Lecture 4 | New Revolutions in Particle Physics: Standard Model

[Music] Stanford University I'm going to do something now I've wanted to do for years I've really wanted to do this okay ladies and gentlemen people out in the audience viewing this from afar we are now going to stop for a commercial break I want you all to go out and buy my book the black hole war my battle is Stephen Hawking to make the world safe for quantum mechanics in that book you will learn about the cruelty of theoretical physicists you will not only learn about Schrodinger's cat but you will learn about Susskind fish and how he flushed them down the toilet audience please yeah come on right now back to our program you may remember that when we last saw our Alice and Bob Alice was crying bitterly because the cruel Bob was forcing her to learn group theory Alice sad to say is not finished Bob is not finished with her tonight tonight we will see Alice suffer more okay are there any questions about the group theory at the moment that now that we're back to the story main storyline any any questions about group theory we're going to we're not finished with it we're really not finished with it yeah there's there's two parts of your expose on group it seems to be some kind of abstract mathematical objects that have some kind of multiplication to pay right and there's also the part on how those groups acts on various objects right can we disassociate the two or is it well they they're completely related is the abstract notion let's so let's take the case of rotation of a spin alright there's a perfect case to study there's the abstract completely abstract notion of a rotation in space and I don't tell you what it acts on I just tell you that is an operation that you can do or physics has to respect symmetry under the rotations of space and so we must then say how does the rotation of space act on the quantities which describe which are physically relevant right if we were talking about classical physics we might just say that the rotation of space R rotates the coordinates of a particle it might rotate the direction of a magnet whatever it does we know how to describe that we describe that by we know how to submit how to describe that in quantum mechanics the states of a system are always represented by a linear vector space there's the abstract notion of a state of a system and there's the concrete representation of it in terms of column vectors how big are those column vectors what's the what's the number of entries into the column vector well that depends on the system but basically it's the number of mutually orthogonal possibilities for that system if the system in question only consists of an electron spin then there's only two states to mutually orthogonal States spin up spin down if it's two electrons and again we're only interested in the spin then there would be four states two up two down one up one down and down one up are if we're talking about a more complex object for example the entire motion of an electron its position as well as its momentum then there are an infinite number of states the number of states that it takes to describe the orbital ok the number of mutually orthogonal states of an electron speaking now about its position is infinite it could be anywhere in space or you could choose to describe it in terms of momentum so the column vectors describing an electrons position are infinite dimensional an amplitude for every possible location or for every possible momentum so there must be representations of the rotation group of this group of rotations which are infinite dimensional matrices and they are they are just how what are the matrices which act on the column vectors which describe the system that you're interested in the size of those matrices will depend on the number of mutually orthogonal States all right so let's come back spin of the electron throw away everything else out of your head spin of the electron two by two matrices the group elements are the abstract rotations they're represented by two by two matrices a spin one particle the spin one particle has three mutually orthogonal states the rotations are represented by three by three matrices in that case those three by three matrices are familiar to some of you they are the three dimensional matrices which mix up the components of three vectors what comes after spin if the three-halves not one-half well first and zero what about spin zero spin zero rotations don't act on at all they just leave it alone okay so it's completely trivial you can think of it in some very useless sense that that rotations are act on spin zero just to leave it alone okay that corresponds to one by one matrices just the unit matrix that's all just the unit matrix does nothing are two by two that's been a half three by three that's been one four by four that's been three halves and so forth so the same abstract group element the same abstract symmetry operation may have many matrix representations of different dimensionality are that's an important fact and we're going to we're going to spend a little more time on it but let's just review for a moment what the symmetries we talked about the symmetries of rotation of space on a spin on a spin 1/2 particle in particular on a spin 1/2 particle and we said those rotation operations become represented by matrices which I call you you because the unitary so these are two by two unitary matrices I won't bother being more specific than just putting dots there those 2x2 unitary matrices act on the state vectors of spin 1/2 again I won't be more specific than writing that let's erase it for a moment and this is called a representation of the group it's in one-to-one correspondence with the group elements matrix multiplication group multiplication or a one-to-one correspondence let's just flesh it out what does it mean for matrix to be unitary it means that its inverse is its hermitian conjugate but we know that the number of degrees of freedom the number of parameters in a unitary 2x2 matrix is how many how many did we say they were eight to begin with four elements each complex that's four elements this is four equations that cuts you down to four but there are only three independent parameters the three Euler angles if you like the three angles describing a rotate the three things describing a rotation namely the rotation angle and the two parameters which determine the direction of the rotation so there are only three independent parameters of a rotation these matrices have one too many parameters and to get rid of that one extra parameter one more condition and that's that the determinant of U I'll just represent that we could write it as Det but I'll just write it as absolute value but it's not really absolute value it's it's a symbol for determinants but that is equal to one now a unitary simple fact incidentally that when you multiply matrices the determinants multiply the determinant of a time's the sorry the determinant of the product of a and B is the determinant of a time's the determinant of B all right it is also true that the determinant of a hermitian conjugate of an operator is the complex conjugate of the determinant of the original operator so here's what we can say let's take the determinant of both sides of this the determinant of both sides of this is first of all the determinant of u dagger u but that's also equal to the determinant of U dagger times the determinant of U and what's the determinant of 1 1 is equal to 1 one here of course means the unit matrix or ones on the diagonal all right now next thing is the determinant of U dagger is just the determinant of well that is the determinant of is the complex conjugate of the determinant of U right and if the determinant of U is 1 the determinant of U dagger is 1 and the whole thing is consistent the only point is that it is consistent it's a consistent condition on use that's when I say consistent I mean that it's consistent with the law of matrix multiplication if you take a bunch of unitary matrices every one of which has determinant 1 and you multiply them together you'll again get a thing with determinant 1 alright so it's a it's a mathematical theorem which may not be too surprising for many of you that the special unitary matrices two by two matrices which have three parameters in them special means that you die to you equals one that the group of those matrices is identical to the group of our rotations in three-dimensional space now it's not quite true okay is there is there nothing that says that the determinant has to be real no permanent of a it just has to be radius one right the determinant of a unitary matrix must be a pure phase okay that's just from the fact that it's unitary okay for the fact that unitary but you can take any art here's the point you can take any unitary matrix let's suppose its determinant is e to the I theta you can then multiply that unitary matrix by well you multiply it by e to the minus I theta over two because when you calculate the determinant you multiply two elements together so if you took you