Unifying Colour SU(3) with Z3-Graded Lorentz-Poincaré Algebra
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Kerner, Richard
A generalization of Dirac’s equation is presented, incorporating the three-valued colour variable in a way which makes it intertwine with the Lorentz transformations. We show how the Lorentz-Poincaré group must be extended to accomodate both SU(3) and the Lorentz transformations. Both symmetries become intertwined, so that the system can be diagonalized only after the sixth iteration, leading to a six-order characteristic equation with complex masses similar to those of the Lee-Wick model. The spinorial representation of the Z3-graded Lorentz algebra is presented, and its vectorial counterpart acting on a Z3-graded extension of the Minkowski space-time is also constucted. Application to new formulation of the QCD and its gauge-field content is briefly evoked.
 Okay, so I'll try to do my best to put it on the level. 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 difference. The works are endowed not only with half integral spin, but apparently they have another variable, which takes on not to but three values, and they are exclusive. 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. The three graded means that there is instead of Z two is based on two. There is one generator who is minus one, for example, and the square is one here the generator is cubic root of unity. And it's square is also cubic root of unity but it's a different one. And finally only the cube is equal to one. And now I would 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 an archive. If anybody is interested, you can look at them. And now let's 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're in, not like electrons or protons or other particles they cannot be observed free. So you see there inside the new plans. There 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. And 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. If they were just obeying Dirac 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. Deeply in the last six catering. I mean when physicists have very energetic electrons, they penetrate inside the new clone inside the 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 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. The electromagnetic force that we know well, strong forces quirks interact with the strong forces and they carry this new new degree of freedom which is called color. There are three different possible states. There are weak interactions so quirks interact with everything they interact electromagnetically weak and strong, while the electrons or other leptons like mew 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. Just the present knowledge, we have not all types of quirks, we have six different types of quirks, of which this type there are three families, and two different states of course, so the, the family that is really well known is up and down this one, the quirks constitute neutrons and protons, and this, these are the most common ones. Then there are strange and strange and charm and top and bottom this works are so heavy that they are observed only in very very energetic collisions, and usually we don't see only this one, 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. And in quantum chromodynamics, quirks are considered like ordinary fermions, but endowed with, sorry, something. But yeah, so they are endowed these colors and this is how they compose particles that are observable, the proton, neutron, these are hyperons, and so on, and only different colors cannot coexist. It is exactly like half integral spin of electron in any atom if you have two electrons, this, beside the spin they have the same energy the same magnetic number the same or momentum, so they cannot have the same spin, they must have two opposite spins. But here, if you have quirks inside the proton or neutron, whatever you choose, they must have different colors this is the new variable. It's like spin but it takes on three values 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 the three grading and not Z two grading. Now I come 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 degrees 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. So this is why he proposed to describe the dichotomic spin variable by introducing two components function to functions, which are called now they're called Pauli spinners. And 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. So, 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 exponent sheet 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. So the Pauli matrices span the three dimensional the algebra, which is algebra of rotations. And they also spend the Clifford algebra of three dimensional Euclidean space. So now how Pauli proposed first, he wrote the simplest Schrodinger like equation in Schrodinger equation you have to the energy is replaced by minus IH time derivative and momentum minus age gradient. The simplest Schrodinger 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. Fantastic we have a linear equation looks like Schrodinger equation for these two components, but unfortunately, it is not Lawrence invariant. It does not obey the Lawrence and invariance because if we square such an equation. If you iterate it, it becomes diagonal. But then we have the relationship, you see, there will be a square. There will be momentum square multiplied by c square. The mass will give us squared. Fantastic, but there will be a double product. And this double product destroys the Lawrence covariance because relativistic invariant is like this. This square. 