Unifying Colour SU(3) with Z3-Graded Lorentz-Poincaré Algebra

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Video in TIB AV-Portal: Unifying Colour SU(3) with Z3-Graded Lorentz-Poincaré Algebra

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Unifying Colour SU(3) with Z3-Graded Lorentz-Poincaré Algebra
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2021
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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.
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[Music] okay
so i'll try to do my best to to put it on on the level so today i will speak it's a privilege to be the last speaker so try to not 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 integer 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 these colors as a
three valued variable into the equation so we'll need to generalize the idea of half integer spinners they'll have much more components and then you will find the spinorial representation of the z3 graded lorentz algebra because everything will become z3 graded z3 graded means that there is instead of z2 is based on two there is one generator which is minus one for example and the square is one here the generator is a cubic root of unity and its square is also a 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 uh what i will be exposing here is uh based on the common work with jesus lukerski from poland from brussels university and there are publications which are on archive if anybody is interested that you can look at them so now let's see this just illustration to to tell you what it is about it's about quarks 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 they are apparently they are inside the nucleus they are as small as electrons if not smaller so you see for example here the nucleon is about thousand times smaller than any atom now the nuclei are in the nucleus they're even ten times smaller but quarks or electrons are even thousand times smaller than nuclei than protons or neutrons but what is also strange that quarks 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 is something different in them this is the
confinement mystery you see the deep inelastic scattering i mean when physicists have very energetic electrons they penetrate inside a nucleon 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 just what i said sorry pass
through 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 quarks interact with 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 while the electrons or other leptons like new mesons they do not see strong interactions they do not have colors they see only weak and electromagnetic interactions
here is 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 of which 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 then they're 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 these ones but there are three different families and three and each family has two work states what is also interesting that we have these three colors
and in quantum chromodynamics quarks are considered like ordinary fermions but endowed with sorry
there's something
i'm not yeah
so uh they are under these colors and th
this is how they compose particles that are observable a proton neutron these are hyperins and so on and only different colors cannot coexist it is 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 can not 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 values and not two and a spin remember to describe this z2 symmetry that there are only two states of spin which cannot coexist so is this plus or minus here to to describe three states of course the natural thing is z3 grading and not z2 grading now i come to the experi 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 poly 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 dig atomic
spin variable by introducing a two component function two functions which are called now they're called poly spinners
and of course on this two component functions you must have hermitian matrices that act upon because all uh material or all quantum operators acting on states should be hermitian in order to have real expectation values so you know very well probably these are the three this is the basis of three traceless hermitian matrices and they must be traceless because if you want to exponentiate if you have the algebra of such things we want to have unitary representation and there is another hermitian the fourth hermitian matrix 2 by 2 which is just a unit matrix but it is not traceless so the exponent will not have the term equal to 1. the three poly matrices span
the three dimension of the algebra which is algebra of rotations
and they also span the clifford algebra of three-dimensional euclidean space so now how paulie uh proposed first he wrote the simplest schredinger-like
equation in schredinger equation you have to the energy is replaced by minus i h time derivative and momentum minus i is gradient so the simplest schrodinger light 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 bytes on two component column so we multiply scalarly by sigma matrices and this is another two by two operator which is hermitian so fantastic we have a linear equation that looks like scheduling equation for this two component spinner fine but unfortunately it is not lorentz invariant it does not obey the laurentian 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 e-square there will be momentum squared multiplied by c squared the mast will give r squared fantastic but there will be a double product and this double product destroys the lorenzo variance because relativistic invariant is like this e square this is the square of uh pseudo scalar this is absolute scalar product of a four vector which is called four momentum energy and momentum in relativity you have four momentum and this 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 so 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 uh produce it like a product of difference by a sum if you have e plus p minus uh multiplied by e minus p then you have e square minus p squared but how to do it well in order to do introduce another
another poly spinner and mix them up you see if we have psi plus which is a power spinner with two a column of two wave 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 we'll get we'll 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
this is 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
lorenz invariant
and of course this two equations
can be written in a more concise form we introduce gamma zero which is sigma three translate with unit matrix this is gamma d x gamma zero and gamma k the three remaining space components are obtained by tensorizing with i sigma 2. why we must put i here because this matrix is hermitian this should be anti-hermitian because the squares should span the minkowski matrix so the square of this will be 1 because sigma 3 is 1 and the square of this sigma k squared gives one but this i will give minus one so we have the proper uh signature of minkowski in space like this so we have the