and multiplied it by e to the minus I theta over 2 you would construct a special unitary matrix all right so the point is the extra phase that's in there is almost trivial you can always get rid of it by a redefinition of U and in fact the rotation group doesn't contain this piece here it's just isomorphic to the matrices with with determinant 1 ok I told you a little fib when I said that su 2 is exactly the same as the rotation groups it's not exactly there's a two to one correspondence instead of a one-to-one correspondence the unitary operators U and minus u if u is a unitary operator u dagger u equals 1 so is minus you whatever u is right in fact is a two to one correspondence the matrices U and - you both correspond to the same rotation in space but that's a fine point that I which is not going to play any special role in in what we say at least for the time being so a simple statement which is almost true is that there's a correspondence certainly there's a correspondence but that there's a one-to-one correspondence not quite true it's two to one correspondence between the unitary matrices two-by-two matrices the special unitary matrices those were determinant 1 and the the group of rotations and so for that reason the electron spin or the electron wave function is called a doublet and it's acted on by two by two matrices now let's talk a little bit about the generators of a group I think we actually talked about it let's be specific about it again the generators of the group when you have a symmetry and the symmetry is represented by a set of matrices like this then there's a concept of the generators the generators have to do with infinitesimal elements of the group infinitesimal element of the group means in the case of rotation a rotation about some axis by an int by a very very small angle in other words it's an element of the group which is very close to unity unity means no transformation elements which are very close which are just very small shifts of angles and so forth are called infinitesimal elements of the group and the generators of the group are very closely related to these infinitesimally small operations now one of the reasons that the infinitely small elements are important and interesting is because you can build up any element out of a sequence of small ones in other words if I want to rotate about some angle by a finite rotation I just think of it as a multiplication of a bunch of infinitesimal ones so if you know the properties of the infinitesimally generators you can put together the entire structure of the group we're not going to do that we're not going to study that today but we are just going to define the generators of the group and the reason we're going to define the generators of the group because because they really are the conserved quantities they are the quantities are that are conserved and which played the role of conservation laws once you know what the symmetry group is ok so let's define them incidentally for what I've written here there's nothing specific to su - this could be su in where n is the number of elements the special unitary n by n matrices there of course not all isomorphic to rotations it's only the su2 group which is isomorphic to rotations of space that's your 3 su 4 su 5 not so easy to visualize okay so here we have the definition and now let's take an element which is very close to the identity it doesn't have to be su to write it U is equal to the identity 1 plus something small and let's put an eye in for simply notational simplicity it'll simplify notations a small parameter epsilon and a quantity which is often I'm not going to call it GG for generator because we've already used G for group element and so therefore I will use T YT beats me I don't know so U is 1 plus I T what is T let's find out what we can learn about T we use the fact first of all that u dagger U is Warren all right so u dagger u that's 1 plus I epsilon T Oh incidentally it's easy to prove that if u dagger U is 1 the new u dagger is 1 I won't prove that for you that that U and u dagger always commute with each other not for all operators but for unit Ares so it doesn't matter which order you multiply all right so it's 1 plus I epsilon T times 1 minus I epsilon T dagger if this is you then this is u dagger I gets changed to minus I when your complex conjugate and this is supposed to be equal to 1 let's retain things only to the lowest order in epsilon epsilon squared is too small for our interest tonight so let's just keep things to order epsilon and that says 1 times 1 cancels the one there and it says that I exelon times t minus T dagger is equal to 0 or to simplify it just says that T is equal to T dagger what's an operator that's equal to its own conjugate permission and hermitian operators in quantum mechanics represent observables hermitian operators represent observable quantities for the case of rotations what are these T's what physical significance do they have there are three of them well first of all why are the three of them they're actually an infinite number of them but there are three linearly independent ones so let me explain why that is the reason is because the rotation group has only three parameters there's only three independent ways that you can rotate you can rotate about the x axis so I could have made an infinitesimal rotation about the x axis I could have made an infinitesimal rotation about the y axis or the z axis and in fact for an infinitesimal rotation if I wanted to rotate about some other axis what I could first do is rotate a little about the x axis and then a little about the y axis and then a little about the z axis another way to say it is that the generators behave like vectors if I want a vector in an arbitrary direction in other words a vector rotation about some axis about an arbitrary direction the way I can think of it is making it up out of the components the component rotations about the x y&z axes so there are three independent directions that you can rotate in and three independent linearly independent T's oh we missed one point come back back up there is this condition here the condition that the determinant is one alright so let me tell you before I get back we'll get back to in a moment the determinant being one the determinant of any matrix which is of the form 1 plus let's call it epsilon times a small matrix the determinant the determinant of M is equal to one plus epsilon times the trace of small M the determinant of a matrix which is close to the identity is 1 that's just the identity and then the small deviation is epsilon times the trace that's something to prove when you go home and prove that but we can see what that means immediately it means that the trace of T is 0 the trace of T has to be 0 in order for the determinant of U to be equal to 1 alright so that's the second condition the trace of T is 0 and it should be hermitian how many independent hermitian two-by-two matrices are there with trace equal to 0 the answer is 3 of them and they correspond to the rotations about the three axes but more abstractly and generally the generators of the group of rotations are the angular momentum operators for a system so that's what angular momentum means in quantum mechanics it is the generator the generators of the rotation group until you act with the angular momentum or 1 plus I epsilon times the angular momentum to rotate a system a small amount and then if you want to rotate it a large amount you just do it over and over again so everything is contained all the information about the structure of the group is contained in the generators in fact it's contained in the commutation relations of the generators those are the important things I won't go into them again we did commutation relations of angular momentum I just want to connect things up a little before for you but the important thing is that the conserved quantities associated with the symmetry are the generators and so their term three independent two by two major seeds satyam there are three independent two by two traceless two by two hermitian matrices yeah I have a name Polly Polly miss yes no three power matrices and then if you went to three by three matrices nearby three unitary matrices unitary traceless and there are eight like the kill mom yeah they over give them are matrices all right we're going to do that a moment that eight is the same eight as the a gluons in Eightfold way actually I did that's a day that's actually a different eight not because it's it's mathematically the same but it's associated with a different SS you three symmetry than then color but yeah mathematics is the same okay so that was a little bit about a little review the sorrows of Alice let's let's let's torch our Alice even more let's go to su 3 su three the pattern is exactly the same except with talk we're talking about three by three matrices in fact I'm not even sure that there's very much more to say about it other than doing some Counting and finding out how many independent generators there are so to do the counting it's very easy a three