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. The momentum 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. You can multiply it by 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, two way functions. But here, 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. What happens if we iterate it now both side plus and psi minus will obey the Klein Gordon equation, which is, of course, it is Lawrence invariant. Of course, this two equations can be written in a more concise form. We introduced gamma zero, which is sigma three, transfer tensor with unit matrix this is gamma Dirac gamma zero and gamma K, the three remaining space components are obtained by tensor is 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 square of this 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. And as 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, the Pauly 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. It's just the same. And they have been discovered of course. So, relativistic invariance, the spinners, Pauly spinners that compose the Dirac spinner, they under Lawrence transformations they transform differently, because they're two different representations of SL to C group, which is covering group of the Lawrence and everything becomes, sorry, everything becomes Lorentz invariant. Now, these two coupled Dirac, Pauly 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 Pauly spinners which are two components. So now let us see, you see how Z2 symmetry acts on these equations on these states, because if you change spin, if spin changes sign, and momentum changes sign. So, the equation remains 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 Z2 cross Z2 group. One Z2 group is this is describing the half integral spin, spin up or spin down the two exclusive states of an electron. So, the Z2 symmetry that has been produced, because we wanted to make it Lawrence invariant. The other is the symmetry is called charge conjugation. It is a symmetry between particles and anti particles. So now let us see how the same thing can be done with colors and with Z3. So 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. So, we attribute colors to them. And, but now they have to interact by a potential in order to understand why they cannot propagate. So there is a special potential that is created very strange one, instead of decreasing with distance, it's increasingly, it increases linearly with distance. The farther you go, the more the forces that push you together grow. That is why but this is, it is 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. So if you have five plus is a Pauli spinner. We will call it red. This one, Chi 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 will call them five minus five minus one this is a Pauli spinner but corresponding to anti color, anti color of red is called cyan cyan. I'll take anti color of kind who was, which was blue is yellow and anti color of green of psi is called magenta. And 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. So red plus green will give you the impression of yellow. But you probably know, you probably observed that these three colors. So you see in yellow and magenta, they are used, not in TV but they are used in your printers, because they are subs, 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. So if you subtract, see, and you'll get red from out of wine. Okay, so now how we'll do. We'll follow the same, the same logic that produced dirac equation out of Pauli equations. So, but now we'll have to incorporate not only see to cross the two but also see three. One Z two for half integral spin spin up spin down one Z two for the fact that there are particles and anti particles. And finally, the Z three symmetry, which describes the fact that we have three different colors. So all in all, the wave function now will have 12 components three times two times two is 12 the dirac particle head for component to do X spinner. 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 will call it J. J is just the cubic root of unity it is e to power two pi i over three. So each time we pass to another color. We have to multiply mass by J. And if pass even more than j square and only after third step we'll have will 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. You see, we start with five plus red. 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. Now 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 to particle. And so you see you have to. We must do six such steps in order to come back to five plus from five five plus to chi minus from chi minus to psi plus from psi plus to five minus five minus chi plus and so on so so you see. So we exhausted all six possibilities that it is the 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. So the first two are the square, and one are third roots of unity but if you multiply by minus one you get six roots of you. Okay, so this is the system we 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. So 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 the 12 column vector on the column of 12 complex functions. So 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. And 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. 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's very interesting because the six power gives you something that looks exactly like a Lawrence in invariant. You remember, E squared M squared P squared, if you put squares here. This was the client Gordon equation. And it is looks like but it is six order. So it is not Lawrence invariant. But if you write it if one writes it in this manner, which looks like Lawrence 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. These factors, they look like Lawrence in violence look this one is a Lawrence invariant. This is E squared minus p squared. This is fantastic. It's like, if you write that this is M squared. Very good. This is a Lawrence invariant quantity, but it is multiplying by two other quantities, which are complex conjugate. And they look like Lawrence invariants, but they are not because they have this to possible roots cubic roots of union, but they're 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 zero grade. This is grade one and grade two and all three. If they are then it becomes the rent invariant. Now how much time do I have. You are until 10 past. 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 at with Sigma three because there's one and minus one and unit matrix two by two. And 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. Because 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 operator would like to make it a 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. 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. Sorry. Yeah. 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 Lawrence invariant, but will show that it is invariant under generalization of Lawrence group, which is the Z three grade that Lawrence group. You see this is exactly a dirac equation but the problem is. Sorry. And 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 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. And 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. 