so we have now created the clifford algebra related with ninkowski metric tensor the sigma matrices created spent the clifford algebra of euclidean three-dimensional space this gamma matrices of dirac they span clifford algebra of minkowski in space and of course the commutators give the generators of the lorenz algebra now 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 paulie was scared of but dirac accepted it and of course he predicted the positrons the 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 now relativistic invariants the spinners poly spinners that compose the dirac spinner they under lawrence transformations they transform differently because there are two different representations of sl2c group which is covering group of the lorenz group and everything becomes
sorry everything becomes lorenz invariant now these two couple there are
uh power 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 cyplus and psi minus are poly spinners which are two components so now let us see uh you see how z2 symmetry acts on these equations on this 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 z2 cross z2 group
one z2 group is this
is describing the half integral spin spin upper spin down the two exclusive states of an electron and there is another z2 symmetry that has been produced because we wanted to make it lawrence invariant the other the symmetry is called charge conjugation it is the symmetry between particles and anti-particles so now let us see how the same thing can be done with colors and with z3 so if we want to describe not only half integer 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 and then we attribute colors to them and but now they have to interact by a potential in order to understand why they cannot propagate so the there is a special potential that is created very strange one instead of decreasing with distance it increasingly it decrea increases linear 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 durack spinners but to give the colors to poly spinners so then you'll have five plus is a poly spinner we will call it red this one kai plus is blue and this one is cyber is this green but we remember that all particles even if they are dirac particles they have to have partners which are anti-particles so we we must have also particles that are anti-colors so there are six other
functions we'll call them phi minus phi minus one this is a poly spinner but corresponding to anti-color anti-color of red is called cyan cn alternate anti-color of kaikou what which was blue is yellow and anti-color of green of psy 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 are additive these colors add up when you look at them they add up for example red plus green will give you the impression of yellow but you probably know you probably observed that these three
colors
c and yellow and magenta they are used not in tv but they are used in your printers because there you subtract them the white page is white and then when you put something you subtract one of the real colors and then you get the anti-colors so for example if you subtract cn you'll get red from out of white okay so now how we'll do
we will follow the same uh
the same logic that produced dirac equation out of power equations so but now we'll have to incorporate not only z2 cross z2 but also z3 so you have one z2 for half integer spin spin up spin down one z2 for the fact that there are particles and anti-particles and finally z3 symmetry which describes the the fact that we have three different colors so all in all the wave function now will have 12 components 3 times 2 times 2 is 12. the dirac particle had four components a direct spinner now you have 12 components
so this is what i just pronounced let us
see what kind of equation can we we'll follow the same logic as in pauli from dirac
so remember that when we pass from particle to ante 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'll call it j j is just the cubic root of unity it is e to power 2 pi i over 3. so each time we pass to another color we have to multiply mass by j and if plus even more than j square and only after third step will 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 phi plus red poly spinner the mass is the same the mass is positive okay but you have to go to the next color and to anti-particle so the momentum acts 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 z3 so here we have mass multiplied by minus j but then you pass through another color into particle and so you see you have to we we must do six such steps in order to come back to five plus from pi phi plus to k minus from k minus to psi plus from psi plus to phi minus phi minus k plus and so on so so you see and we exhausted all six possibilities the it is z2 cross z3 and z2 cross z3 the simple product of z2 by z3 is z6 so these are all all six order uh roots of unity jj squared and one are third roots of unity but if you multiply by minus one you get sixth root of u okay so this is the system we have to investigate and i remind that this this five plus five minus this this are poly 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 twelfth dimension and this is also trend-dimensional because each of these 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 these 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 claim klein gordon equation here it is not possible because we have this entanglement of six different wave functions with three colors 300 colors so in order to get rid of all mixed of all combined double products we have to go to sixth to sixth power and this is very interesting because the sixth power gives you something that looks exactly like uh laurentian invariant you remember e squared m squared p squared if you put squares here this was the klein gordon equation so here it is looks like but it is sixth order so it is not lorenz invariant but if you write it if one writes it in this manner which looks like a lorenz invariant but it's not of course then you see that it can be decomposed that this is a product this is a product of three different factors and these factors they look like lower energy variance look this one is the lorenz invariant this is e square minus p squared this is fantastic it's like if you write that this is m squared very good this is the lawrence invariant quantity but it is multiplied by two other quantities which are complex conjugate and they look like lorenz invariants but they are not because they have this two 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 is a z3 graded lawrence group one is zero grade this is grade one and grade two and all three if they are then it becomes lorenz invariant now how much time do i have
gleb you are uh until um until then first okay 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 uh mass and another was for momentum so if you in one if we introduce this two traces matrices three by three we call it b and we call it q three then the this twelve by twelve mass matrix can be written like this b tends to it with sigma 3 because there's 1 and minus 1 and the unit matrix 2 by 2. and the momentum was q 3 it was like this there was sigma one because they were up diagonal and there was this little two by two momentum operators with poly matrices so now uh it's interesting that these two matrices they are both this is traceless and this is traces of course if you take the enveloping algebra they generate the lee algebra which is uh 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 1 minus 1 minus 1 j minus j and j squared minus j squared and this is also this off diagonal matrix so now