by three unitary matrix has how many elements Emily starts with 1818 independent nine independent complex numbers that's that's 18 but then we have u dagger you one now this really is a equation for each element this is a product this is a matrix product so we can think of it as I J and this is Delta IJ over here how many equations is this this is nine equations so we have 18 minus 9 is 9 but then one additional equation that the determinant is equal to one that's eight equations and so we can guess that there are eight different if we start with the if we start with the unit matrix in how many distinct directions in the group space can we deform away from the identity and the answer is 8 the implication being there are eight linearly independent 3x3 traceless hermitian matrices traceless hermitian matrices the number of parameters is 8 so and we are going to come back to that [Applause] but before we do let's talk a little bit about our representations combine to form new representations now what we're talking about here is combining systems together to form new systems in the case of spin we might be talking about combining more than one particle together to create some sort of composite for example and that composite would also have a spin let's not worry about the motion of the particles let's just think of them as things that we staple on top of each other so they're nailed down to each other but each one has a spin so each one has a spin and let's suppose that we take two such particles we put them on top of each other to half spin particles what do we get well we get four possible states but those four possible states can be thought of as a a basis in there for particle States can be thought of as a spin zero object and the three spin one objects that about 1/2 the two half spins can make we take two half spins we can certainly make a spin one why we can line up the spins in the same direction so the Z component of spin can be one it can be minus one and it can be zero but it can be zero in two ways all right but still it's been one particle only has three states so one of those two ways of making 0 must not be part of the spin one multiplet what can it be it can only be spin zero why spin zero because the kid is nothing for it to transform into so there are three combinations which form the spin one multiplet and one which forms the spin zero multiplet when you combine spins together that way there are two possible total spins you can make so if you were to take an atom made out of made out of two half spin particles and ignore the orbital motion altogether one possibility is that the spins are in the same direction that would give you spin one and the other possibility is that they are in the opposite direction in which case it would be spin zero it's ortho and para hydrogen in particular for the hydrogen atom where the two spins would be the proton and the electron right so you combine it's important to know how to combine spins and to know what you get what possibilities you get when you combine them together what happens in SU 3 now we want to jump to su 3 and discuss some very specific representations of SU 3 the important ones which we will discuss first of all there is the triplet the triplet is simply if u is a 3 by 3 matrix then it has to act on a thing with three components in the present context the three components represent the three colors of a quark so we're talking now for the moment about one quart now three quarks one quark and that quark can be yet red yellow red blue or green it can be red blue or green this is the amplitude that it's red this is the amplitude that it's blue this is the amplitude that it's green and what does the group do to them it mixes them up it mixes up red blue and green takes any particular state and sends it to some quantum superposition of states of mixed colors shall we say so if we took a red quark that would be like an up spin 0:01 and we rotated it by this su3 operation we would get something of a mixed character which would have amplitudes for being red blue or green some quantum superposition so that's that's the meaning of these operations but it's interesting and important to know what happens if you combine representations together let me go back a step let me go back ya know now we want to talk about various representations of SU 3 the representations of SU 2 are the spin States it's been 0 1/2 1 3 halves to 5 halves and so forth those are the representations of rotations we want to know something about the representations of SU 3 and I'm not going to tell you very much just a very little bit there are rather two and three important representations for our purposes one of them is just a three-dimensional representation and it can be thought of as the three components the three things that go here could be thought of as the three field operators for let's call them quark red quark blue quark green we can either think of these matrices as mixing up the states of a single quark or we can think of it as mixing up the field operators that create single quarks red blue and green now next this is all this representation by 3x3 matrices that acts on the quarks that representation is called the three in the same language you would use the same language you would call the half spin particle of SU 2 you would call it the two two four two states so the three is a representation that's identified with the quark field itself or with quarks now there are also anti quarks antiquarks their field operators are the complex conjugates of the quark operators so you can ask yourself if there is a matrix which acts on Q to give some new thing we could complex conjugate everything we could complex conjugate everything and we could say let's let's write it if let's call it u u is a three by three matrix if it acts on the column vector Q I'm now being a schematic you as a three by three matrix q is a column vector if it acts to give let's call it Q Prime the rotated Q then what is it that acts on the complex conjugate or the hermitian conjugate field operator that's the object which represents anti quarks well it's quite obvious it's going to be the complex conjugate the not the hermitian conjugate the set of elements if u is a bunch of elements then u star is nothing but the complex conjugate elements go to star let's call it star to be consistent so there is a second representation of su3 which is the set of matrices which are the complex conjugate represent which are the complex conjugate matrices for every matrix u there is a complex conjugate and that defines a second representation it's called the complex conjugate representation it is the transformation properties of anti quarks and that representation is called the three bar this is very abstract but what is it in the case of electrons what does this correspond to in case of electrons it is simply the fact that if the electron wave function gets multiplied by e to the I theta then the positron wave function gets multiplied by e to the minus I theta they transform with complex conjugate group elements so you would say that the electron the positron are complex conjugate representations of the u1 group that's the language the quark and the antiquark are complex conjugate representations of the group su 3 it would be the complex conjugates of the al-ahly splits in matrices for the case su - yeah maro su - is a special degenerate case where the su with a complex conjugate is equivalent to the original matrices by another matrix and it's a special case it's called a real group but this is this is not important for us now the 3 and the 3 bar representation are quite different I mean they look very similar to each other but they are distinct they should not be thought of as the same representation of SU 3 they're quite different and in fact the generators the generators of the three bar representation are the negatives of the generators of the three representation that's not too hard to prove that they are the that means they carry the opposite color if a quark carries a red color then the anti quark carries a - red color has the opposite value for the generators all right so these are the quark and the antiquark now what happens if you take a quark and a quark to quarks or a quark and antiquark just like you could take two electrons and make a spin one which was a triplet not got to do with this three but thus the triplet of SU - just like you can make a spin one or spin zero you can do some similar things with with three and three bar you take two quarks and put them together you can call that three times three bar sorry three times three two quarks are three this is just a notation three times three means two quarks sometimes this is represented by a circle to indicate that you're not actually multiplying numbers you're simply building a space which is now a space of two quarks it has nine independent states red red red blue red green and so forth all nine states and if you work out what representations of SU three appear when you multiply three by three the answer is first of all there