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# New Revolutions in Particle Physics: Standard Model | Lecture 3

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Steiger University right let's
that it doesn't change the leader of the that's the property of a rotation changes nobody's Lane incidentally would take the components of the vector to be real numbers from armored real numbers and matrices are also composed of real elements so was the condition the condition is the condition for this to be a rotation is that every possible V prime be
prime being the result of rotation that via
i.e. VERY this means the sum EX square Was BY squared was busy square but the link the square of the length of should be the same as the square of the of prime but what that would see what that says report out detail this is on our party Jerry but what put VJT here as a matter was something it doesn't matter what we call a J & R I J V J is equal to be prime direct all review to be I want the idea was that this is a front let's write the same equation again R I K E V Katie equals prime I've seen saying you never know this dummy index the summation or complete with this 1 here but this 1 is the same as the 1 now I wanted the ozone a form prime IV prime relies on multiply this by this In some will I get there will get be primary primary equals I J. J. R. I K b these with every repeated next some for way just a rival differently RI Jr K V J K this should be the same as what this should be the same as VIE I that here physical there's already here 1 of the condition that this is able to this every possible right right 2 steps argues that on a J E R I K D is equal to J. K another is if this were true RJR RIK some but why was the chronic Delta it would tell you that said J. equal to Katie and this would become a square so that's a condition for a rotation matrix they can be written in a different way to be written about transpose are 10 times transpose is equal toward the unit operator and this is the representation are planned transpose so our rotation matrix satisfies here right are transposed times are is equal to warn Missouri remember trends poses Transco's adjust the changing rows and columns ah transport was ours equaled 1 actually wheat immediately now know what universe are any rotation matrix years is just a trans replacing group as the property and that's his that's basically it's defining property arbitrary bird the defining property of the matrices rotation matrices if you construct for every rotation such a matrix and you multiply the matrices you will find the rugby um the matrix representation of these multiplication table of our our our rotation and of doing this is an example car alone there are 3 by 3 representation three-dimensional representations of the rotation can we think of the three-dimensional representations of the retention group has in any way being related to the quantum states of something well we studied spin describes a movie angular momentum of particle and so forth and you could rotate the of a particle you could take a particle and you can rotate it you compose magnetic field rotate the magnetic field and makes up the state's who makes up the states last week discovered that they were various angular momentum possibilities for there was perhaps warned 3 of the vote was been 0 0 have warned many states does a spin 1 particle have 3 what happens when you take the stand I was what does that mean that means that it's been 1 particle Justice pinot nothing but spinner it can be described by a column vector with 3 entries representing the input for any 1 of the 3 states of the state suppose you want a rogue state that well To wrote that that's been you must multiply it by some kind of rapacious some sort of operator that rotated the papers been state scrapped restates the rotation matrices of a rotation operators which act on the state vector the mix up the components must also a three-dimensional representation of the retention growth package just that right there is just that so it often 1 speaks about spin 1 particles being vector particles that means the 3 states I'll be thought of as being in one-to-one correspondence with the 3 components of a vector and mix up among themselves the same way as the components were quantum states it's been 1 particle but I want that I'm interested in this name of article members but I have particle I want to know how the state and anybody want arrest cigarettes grant a lot of material or try to keep it as simple as it possibly can act as
we encountered 2 different kinds of groups Our discrete groups and continuous groups growth our composed of 1 and he has only 2 elements redundant regroup and bits of discrete uh you is flip 1 must let the was the rotation Gould the multiplication might lead the I continuous fight can be a continuous variables the continuous and beat are elements of course in the rotation group there are also a continuous infinity of elements sold the kind transformations in the kind of symmetries a and think about physics break up into 2 kinds the discrete symmetries and the continuous cemeteries rotation being a continuous symmetry reflection of being in discrete symmetry these groups that are associated with continuous symmetry is really quarterly groups LORE They were 1st buried in mathematics by by mathematician friendly or German
mathematician I guess the 19th century right from which is mainly being continuous groups were now and then we might think of it the discrete group OK armed will yuck I said that first one particle which has 3 states be mathematics of rotating around states it was just the mathematics of three-dimensional representation of rotation but then you could have what about may have part of a first-order 0 particles what happens to a spin spin 0 when you rotated nothing results that so the year of these symmetry group as applying to spin 0 particle rotation really only has 1 element the unit element and nothing else and its representation just completely represented by nothing more than you who operate completely trivial system the spin 0 particle the spent a half the article also are already has 2 states that given a state arms bitter have particle was a need to be given a state it means 1 0 0 bats to be identified with up 0 1 is to be identified with and general quantum state has a complex number here have scored out for 1 an alpha to so complex numbers only