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. And they span a very interesting ternary algebra I will not. You see these combinations. The skew, 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 in the middle of this so we have eight different generators, which generate this algebra. This is a base basis of SU three algebra, and this basis was already started by Victor cuts in 25 years ago. And then 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 algebra and then we find the Z three generated the symmetry which is SU three, which is fine. 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. 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. Now this is the equation which is written now the gamma matrices are like this. We must 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. So the generators of Lawrence algebra, they have to be also because they will commuted them. So we have, we must take them from 12 by 12 matrices. So, I'll speed a little bit. Yeah. So let us start with this kind of commutators. So gamma k 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. 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. I'll 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 if you have the grades add up. For example, if you take zero grade with zero grade here will 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 the 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. And 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. And now the problem is how they act on gammas on our color direct matrices. And now the most important things comes. We will, in order to simplify the problem, we will use the same method. And now the problem is how they act on gammas on our color direct matrices. And now the most important things comes. The old 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 which can be one or zero zero is the unit matrix, and 123 are poly matrices. 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 direct equation. This was this to 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. And if you take this. 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 then you see, 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. And finally the final result is the following. 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. We have to choose this can be chosen from a should not be equal to be there chosen from this set from this set. So, three and no. 456, 788. Yeah, no three and no. Which one. This one is 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. Of course, 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 the rack equation generalizing it, making it C3. We have 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. So 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, is the three graded covering 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 a glib. Yeah, to be the chairman as asked by the live. Are there questions. I don't know how to presentation. I don't know how to I can see. I don't have the. Can you see me better. Yeah, now I can see you. Okay, are there questions. I have a small questions. Your sectors are in number three, because you, you were a grade by Z three. Yeah. Is it related by your to your ternary previous work. Yes, yes, yes, of course it is inspired by it. Yeah. Okay. Yeah, this is, yeah, because we started this ternary algebras. But finally, here it is, it is simpler because there's the variables are not Z three graded. The variables are just complex functions and complex matrices. What is 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's this of course. Anyway, your slides are in the 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, your slides as soon as possible so that we could put them on the program in IHS. And I thank you for making talks. I thank you for attending.
 Okay, so I'll try to do my best to put it on the level.  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  difference. The works are endowed not only with half integral spin, but apparently they have another variable, which takes on not to but three values, and they are exclusive.  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.  The three graded means that there is instead of Z two is based on two. There is one generator who is minus one, for example, and the square is one here the generator is cubic root of unity.  And it's square is also cubic root of unity but it's a different one. And finally only the cube is equal to one.  And now I would 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 an archive. If anybody is interested, you can look at them.  And now let's 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're in, not like electrons or protons or other particles they  cannot be observed free.  So you see there inside the new plans.  There 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.  And 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.  If they were just obeying Dirac 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.  Deeply in the last six catering. I mean when physicists have very energetic electrons, they penetrate inside the new clone inside the 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 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.  The electromagnetic force that we know well, strong forces quirks interact with the strong forces and they carry this new new degree of freedom which is called color. There are three different possible states.  There are weak interactions so quirks interact with everything they interact electromagnetically weak and strong, while the electrons or other leptons like mew 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.  Just the present knowledge, we have not all types of quirks, we have six different types of quirks, of which this type there are three families, and two different states of course, so the, the family that is really well known is up and down this  one, the quirks constitute neutrons and protons, and this, these are the most common ones. Then there are strange and strange and charm and top and bottom this works are so heavy that they are observed only in very very energetic collisions, and usually we don't see  only this one, 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.  And in quantum chromodynamics, quirks are considered like ordinary fermions, but endowed with, sorry, something.  But yeah, so they are endowed these colors and this is how they compose particles that are observable, the proton, neutron, these are hyperons, and so on, and only different colors cannot coexist.  