is an antiquark representation that's going to prove to be interesting that if you take two quarks you make something whose color is the same as an anti quark that's a little bit odd don't you think but it's true it's true anyway and one more representation called the six dimensional representation of SU three we're never going to need the six dimensional representation but there is a six dimensional representation of SU three I can tell you what these things correspond to if you think of the quarks as just indices color indices then you can combine the two coils let's think about the yeah you can bind the two quarks States symmetrically or anti-symmetric Li you can combine them symmetrically or anti-symmetric Li anti symmetrically gives you the 3 bar symmetrically gives you the 6 but this is not terribly important point is if you make two quarks somehow in the laboratory the transformation properties of the to quark system under this su 3 there will be one term or one way of combining them together three independent states of that system which will be like an anti quark and the other one will be this mysterious six which we don't give a damn about we'll see why we go on now what happens if we take a quark and an antiquark if we take any a quark and an antiquark what we get is different then we get when we get a quark and a quark we get again six and three is nine States but it's not the same nine possibilities first of all we make an object which is a warning what is a warning that's a thing which is completely invariant under the color operations roughly speaking it's a red plus Anantha a red anti red quantum superposition of red anti red blue anti blue and green anti green that's one thing and that's a single representation and it simply does not transform under the su 3 group it's called a singlet an su 3 singlet and one way of saying it is it's the analog of combining together two electrons to make a spin zero state it's like a spin zero state it does not transform under the rotations doesn't transform at all this is one state so how could it transform is nothing for the transform into so when you take a quark and an antiquark there are nine states one of which one linear combination ready anti red plus blue anti blue plus green anti green that just forms a singlet and then there are eight others and those eight others are called the eight dimensional representation of SU 3 that eight incidentally think about that eight for a moment we've seen eight and another place it was the number of generators just like the number of angular momenta when you rotate space the angular momenta rotate into each other when you rotate in color space the eight generators get mixed up with each other the way the eight generators transform is called the eight dimensional representation of SU three this is sometimes called the adjoint representation the words don't matter it's just a representation that transforms the same way as the generators and that's what you get when you multiply 3 cross 3 bar finally one other operation what do you get if you take 3 quarks 3 cross 3 cross 3 you take three quarks there are 27 states well you can do it in pieces you can say if I combine together these two quarks what I will get is either a one oh sorry is either a three bar which is an anti quark or a 6i so two of these quarks together will necessarily either transform as a anti quark or as a six a six is a none as a peculiar representation which I said we won't care about very much now what happens if you take this product and now cross it again with three in other words we see what we get when we combine two quarks here's what we get and then we combine it again with another quark we will either get what appears if we take a quark and an antiquark but we know what that is that's one plus eight the plus here just means or you either get a one the singlet or an eight just like saying you either get spin one or spin zero that's what this plus means or when you multiply 6 times 3 I think you get an 8 plus a 10 but this again is not very interesting to us we don't care very much about that the interesting thing and the most interesting thing is when we multiply 3 quarks together we get the possibility again of a singlet of a state which is completely colorless it has no transformation property under the color it's neutral it's just like taking a plus charge and a minus charge which cancel each other we get the singlet which has no su 3 transformation at all and it can be thought of as a combination which simply is invariant under the su 3 group under the su 3 of the mixing up the quarks what is it can we guess what it is like I'll tell you what it is it's just a red a green and a blue quark but you have to be careful it's a red green and blue quark anti symmetrized not important it's a red green and blue quark a red green and blue quark are the singing of the singlet then there are other combinations red red green and so forth they form the remaining states here but the most interesting one for our purposes will be this singlet here so we see that when we take a quark and antiquark just with regard to the colors you can combine the colors in a particular way to get a singlet plus other stuff and when you take three quarks you can combine them together to get a singlet plus other junk is the probability of these other things so long that we don't consider it the question is what the energy of them is and I will tell you the energy of them is infinite but we will come to that right the energy of them is infinite and so they don't appear in the spectrum now how can the energy of them be infinite and that's that was a great puzzle which we now understand that I'm going to talk about a little bit right right but now ah there is one other kind of particle in nature in sorry there's many other kinds of particles in nature but there's one of the particle in quantum chromodynamics we're talking about quantum chromodynamics QCD the theory of quarks and gluons but of course we've left out is the gluon the gluon is an object which I think I've told you before behaves as if it were a quark and an antiquark with respect to the symmetry it doesn't behave like a quark and an anti quark with respect to what happens if you hit it or anything just with respect to the color symmetry its colour properties are the same as a quark and an antiquark but only the eight it transforms as the eight dimensional representation it transforms the same way as the generators of the group so again it is this octet which is separate from the singlet here it's the remaining states here and that's what a glue on is a glue on is a thing with two indices like a quark and an antiquark except a quark and an antiquark arranged into this octet of SU 3 so it's not neutral it itself is not neutral so let's let's quark postulates if you like of quantum come off QCD it's got an su 3 symmetry which i won't write down the quark transforms as a 3 sorry the anti quark transforms as a 3 bar and the gluon or the gluon field as an 8 because these equations don't really mean anything particles can't equal numbers particles or particles numbers are numbers this just means the quark field transforms as a triplet the anti quark as the anti triplet complex conjugate and the gluon is the eight which is a piece of a quark antiquark this remaining possible gluon doesn't exist in nature do we know why not completely but the important point about it is it doesn't mix up with the other components and so throwing it away is a consistent a mathematically consistent thing to do because it doesn't get mixed up with the other components when you transform the SU 3 group that's that's group Theory has applied the quantum chromodynamics applied to quarks and gluons are another postulate of quantum chromodynamics and it's not really a postulate it's a dynamical put of the theory but let's take it as a postulate and then explore what it means in a moment and that postulate is all of the particles of nature the real particles in nature not quarks and gluons which are never seen singly but are always seen in composites that the particles which can be separated off and examined individually can mail of particles of finite mass the real things that occur in the laboratory are all particles real particles I don't know there's nothing not real about a quark so I hate the quantum the real particles the unconfined particles the free particles the liberated particles or whatever you wish to call them free particles are always transforming under the one in other words they are singlets all the real particles in nature are singlets there is only two independent ways well I'll tell you what the ways of creating singlets are this mathematical fact now there are only two fundamental ways of making singlets out of quarks one of them is to take a quark and antiquark all right that creates a singlet the other is to take three quarks now you can do other things you can take a quark and an antiquark and juxtapose it with three quarks but the elements the basic or you can take three quarks and another three