation, this equation, naturally introduced this two matrices, these two matrices, introduced the the algebra and then we find the Z three generated the symmetry which is SU three, which is fine. Now the problem is that we cannot produce the Clifford algebra with these gamma matrices. We have this gamma zero and gamma k, but they do not anticommute like Dirac matrices. No good. So the problem is how to implement the action of Lawrence group on these matrices. There are only two, but there are many other. So how many we don't know, don't know yet. So the equation which is written now the gamma matrices are like this. And let us try to introduce the generators of Lawrence group which will act on them. Of course, these matrices are you remember they are 12 by 12 matrices. The generators of Lawrence algebra, they have to be also because they will commuted them so that we have, we must take them from 12 by 12 matrices. Now I will show a speed a little bit. Yeah. So let us start with this kind of commutators. Gamma j gamma k gamma j gamma zero. And of course these are new matrices which should be interpreted this are the generators of ordinary space rotations, and these are generators of Lawrence boosts, which makes up time and space. Now this you see that they, these generators that we have produced, they satisfy the exactly what they should satisfy this is the Lawrence algebra ordinary. But the problem is that if you take further commutators, then you get something more. In fact, by commuting more and more we get new generators, which we called. You see with q2 with q1 and so forth. So finally, we get the following graded group. Yeah, graded Lawrence algebra. Well, I will skip the construction. I'll show you the result. The result is that you have the same commutation relations, like with ordinary Lawrence algebra, but they are graded. So you see the grades add up. For example, if you take zero grade with zero grade, here you'll have zero grade two. But if you have something that has Z three grade one with Z three grade one, it will be give you Z. Three grade two, two and two will give you one zero and one will give one and so forth. So these things are taken module three. These are the other. So this is the full set of this graded Lawrence group. Sorry, graded Lawrence algebra. These are generators of ordinary algebra. And these two are generators of grade one part and grade two part. Subalgebra, these things are not sub algebras because they map when you take commutators here they put you here commutators here put you here like it should be with Z through grading, but the whole thing is an algebra the whole thing is the algebra. Now the problem is how they act on gammas on our color dirac matrices. And now the most important things comes, we will, in order to simplify notations, the all possible gamma matrices will be constructed like this you have one of these three by three matrices which are generators, one of the poly matrices and sigma mu which can be one or zero zero is the unit matrix and 123 are poly matrices. Good. So now we start to expand of course there are many many different commutators to be taken. I don't show you what what is the result but the result is we start with these two. Remember these two matrices gamma zero and gamma I was what we got when we constructed this colored dirac equation. There are two 12 by 12 matrices and only one equation, but the problem is that if we commute them with different generators will get will create more and more similar matrices. But what is amazing that after all these commutations. This are the rules. This is with K zero but we have to commute them with K one, J one, and so forth, in order to produce more and more. These are Lawrence doublet's because we have if you have 23 and eight to they will produce from 83 and 22 because they transform into each other. And finally, you see we have already a doublet because we have matrix and the matrix that is obtained by interchanging the color terms in the first but they all represent the same equation. The final result is the following that of course the generators Q3 and Q3 bar were employed in the construction of Lawrence. Generators be they also they appear in the first matrix. So simple. We are in the combinatorics so this is at least one combinatorics here. We see that all gamma matrices that can be produced are as follows. So choose this can be chosen from a should not be equal to be there chosen from this set from this set. No three and no 456. Yeah, no three and no. This one is missing. Okay. Anyway, what is, first of all, we see that we can have as many as 42 different realizations of this thing, but after completing the all commutators out of this 42, we get only six possibilities. Six gammas and six gamma tilde which are the conjugates. This means that there are only six. Six possible different quarks and six possible anti course. This is very interesting because we have exactly what was predict we predict. It is prediction into the past and not in the future, because it was it is already known, but we somehow we constructed it from the imposing the colors on Dirac equation generalizing it, making it C3 C3 graded. We see that it can be done. This Lawrence. Lawrence invariance imposes new degrees of freedom and this new degrees of freedom are exactly six exactly like this what is observed that is not only, you have not only color quarks, but you have six different color quarks. You have two families, and in each family, you have two flavors, as I say, up and down charm and strangeness, and top and bottom. So you have six, and of course on take works against six. This is the result which is came from in position of Lawrence invariance and of course this Lawrence group is interesting in itself because it is the C3 graded covering of Lawrence and with this three different items but I think I will stop here because the time is over. So thank you for your patience. Thank you dear Richard. I take the turn after Gleb to be the chairman as asked by Gleb. Are there questions and this presentation? I don't know how I can see. Can you see me better? Yeah, now I can see you. Okay. Are there questions? Well, I have small questions. Your sectors are in number three, because you were a grade by C3. Yeah. Is it related by your to your ternary previous work? Yes, yes, yes, yes, of course it is inspired by it. Yeah. Okay, okay. Yeah, this is because we started this ternary algebras. But finally, here it is simpler because there's the variables are not C3 graded. The variables are just complex functions and complex matrices. Are these graded? Are the, it is grading comes because the matrices are different. You have different. Any question and more question, remark or comment? People are tired. People are tired. Okay, yes, you can show the archive if you. Yeah, yeah, yeah, there is this of course. Anyway, your slides are. Your slides are accessible. There's much more on the program. Of course, there are papers published. So I take the opportunity to, for my closing, please send your slides as soon as possible so that we could put them on the program in IHS. Thank you for making talks. Thank you for attending.