It is exactly like half integral spin of electron in any atom if you have two electrons, this, beside the spin they have the same energy the same magnetic number the same  or momentum, so they cannot have the same spin, they must have two opposite spins. But here, if you have quirks inside the proton or neutron, whatever you choose, they must have different colors this is the  new variable. It's like spin but it takes on three values 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 the three grading and not Z two grading.  Now I come 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 degrees 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.  So this is why he proposed to describe the dichotomic spin variable by introducing two components function to functions, which are called now they're called Pauli spinners.  And 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.  So, 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 exponent sheet 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.  So the Pauli matrices span the three dimensional the algebra, which is algebra of rotations.  And they also spend the Clifford algebra of three dimensional Euclidean space.  So now how Pauli proposed first, he wrote the simplest Schrodinger like equation in Schrodinger equation you have to the energy is replaced by minus IH time derivative and momentum minus age gradient.  The simplest Schrodinger 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.  Fantastic we have a linear equation looks like Schrodinger equation for these two components, but unfortunately, it is not Lawrence invariant. It does not obey the Lawrence and invariance because if we square such an equation.  If you iterate it, it becomes diagonal. But then we have the relationship, you see, there will be a square. There will be momentum square multiplied by c square.  The mass will give us squared. Fantastic, but there will be a double product. And this double product destroys the Lawrence covariance because relativistic invariant is like this.  This square. 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.  The momentum 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.  You can multiply it by 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,  two way functions. But here, 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.  What happens if we iterate it now both side plus and psi minus will obey the Klein Gordon equation, which is, of course, it is Lawrence invariant.  Of course, this two equations can be written in a more concise form. We introduced gamma zero, which is sigma three, transfer tensor with unit matrix this is gamma Dirac gamma zero and gamma K, the three remaining space components are obtained by tensor is 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 square of this 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.  And as 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, the Pauly 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.  It's just the same.  And they have been discovered of course.  So, relativistic invariance, the spinners, Pauly spinners that compose the Dirac spinner, they under Lawrence transformations they transform differently, because they're two different representations of SL to C group, which is covering group of the Lawrence  and everything becomes, sorry, everything becomes Lorentz invariant. Now, these two coupled Dirac, Pauly 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 Pauly spinners which are two components.  So now let us see, you see how Z2 symmetry acts on these equations on these states, because if you change spin, if spin changes sign, and momentum changes sign.  So, the equation remains 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 Z2 cross Z2 group. One Z2 group is this is describing the half integral spin, spin up or spin down the two exclusive states of an electron.  So, the Z2 symmetry that has been produced, because we wanted to make it Lawrence invariant. The other is the symmetry is called charge conjugation. It is a symmetry between particles and anti particles.  So now let us see how the same thing can be done with colors and with Z3.  So 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.  So, we attribute colors to them. And, but now they have to interact by a potential in order to understand why they cannot propagate.  So there is a special potential that is created very strange one, instead of decreasing with distance, it's increasingly, it increases linearly with distance.  The farther you go, the more the forces that push you together grow. That is why but this is, it is 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.  So if you have five plus is a Pauli spinner.  We will call it red. This one, Chi 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 will call them five minus five minus one this is a Pauli spinner but corresponding to anti color, anti color of red is called cyan cyan.  I'll take anti color of kind who was, which was blue is yellow and anti color of green of psi is called magenta.  And 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.  So red plus green will give you the impression of yellow. But you probably know, you probably observed that these three colors.  So you see in yellow and magenta, they are used, not in TV but they are used in your printers, because they are subs, 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.  So if you subtract, see, and you'll get red from out of wine. Okay, so now how we'll do.  We'll follow the same, the same logic that produced dirac equation out of Pauli equations. So, but now we'll have to incorporate not only see to cross the two but also see three.  One Z two for half integral spin spin up spin down one Z two for the fact that there are particles and anti particles. And finally, the Z three symmetry, which describes the fact that we have three different colors.  So all in all, the wave function now will have 12 components three times two times two is 12 the dirac particle head for component to do X spinner. 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 will call it J. J is just the cubic root of unity it is e to power two pi i over three.  So each time we pass to another color.  We have to multiply mass by J.  And if pass even more than j square and only after third step we'll have will 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.  You see, we start with five plus red.  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.  Now 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 to particle. And so you see you have to.  We must do six such steps in order to come back to five plus from five five plus to chi minus from chi minus to psi plus from psi plus to five minus five minus chi plus and so on so so you see.  