quarks three quarks in a singlet and three other quarks in a singlet that will also be in a singlet but all of the objects that you can make up have the quantum numbers of combinations of things made out of three quarks or a quark and an antiquark what about what about the gluons themselves although the first of all before we do it let's give these things names what is an object made out of three quarks or baryon a baryon right baryon they always have F spin and have spin when I say half spin that can be one half three has five half seven halves why because they're made out of three half spin particles what about the quark anti quarks this is these are the ones of course in the 3 and 3 cross 3 cross 3 there's the triplet sorry the singlet I'll call baryons the singlet and 3 cross 3 bar or they called quark and antiquark des armes now we left out one thing can we make things out of the gluons can we make things out of the gluons yeah there is a thing that you can make just out of gluons in fact if you take an 8 cross an 8 I told you roughly what happens if you combine quarks together you could ask what happens if you combine gluons together or gluons with quarks I'm not going to go through the whole list of multiplication tables that you get when you combine all sorts of things but they're certainly at least one more interesting thing and that's an 8 across an 8 this is simply two gluons to gluons no it's not obvious you get 63 states altogether 8 cross 8 one of them is a singlet well let's say 63 64 states that's a 63 yeah there are 64 states and a cross aid and the rest of them form 63 things which are a combination of eighths and other things eight appears a number of times I forget which I don't remember which representations appear but there are 63 other states none of them are single it's just one way of combining together gluons to form singlets so you might ask is there a particle which is composed of two gluons just two gluons bound together into an su3 singlet and the answer is yes the all glueballs they're not made of quarks they don't have any core content in them they're fundamentally just a pair of gluons you can also make three gluons incidentally it's possible also in it's also possible to get the more complicated things but there are Global's so the spectrum of hydrants the spectrum of strongly interacting particles which come from quarks and gluons consist of baryons mesons and glueballs all of them are color singlets color singlets is the watchword now another fact another fact you can check this yourself easily I think we've actually checked it before remember now all we can make is quark antiquark quark quark quark or glue on glue on and combinations of those what kind of electric charges do we get if we combine three quarks well I think we can get start of spin I think we can get charge to let's say we can get the zero one and two we can get to by taking two three up quarks three down quarks sorry with three down quark of three down quarks is minus one so we down quarks is minus one and you're minus one zero one and two but the important point is you cannot make a fractional charge with three quarks there is no way to take three quarks and make a fractional charge what about a quark and an antiquark well again you can take an up an anti up or 2/3 and minus 1/3 now 2/3 and 1/3 you know where you know top story is and you can again make integer charges it's the same as the charges down I did tell you because the way that I group them oh yeah it's same as a down all of the quirks are either up like or down like and they come in three families the up two down the charm is strange and the top and the bottom but they're three replicas of the same thing so all you can make is integer charge particle and so from the point of view of color the fact that you only have integer charged particles nature is the same statement as all the particles combined always into su3 singlets the color singlet single is like like a carrot tangle electrons is that any significance here yeah other than the way you transformer murder well that is the way you make such states is to entangle them very definitely they are certainly entangled does it have any I'm not sure well you you can't separate them because they get stuck together but but you know you can separate them as far as you like and just cost you a lot of energy so in principle yes they're entangled entangled objects not easy to manipulate you wouldn't want to try to do an experiment that type women not the details of how you deaf or what what the operations are all right left hand sides got three plus three yeah which is the group 3 group representation 3 okay so let's let me let me tell you more mathematically what is going on here yeah okay let's take su 2 first all right and take two spin a half's we can think of the two spin a half's in terms of the field operators which create their particles that's the easiest way to think about it for the moment alright so let's just call the field operator sigh and there are two components to it the sigh which creates a up spin and the sigh which creates a down spin let's call them sigh sigh I where I can take on two values either up or down now supposing we want to create two particles if sigh creates a particle or side dagger doesn't matter if sites a particle how do we create two pause so I creates a particle we can think of the two components we can think of it as side one and side two açaí up inside down and the su 2 operations act on this now that's one particle how do we make two particles well we act twice with sorry but we can act with sigh up twice that creates two up particles we can act with side down twice that creates two down particles or we can do it the one of each so let's write sigh I sigh J in fact these two operators don't really have to correspond to the same particle let's even we could even think of one of them as being the electron the other being the proton and a hydrogen atom for example let's see let's even give them different names ah sine Phi they might be the same they might not be but let's let's allow them to be different for the moment okay so this now is an object with two indices it's an object with two indices it's kind of like a tensor if we were thinking about vectors in three-dimensional space tensors are objects with two vector indices this object is like a tensor in that it has repeated indices now su2 can operate on it because su2 operates on psy and an ax on Phi this is a collection of four objects up up up down down up and down down so we could lay out those for psy upside up so I Phi up side up Phi down and so forth and so on we could lay them out in an array and then ask what happens when you act with su 2 on this composite object here well they get mixed up with each other they get mixed up with each other for the simple reason that the size get mixed up with each other and the Phi's get mixed up with each other and so certainly the side times Phi's are going to get mixed up with each other and it's going to be a four by four matrix which represents the same action as just going through size and Phi's separately and rotating them by the group action okay now you look at these at the way these things mix up with each other you look at the way they mix up with each other and you come to a discovery you discover that the combination let's see what it is sigh up fide down - sigh owned by up doesn't mix with anything else under this operation it just goes into itself it just goes into itself this is an easy thing to check you just see how su 2 acts on side up and mixes it with side down you see how it acts on five down and you check what happens to this particular combination and the answer is nothing so this is a singlet this is one the other three possibilities where are they they are side up side down plus side down fire up this is a thought I'd have not put in square roots of two but you could put in the square roots of two if you wanted this is orthogonal to this and then there's sy up Phi up and side down Phi down what happens when you do the SU 3 su 2 rotations on these states they mix into each other in particular ways they will mix into each other this singlet does not get mixed with these three states at all and these three states mix up among themselves as if they were a spin one object they get mixed among each other and this is then called the three spin one has three states these three mix up among themselves this doesn't mix up with anything so when you combine two half spin States that's are when you combine two half spin States you make a 1 or you make a 3 depending on which combination you take you make a 1 or you make a 3 you make spin 1 or sorry you make spin 0 or spin 1 that's the sense in which these equations are being used here when you combine together a quark and an antiquark there are nine different ways to do it one of the linear combinations behaves in this way here and it's a singlet and the others transform into each other eight objects eight additional objects will transform into each other so that's that's the meaning