in order to make it
again like dirac equation we'll 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 we would like 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 b dagger and sigma three then you'll have one here unit matrix here and this will be just for
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 dirac operator the only difference is that only six power is proportional to 112 so this is the diagonalization of the system because now each of these components satisfies the same equation all 12 components satisfy the same equation but unfortunately this equation is not lorenz invariant but we'll show that it is invariant
under a generalization of clearance group which is the z3 great deterrence group so you see this is exactly a direct question but the problem is
sorry
of course one can say there are many different choices the problem is why we choose this one it depends because uh 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'll have a different representation of the same color dirac equation now the question is how much how how many such dirac equations are possible because there are eight different
there are eight different generators you see this six matrices traces matrix traces hermitian matrices uh they all span space of uh but this is not complete you have still two other traceless matrices which are diagonal but we 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 their hermitian conjugates will be given grade two and grade zero will be
yeah grade zero will be two diagonal
matrices but traceless one will be called b another b
dagger her mission now this is they span a very interesting ternary algebra i will not you see these combinations the skew z z3 commutators are zero and the anti-commutator
see the you have three different permutations and they are all proportional to one but this is this is the tensor one j j squared so this is this is called ternary cliff algebra and
of course you have the same for complex for hermitian conjugates with a hermitian conjugate of this spinorial metric or these are the two matrices that were not the numerators so we have eight different generators which generate su 3 algebra this is the base basis of su-3 algebra and this basis was
already studied by victor cuts in 25 years ago so we have this symmetry su 3 you see
this is interesting because we started with z3 we produced an equation this equation naturally introduced these two matrices these two matrices introduced the the algebra and then we find the z3 generated the symmetry which is su 3 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 anti-commute 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 now this is the equation which is
written now the gamma matrices are like this and let us try to introduce the generators of lorenz group which will act on them of course these matrices are you remember they are 12 by 12 matrices so the generators of lorentz algebra they have to be also because they will commuted them so they 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 g gamma k gamma z gamma zero and of course these are new matrices which should be interpreted these are the generators of ordinary space rotations and these are generators of lorenz boosts which mix 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 lorenz 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 the graded lawrence algebra well i'll skip the
construction i'll show you the result the result is
that you have the same commutation relations like with ordinary lorenz 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 z3 grade one with zeke regrade one it will 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 taking modulo three
these are the other so this is the full set of this graded lorenz group sorry graded lorenz algebra these are the generators of ordinary algebra and these two are generators of grade one part and grade two part this is a sub algebra these things are not subalgebras 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 and now the problem is how they act on gammas on our colored 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 one two three 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 these were these two 12 by 12 matrices and only one equation but the problem is that if we commute them with uh different generators will get will create more and more similar matrices but what is amazing that after all these
communications
you see like this this has the rules this is with k0 but we have to commute them with k1 j1 and so forth in order to produce
more and more these are lawrence doublets because we have if you have 2 3 and 8 2 they will reproduce from 8 3 and 2 2 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 they all represent the same equation and finally the final result is the
following
that of course the generators q3 and q3 bar were employed in the construction of lorenz generators b 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 uh a should not be equal to b they are chosen from this set from this set no three and no uh four five six seven eight yeah no three and no uh which one this one this is missing okay anyway what is first of all we see that uh 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-cores this is very interesting because we have exactly what was predicted 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 reconstructed it from the imposing the colors on the rock equation generalizing it making it z3 c3 graded we see that it can be done this lorenz uh lorenz invariance imposes new degrees of freedom and these 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 three families and in each family you have uh two flavors as they say up and down charm and strangeness and top and bottom so you have six and of course anti-quarks against six so this is the result which is came from imposition of lorenz variance and of course this florence group is interesting in itself because it is a z3 graded covering of lorenz and with with three different items but i think i will stop here because the time is over so thank you for your patience thank you dear uh richard i take the turn after gleb to be the chairman as asked by gleb uh are there questions and this uh presentation i don't know how i can see ah i don't have the can you see me better yeah now i can see you okay are there questions well i have a small questions uh your sectors are in number three because you you were grade by z3 yeah uh is it related by you to your ternary previous work yes yes yes of course it is inspired by it yeah okay okay yeah this is uh yeah because we started this ternary uh algebras but finally uh here it is is simpler because there's the variables are not z3 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 uh if you yeah yeah yeah there's this of course okay it anyway your slides are yeah this slice is accessible there's much more on the program there are of course there are papers published so i take the opportunity to for my closing please send your your slides as 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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