So we exhausted all six possibilities that it is the 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.  So the first two are the square, and one are third roots of unity but if you multiply by minus one you get six roots of you. Okay, so this is the system we 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.  So 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 the 12 column vector on the column of 12 complex functions.  So 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.  And 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.  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's very interesting because the six power gives you something that looks exactly like a Lawrence in invariant. You remember, E squared M squared P squared, if you put squares here. This was the client Gordon equation.  And it is looks like but it is six order. So it is not Lawrence invariant. But if you write it if one writes it in this manner, which looks like Lawrence 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.  These factors, they look like Lawrence in violence look this one is a Lawrence invariant. This is E squared minus p squared. This is fantastic. It's like, if you write that this is M squared. Very good. This is a Lawrence invariant quantity, but it is multiplying by two other quantities, which are complex conjugate.  And they look like Lawrence invariants, but they are not because they have this to possible roots cubic roots of union, but they're 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 zero grade. This is grade one and grade two and all three. If they are then it becomes the rent invariant. Now how much time do I have.  You are until 10 past.  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 at with Sigma three because there's one and minus one and unit matrix two by two.  And 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.  Because 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 operator would like to make it  a 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.  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.  Sorry.  Yeah.  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 Lawrence invariant, but will show that it is invariant under generalization of Lawrence group, which is the Z three grade that Lawrence group.  You see this is exactly a dirac equation but the problem is.  Sorry.  And 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 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.  And 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.  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.  And they span a very interesting ternary algebra I will not. You see these combinations. The skew, 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 in the middle of this so we have eight different generators, which generate  this algebra. This is a base basis of SU three algebra, and this basis was already started by Victor cuts in 25 years ago.  And then 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 algebra and then we find the Z three generated  the symmetry which is SU three, which is fine.  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.  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.  Now this is the equation which is written now the gamma matrices are like this.  We must 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.  So the generators of Lawrence algebra, they have to be also because they will commuted them. So we have, we must take them from 12 by 12 matrices.  So, I'll speed a little bit.  Yeah.  So let us start with this kind of commutators.  So gamma k 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.  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.  I'll 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 if you have the grades add up. For example, if you take zero grade with zero grade here will 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 the 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.  And 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.  And now the problem is how they act on gammas on our color direct matrices. And now the most important things comes.  We will, in order to simplify the problem, we will use the same method.  And now the problem is how they act on gammas on our color direct matrices. And now the most important things comes.  The old 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 which can be one or zero zero is the unit matrix, and 123 are poly matrices.  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 direct equation.  This was this to 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.  And if you take this. 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 then you see, 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.  And finally the final result is the following.  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.  We have to choose this can be chosen from a should not be equal to be there chosen from this set from this set.  So, three and no.  456, 788.  Yeah, no three and no.  Which one.  This one is 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.  Of course, 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 the rack equation generalizing it, making it C3.  We have 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. So 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, is the three graded covering  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 a glib.  Yeah, to be the chairman as asked by the live.  Are there questions.  I don't know how to presentation.  I don't know how to I can see.  I don't have the.  Can you see me better. Yeah, now I can see you.  Okay, are there questions.  I have a small questions. Your sectors are in number three, because you, you were a grade by Z three.  Yeah.  Is it related by your to your ternary previous work.  Yes, yes, yes, of course it is inspired by it. Yeah.  Okay. Yeah, this is, yeah, because we started this ternary algebras.  But finally, here it is, it is simpler because there's the variables are not Z three graded.  The variables are just complex functions and complex matrices.  What is 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's this of course.  Anyway, your slides are in the 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, your slides as soon as possible so that we could put them on the program in IHS.  And I thank you for making talks. I thank you for attending.
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