of these equations here how many of each kind of combination you get this multiplication paper it should be two plus seven state again okay so you have to know some good products three for the b2 plus seven or you have to know what the representations of SU three are we did a bit of work to find out what the representations of SU two were and they were all the spin half and spin in integer spin particles we could spend two more weeks studying su three and find out what the representations of SU three are what are what matrix representations are there not there are not matrix representations of every dimensionality there are matrix representations for su 2 that happen to be for every dimensionality a matrix representation of su 2 for su 3 not so there's the singlet there's there's the triplet there's the anti triplet there is the octet the eight there's a ten is a fifteen twenty the whole bunch of them but not every number appears in the possible matrices which have the same multiplication table as su three so that's it that's a bit of work to prove we didn't want to go there I just went there by saying all right here's what you get and but the but the real interesting point is the singlets that appear the ways that you can combine quirks together to get combinations which don't rotate or which are the analog of electrically neutral why do I say electrically neutral because electrically neutral are exactly the objects which don't transform under that u1 the things which the phase cancels out of remember go back again to electrodynamics the things which are electrically neutral are the combinations where all the phases cancel out if an electron has an e to the I theta and a positron and e to the minus I theta then an electron and a positron the combination of them are neutral don't transform at all two electrons that gets an e to the 2i theta so it's not called neutral or it's not called or transforms under the group so singlets are important but you should just think of them as neutral objects with respect to the two the transformation properties and I mean just possibilities you could just show with that singlet that that transformation just one example there is that the oh we can go through it yeah it is a little too much but um one simple way to see it I don't know if this will help you at all if we write just write this object here Epsilon you know what epsilon IJ is it's the two-by-two matrix which is the anti symmetric matrix yeah and contract the indices this way it's that that's equal to this that's equal to this and it's easy to prove that this is a oh okay this yeah I'll just spend a moment at it in every dimensionality for matrix matrix dimensionality there is always an epsilon symbol which has a certain number of indices for two by two matrices the epsilon symbol and the epsilon symbol is the anti-symmetric the fully anti-symmetric matrix compose that of zeros and ones okay all right epsilon IJ is the two-by-two matrix what about for what about for the 3x3 matrices in a three dimensional and it's epsilon ijk requires three indices alright so I'll tell you right now if you take an su 3 epsilon ijk and let's call it what did we call a quark fields q qi q j qk then you make the three quark singlet which has no color I J and K have to all be different from each other otherwise this is zero so that's why I said a red a green and a blue make a singlet okay well uh we don't wanna make Alice suffer too much tonight for what it's worth I just happened to bump into this in this other book that eighty cross a equals twenty-seven so let's write it down okay eight plus eight all right so that means when you when you take two eighths and put them together in other words two gluons you first of all make a singlet what else well how does this again eight plus eight plus eight okay I'll tell you what this means in a minute plus ten plus ten okay all right so let's see is that right the 10 + 10 is 20 and eight native this should make a well 8 times 8 is 64 oh it's no good plus 27 it's not even filled yeah well what do you serve it at the end you say there's a plus of 27 does that make 63 I can't tell 64 is that 64 okay good all right all right so for your information this is a fact I know I know the representations of SU 3 up to 27 if that I think there's a 35 but I don't remember but another 10 we don't know we're not up to 60 no okay all right so what this means is first of all there is one and only one combination quantum quantum superposition of states of two gluons which makes a singlet one and only one and that's the blue ball then it turns out there are two orthogonal ways of making octets - that's sixteen states all together but they form two groups and the first group mixes into themselves without mixing into the second the second group mixes into itself without mixing into the first and both of them form octet different combinations different patterns of symmetry of the wave function of the of the state of the - core of the two gluon system then there's something called a Ken and again there are two distinct ways of combining the gluons together to make tens but who cares about tens because they don't appear in nature anyway and then there's a twenty seven dimensional representation how about 3 cross 3 cross 3 ok so there's a singlet I don't know I don't know we had them written down but I don't remember it fair to say that that what happens if you change the sort of the basis of the thing to thee let's go back to this spin - Caze - thee for the singlet and the triplet and the - up up up down down very don't worry here yeah alright so you have that and you those four things as as a basis than a matrix which describes the operation factors into a esmad has one zero zero sir and then I see yes and a lower part that is that move that moves that exchange moves two three among themselves so the make you can describe it in terms you have a lot of matrix looks if you just work in terms of the basis vectors up up up down up up up up down down up and down down then there's a certain set of four the 4x4 matrices which represent the group alright but if you work in terms of these linear combinations this one this one this one and this one then the matrices are exactly the property that you say there's a one over here and then one 3x3 matrices down yet the same thing for this other case here is the one yes it's that's right that's right the matrices the big ape the big 64 by 64 dimensional matrices break up into a bunch of blocks right let's block diagonal in a particular basis right in the basis that's right that's right that's in fact what what you're saying is that there basis driving weeks that in which this complicated matrix that simplifies into but black bag that's right so there are sixty four states so the SU three operation has to be represented on the two on the two gluon system as 64 by 64 dimensional matrices right but in some basis those 64 then by 64 dimensional matrices are a 1 and 8 by 8 matrix over here another 8 by 8 matrix over here then a 10 by 10 I don't have enough room another 10 by 10 matrix and then a 27 matrix down there in a particular basis that's exactly right does it mix the other ones that's right each that's right in the right bases they don't get mix this one doesn't get mixed with this one doesn't get mixed with this one they only get mixed together as an 8 dimensional representation of the total basis that basis you mean for this for a piece of it like this well well when you multiply I'm not I'm not sure what the what the right right buzzword I think you're looking for a particular term which escaping me right now like I don't think such a term exists but I know what you mean is don't exist in nature yeah there's a well maybe you shouldn't say you couldn't care about them but they won't appear as real particles in the laboratory we're going to talk about what what creates this peculiar situation that you can't have a freak work by itself but we'll come to that so let's talk about it a little bit incidentally the quantum chromodynamics idea goes back a long ways it goes back a some time I know who it's doing embu to your children number and number had the idea quite a 1962 quite maybe good ten years before quantum chromodynamics became the standard theory of of quarks and gluons in fact the whole idea of color goes back to him he didn't call it color I don't know what he called it and the idea of gluons also in the same idea and his idea was fairly simple he understood that if you made color singlets then you would not get fractionally charged particles that was his motivation what do you have to add into quantum chromo into the theory of quarks in order to have some dynamics which will forbid particles of fractional charge so he realized that if the quarks carried this kind of color quantum number and if the color always had to be a singlet then you would never get you will never get fractionally charged particles he had an answer to the question of why particles should only occur in singlets and it went something like this he said right now we now we