specify about them we specified nothing more than that the sound Our the squared and magnitude of them powerful 1 star 4 1 suffered two-star offered to should be equal 1 that's all Orville that it's free To our through to choose for if we chose a powerful 1 Eagle One and offer to equal to 0 then we're talking about a Spain's pointing in Op direction now top of classes relative to some definition of an axis of let's pick our axis vertical z-axis then this state represents an spin this state represents a balanced and in general this combination here represents a spin state in some direction it's been oriented and some direction which is determined by powerful enough to know that the bill the algebra figuring that out but we can't say What if we take a given the spend state How 1 now and we we or do we just rotate coordinates although you physically rotate or was instead we physically rotate spin the another taxes orders whatever the whatever the configuration was we rotated where we get well we have operate on to get something forewarned prize for 2 prime where we operate on the states to get new states we operate with operators or with matrices again matrices in the matrix representation so this gotta element drugs Co you you warned 1 you to you
dagger plans you as you go toward but an insect all
really guarantee there is an 8 cool by 2 representation of rotations in space complex matrices and unitary This is a condition for a unitary matrix many components the unitary matrix feel about malaria every it's a complex matrix 8 mu components right 8 Rio components for complex compartments but not quite now there's a there's a bunch of relationships he Mahoney relationships is this factor for Pfizer's Florida because this is a matrix equation and there's a comma and as a separate equation for each component of 4 components this is former real equations 0 for really for real equations and so there are 4 0 really equations among 8 unknowns there are 4 independent parameters on unitary toe but to make it now that's one-to-many because we've already found that rotations humbly corrupt parameters characterize rotation 3 be at very unit vector which is to any angle groceries will want to many components yet 1 too many degrees of freedom from actually described the rotation but it was an obvious candidate for a for 1 more issue on obvious but there's a candidate for 1 more restrictions that we could put on these unitary matrices which will cut the number of parameters by 1 or more and that is to set a determinant of them equal toward why the determinant well on 1st of all you could see that that at the determinant of the product of 2 matrices is just a product of determined determined that they have determined the bees determinant of any kind so if we want to set determinant of every you equal to 1 then at least that would be consistent because we multiplied to to use together and now they knew we would still have the determinant of 1 so it is actually appearance is very easy to prove you compose services a one-liner basically that if you would tank only the matrices Reulet Kerry matrices which had determined 1 that they form a close grow setting determined equal the 1 not destroyed the group property simply be because 1st of all you will operate as determined 1 man since the product of determinants is a determinant of the product there still is a product structure of the group sold want additionally equation that 1 can impose without destroying the group property that's just right as a determinant of you determinant on by a pair of brackets like that is equal to 1 it's also true that determined the viewed daggers cyclical won their complex hundreds of each other so the group of you Terry matrices tool bite to unitary matrices with the property that every determinant is equal to 1 is 1st of all a 3 parameter group has the same number of parameters of the rotation group in fact it is the rotation group but the subtleties wandered into it is the rotation group as saying multiplication table as the rotation group it is the rotation group but it doesn't act that there is in acts on the state of us that may have system so it discovered something we discovered that there are 2 by 2 representations of the rotation group the journalist daggers the same not complex conjugate a bit of today but not 1 if let it it you would not want cup but is not but it yes way Dad plus mud being your bitch additional shift that determined that you times the determinant of daggers war that's as the determinant of dagger well not to be eaten I paid for my size yard now we have sex that we fixed it be determined is equal to 1 the group prior definition the group I of bite In a unitary matrices is called you end the a special case is 1 by 1 unitary matrices as you 1 of the group unitary matrices with determinant 1 is called the special unitary groups as in SC you in special unitary group that plays an enormous role in physics as you can special unit carried in by and matrices so this group here which acts on 2 components of us is S U 2 that's as you to let's go 1 step further before we turned the quirks 1 step further very happy with this where it sent a distress classes it spin matrices segments dog erupts shoe showed just plus a sigh it's or with young is a good question I will tell you play right now and were still there are those matrices best 1 joined they were actually does have Polly matrices I was so you remember what Powell images of workers I think about group elements which are very close to the identity that means rotations by a very small finals rotations by a very very small angles they operators of matrices should be very close to the unit operator so we should be able to write those the special case small rotation we should be able to write them as having unitary matrices which are close to 1 1 plus something small and indicates small but epsilon epsilon times some matrix of paint time some tricks are let's 1st asked What is that conditions that Liao how we wonder what's the condition that you is unit I'm so that's fine now let's multiply you times you dagger you Dyazide you gotta that's multiply by you again 1 plus Epsilon and that we permission conjugal 1 is just 1 taking epsilon quality will take epsilon Toby real from moment but there's just a small number is just a small number indicates more almost anything complex going to these ends here and this is going to be 1 plus that Mr.