need another postulate and the other postulate is the connection or the interaction between quarks and gluons the postulate is that gluons play the same role in quantum chromodynamics that photons do in electrodynamics but photons couple to the electric charge the source of the photon field is electric charge the gluons there are eight of them what are the sources of the gluons what is the source of a gluon unless there must be eight quantities eight conserved quantities which you can think of as being similar to charge and each one of them is a source of that particular gluon of that particular species of glue on the eighth the eighth gluons correspond to fields which are similar in some sense to to the photons or to the electromagnetic field the electromagnetic field is sourced by electric charge what is the source of the gluon field well what eight quantities do we have available the eight generators of SU three which are analogous to angular momentum for su 2 angular momentum is a conserved quantity it's additive if you have two objects you add their angular momentum ah and color really means when you speak of the color of an object you're speaking about how it transforms under the group but the actual thing which radiates the thing which radiates the gluon field is the color itself or the color itself or the generators of the color Group eight of them and each one can emit the appropriate kind of glue on the first meaning of this is that if you take a colored object such as a quark it's going to be surrounded by a glow on field so much like electrodynamics are the quantum chromodynamics replaces the electromagnetic field with a quantum chromodynamics ield but the only thing new about it is that there are eight such fields and the eight such fields transform the same way under that same eight that's associated with the eight gluons okay so that's the first thing now the second thing is to know something about about the dynamics of electrical electromagnetism let's begin with a charge and an anti charge a charge in an anti charge attract each other the meaning of that is that if you bring them close together the energy goes down right so a charge and an anti charge have less energy if they're close together than they are then if they far apart that's why they bind that's why they bind because the energy of a neutral system is less than the energy of the two charged brother the energy of a neutral system is lower than the energy of a charged system as a rule for example the energy of two plus charges is much more than the energy of a plus charge and a minus charge how do I know that because the two plus charges repel each other you'd have to do work to push the two plus charges together you get work out of letting a plus charge fall towards a minus charge so here's an example where a system with a net charge has more energy then a system in which the charges cancel out another way to think about it is if you have two plus charges the lines of force have to go somewheres and so there's knit field out beyond the object that net field has field energy field energy is always positive it's proportional to the square of the electric field and so this field energy stored in the fact that there are two plus charges here if there was a plus charge and a minus charge there might be field in between them and yeah and let's suppose you brought them very close together there might be a little dipole field in there but there would be no field on the outside there would be no large field energy on the outside and so the field energy or the field energy stored in a neutral system is less than the field energy stored in a in a charged system the larger the charge of the electron the more profitable it is let's say to have a neutral system energetically or the less profitable or the more expensive it is in energy to have a net charge system just the bigger the electric charge if we have if the electric charge of the electron was a thousand times bigger than it is the self energy of the electron due to the electromagnetic field would be much bigger than it is in fact the repulsive force between two plus charges would be much much stronger than it is in the amount of energy it would take to assemble two plus charges would be very much bigger than the energy needed for a plus charge and a minus charge the huge charge of the electron would then make it very very prohibitive to have net charge and very very efficient low energy to have all charges cancel out so if the coupling constant of electrodynamics were large very large we would be very used to the idea that the low energy particles the low mass masses energy of course the low mass particles in hhor-- the low mass objects in nature would be electrically neutral if the charge was big enough we might never have discovered individual electrically charged particles they would be confined it would just be to prohibitive an energy to pull them out of an atom for example so what Nambu said is look if I have this quantum chromodynamics and if color the color generator or the color degree of freedom is the source of the gluon field then if we make the charge the numerical coupling constant the the analog of electric charge if we make it large enough then it will be simply impossible to pull apart these quarks into non singlet States a singlet state as the analog of neutral pulling them apart you would be left over with net amounts of color which would cost you lots of energy so he understood that when systems were in the color singlet state they were fully attractive and pulled themselves together when they weren't there was always an element of repulsion that that meant that cost an enormous amount of energy to pull them together so that was namboze explanation he said quantum chromodynamics has a natural way of making sure that the real particles in nature are integer charged and that quarks always come in either the quark antiquark combination or the three core combination r and it was essentially correct it turned out to be correct but the dynamics is a little more interesting than that so let's talk about the dynamics and why the dynamics is more interesting any questions up till now okay why is the dynamics more interesting than that and the reason the dynamics is more interesting than that is because the analog of the photon the gluon is charged carries color it the glue on itself is not neutral with respect to the remember the neutral piece was this ninth member that we threw away the eight gluons themselves are not neutral they transform under the eight it's as if they had a color charge that is why gluons don't appear in nature as objects the color they they themselves are colored and because they're colored they have a big color field around them and they're prohibitive in energy but this is interesting what it says is that the gluon itself is charged in a way that the photon is not the fact that photons are not charged means they don't interact with each other they're not sources a photon is not a source of another Photon electrodynamics or at least classical electrodynamics is a completely linear theory meaning to say that the electromagnetic waves just pass right through each other that's because the electromagnetic wave itself is not charged and doesn't doesn't influence other electromagnetic waves so electrodynamics the photon and the electromagnetic wave are very simple and linear but in quantum chromodynamics it's much more complicated the analog of the photon itself is charged that means that the analog of an electromagnetic wave intersecting another electromagnetic wave will be a serious interacting phenomenon in fact even just a single wave of a glue on one part of it will interact with the other one part of it may have some color the other part of it may have some color those colors will interact with each other and the dynamics of gluons is far and then the not just gluons but the field analogous to the electromagnetic of the Maxwell field is far far more complicated and dynamically interesting but also much more difficult to understand turned out that the pretty much the important thing that happens which is new can be summarized fairly easily and I'll summarize it for you on the blackboard in terms of pictures the equations that go with this there are equations but but the pictures I think are better than the equations there's a quark an area and there is it's quantum chromodynamics ield around it now in electrodynamics the different pieces of the photon field or the electromagnetic field do not interact with each other and so there's no sense in which one field line repels or attracts other field lines they're just they're just completely independent they don't interact with each other in quantum chromodynamics the gluon field itself is not neutral and therefore the gluon field interacts with other pieces of gluon field and the result is very simple it turns out to be fairly simple the lines of flux basically attract each other they attract each other and they attract each other in a way which makes a rule about lines of fluxes they don't end except on another charge the lines of flux coming out of a quark or coming out of a color charge attract each other and in the process of attracting each other energetically they want to form tubes they want to form tubes of flux with the flux running right down the tube it is more or less you can either imagine that the lines of flux are attracting each other or that empty space is repelling the lines of flux and pushing them and squeezing them into a tube that's what happens that's pretty much the main effect of the nonlinear interactions between the gluon field that it makes it energetically favorable to pick those lines of flux and and bundle them into tubes let's suppose that's true now now supposing we have a quark and an antiquark we have a quark and the antiquark lines of flux come out of the quark and go into the anti quark if this were electrodynamics those lines of flux would separate and form the nice dipole pattern that we're all familiar with anybody who's ever looked at a magnet then right all right the field energy associated with this big dipole configuration is relatively small because the field gets weak far away in this configuration there's a uniform field no matter how far you take these particles from each other there is a uniform field in between just exactly as if you took an electro electric or magnetic field and pushed it through a tube there would be a uniform field the lines of flux don't end and because there's a uniform field there's a uniform energy per unit length what's the result of that the result of that is if you try to separate a quark from an anti quark the energy just grows linearly with the distance between them as you pull this out it's like the gluon field was really sticky really Turkish taffy except accussed happy would get thinner and thinner as you stretch that out huh so it's not quite like chewing gum of Turkish taffy it's like a kind of chewing gum that as you separated it always created more chewing gum in between so that it never got thinner and thinner there's always a fixed number of lines of flux coming out of every charged object and they don't disappear so this tube it's called a flux tube or flux ID and the flux ID has a uniform charge per unit distance any charged object which means any object with nonzero color any none singlet any object which is not a singlet here it is it puts out lines of flux those lines of flux bundle themselves and unless they are unless they in an object of opposite color that object is just going to have an infinite charge because those lines of flux have to go someplace they'll end in infinity the cost and energy will be infinite that's the basic dynamics of quantum chromodynamics linear flux tubes like this which are the ingredients which keep quarks from flying around freely if you hit a quark real hard inside amazon this is a picture of a sort of extreme picture of amazon a quark and an antiquark you hit that quark hard it goes flying out but it simply can't get away the field energy just increases and increases and increases linearly with distance until you've run out of kinetic energy and then it gets pulled back or something else happens what's the other thing that can happen right yeah right the other thing that can happen is that the string can break if this is a quark and this is an anti quark then spontaneously out of a vacuum particle pair production can create an anti quark over here and a quark over here but you still haven't created a quark you've created two massan's so you give one quark in amazon a give good shot you hit it really hard it tries to escape and either it can't escape because it runs out of energy and gets pulled back but actually more likely what will happen is a quark and antiquark will be created in between and you'll just create two Amazons if this quark over here is still going too fast it may still try to escape at me to escape this one over here what will happen and so what happens when you hit a quark really hard is that inside Amazon is that a whole bunch of mess ons go flying off forming what is called a jet oh yeah oh yes absolutely you can get anything as long as it's neutral right this is the picture of a mess on the other ingredient that is in quantum chromodynamics is that three quarks can exist this doesn't explain three quarks the three quarks are explained by a mathematical fact that the flux lines can come together in threes quark quark quark and [Music] but that's not too surprising because three quarks again make a neutral object three quarks again make a neutral object and that means that if you have three quarks there'll be no net lines coming out of it so somehow the lines of force that come out of each one of them have to be able to cancel each other that's quantum chromodynamics it's based on the SU three group and it's called a gauge theory a gauge theory is simply any theory where the conserved quantities are coupled to a Maxwell like field or coupled to a field similar to photons it has a deeper meaning than that but for our purposes anytime there is a conserved quantity analogous to electric charge which functions as the source of a photon like field photon like field blue on like field that's called a gauge theory the gauge theories are always based on symmetries why are they based on symmetries because they're based on the idea of conserved charge and conserved charge means a symmetry of some kind incidentally why what's the connection in between conservation and lines of flux conservation and lines of flux are intimately connected if you have a rule for example in electrodynamics that every charged particle has to be the source of lines of flux and that lines of flux can only end on charged particles then charge has to be conserved there's no way none of these lines of force go off to infinity well you could say I could make this charge suddenly disappear if I allow the lines of flux to simultaneously all disappear but that would violate the speed of light constraint you would be able to send the message arbitrarily far distance if by simply removing that charge you suddenly changed the field at infinity the fact that you cannot send a signal faster than the speed of light tells you that you can't remove this charge why because it's sort of anchored by the flux lines at infinity and if you could remove it either one of two things either you would send the sudden signal far away that the charge wasn't there by removing the field or you would be left over with these lines of flux that don't end on charges so if lines of flux have to end on charges the next tantamount to saying that the sources of the field are conserved which in another language says there must be a symmetry so some symmetries are connected with photon like fields which give rise to lines of flux those theories are called gauge theories and you can't even imagine without breaking the rules of a theory what it would mean for the charge not to be conserved because the rules of a theory a patch the charge particles to the to the lines of for our force okay that's that's quantum chromo then Amyx now we've actually gone either depending on how you count either 2/3 or 3/4 of the way through the standard model we've gone through 9 generators if you like the 8 generators of SU 3 and what's the other one the electric charge per pound which is not the ninth generate energy 3 the electric charge those are nine conserved quantities they coupled to the 9 gauge fields the eight gluons and the photon the standard model has three more generators which we will talk about so that's we've got 9 out of 12 that's 3/4 the other way of camping is to say we studied su 3 that's quantum chromodynamics we've studied you one that's electrodynamics the standard model is su 3 course su 2 cross u 1 we're missing the su 2 the su 2 not that's not angular momentum there's another ingredient in the standard model called su 2 we'll pick up it will start to take up su 2 next time we've covered a lot of ground and I think in about two more lectures we can probably finish off the basic structure of the standard model su 3 cross su 2 cross u want u 1 put it together into a package and and I think the other element that we need of course is the Higgs field the Higgs field plays an important role there but um I would say within about three lectures we will have the working parts of the standard model and then we can start to explore what the puzzles of it are why why people think we have to go beyond the standard model and why people think there are going to be interesting new things discovered at certain I think that's what we want to get and okay good for more please visit us at stanford.edu

Channel: Stanford

Views: 59,355

Rating: 4.9010601 out of 5

Keywords: Science, physics, quantum mechanics, rope theory, rotation, spin, vector space, spin up, spin down, electrons, matrix, matrices, mutually orthogonal states, parameters, degrees of freedom, conjugate, determinant, unitary operator, angular momentum, gluons

Id: n88Qe-aAAYE

Length: 101min 32sec (6092 seconds)

Published: Thu May 13 2010