Beagle Hawaii I only multiply this out our only keep things to linear order in epsilon would drop things the quadratic Epsilon and that's says that Epsilon In class and that there is equal to 0 we have 1 times 1 over here which cancels a sworn warning quadratic things we don't care about We have epsilon plus a dagger is equal to 0 and the ones canceled another words where it says is however and there to be minus intermission conjugate Beverly and this is quote entire her mission the thing which has this property called he died her mission we make something remission of something that intermission is multiplied by would so what's think about putting a nite in here let's go from the beginning in providing here where we do over here might decide permission conjugation always involves complex conjugation among other things if we did that there we would get in minus and Baghdad equal 0 because minor signed and then we will find it in equals and that there never was a free Inc. eyes the definition then again would be permission and that is our permission to by tool matrix right Mr. permission to buy 2 matrix How many emission toe by 2 matrices are there How do you make of mission to buy to make it all something else what about that could determine what's the conditions that could determinant is equal to 1 the Zoe trickier found there was a condition called I know New York was a right far away it's 1 plus Ali epsilon earned in the minors and there and in order epsilon squared sought forget order epsilon scored as the determinant of 1 plus a small matrix you know what they determined the 1 plus a small matrix but I'll vote it is forces former said he eats are due for a player who could do that and armed young that's where the let's look at let's do it 1 1 and that a small matrix plus and 1 more of a plus and 1 1 plus a 1 Our and to warn and warned plus an 2 2
believed in fact mathematically absolutely required physics is invariant under this tree operations another words kind of mixing up a quarks which of the colors of the quirks which is analogous to the mixing up Avi components of up-and-down also analogous to the mixing up of the directions 3 directions a space this is not would never be rotations is a special unit matrices Hyundai special unit developed counter member special unitary the cap member parameters a mini parameters to describe a special unitary 3 by 3 matrix he said that of a Rasht 18 right Mr. Clean Hands February why 18 you've got the 9 elements 9 complex elements that you have these equations you dagger you equals 1 9 such equations 1 for each entry in the matrix that's 18 minus 9 and I was the determinant equaled the 1 18 minus 9 members 1 is 8 independent 8 hits in 8 parameter grew just like the rotation group was a 3 parameter group sodas in April and our fairway rotations you want that thank you wondering what it has to do with the fact that a glue ones aren't you to say that same that our comfort by 499 you see through away to determine that doesn't make any sense at all but still anyway have the operative his sister of gap so special military 3 by 3 matrices are a particular representation of S U 3 pursuit 3 is the symmetry
group basic symmetry
group of quantum chromodynamics and just like quantum electrodynamics was in variant of this operation here and that told that's that only the only Nova Granzien only things with the same number of our societies and sigh daggers can enter the idea that the Lagrangian is invariant under S U 3 also has profound consequences for the structure for the symmetries conservation laws so the role is every possible terminal Grosjean of quantum chromodynamics work physics should be invariant under this group here will come back start talk about how you construct quantum chromodynamics a little bit I think finish for tonight and I'm certainly finished ahead intended to go a little bit further I might consider
you a little more but I think that was quite a lot I don't know how many of you followed how many of you didn't follow but I hope you at least get the flavor if not the color of work will go the further with this but they will get back some things to command the without much mathematics formal or
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