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

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Steiger University right let's

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talk about rotation those based language repressions of space are asymmetry and they can act on various things but 1 of the things it can act on sectors vectors might be a vector field of it might just be Somerset Victor and space velocity vector whatever Our rotations act on vectors and they rotate vectors from 1 toward other world look With can be combined with take object rotated about an axis by a certain angle and by another axis by angle and the result is an overall rotation about other acts by some of the anger or so an abstract terms are the rotations door formal group puts think about 4 minutes of the rotation operations have a group structured on Nagin taking a set both coordinates the incidentally heavily parametritis rotation rotations can be parameterized victories we won't worry about the fact that rotating by more than for the moment but more than 2 pi brings you back but worry about that from rotations can be 1st of all they have an axis of rotation about an axis let's take all rotation stuff always be right here the rotations right-handed weapons if we have a rotation which is left left-handed where can I don't think of it as a rotation right in the rotation by negative angle or we could think of it as a right here the rotation by the opposite taxes doesn't matter because right so rotation has an access but Coletti unit vector in and it has and Angola to rotate data I really parameters does it take to describe an angle you that yeah thanks 2 parameters describe unit vector of longitudinal attitude atmosphere if vectors of fire an arrow pointing units via so still angles and another angle for how much you rotate about 3 angles or 3 parameters that describe a rotation of 3 parameters family of objects now when you rotate an object if you rotate an object original object as you you have a rigid object and a rotating about an axis in Europe about some other taxes the result is all the ways equivalent to a rotation by yet some other axes and some other angle so the product of 2 rotations 1 done sequentially 1 then another it is always a mother of patients the answer is yes there is a product structure to rotations No. 1 there is an identity operation for rotation is just no rotation of although there is in the interests any rotation about any axis universe of it is just the rotation by the opposite angle about the same taxes all our nuclear and if you like think of the same rotation about the opposite axis for VersaCAD rotation about the axis is always exactly the opposite 1 which I'm does it so the their structure Rosario dabs of infrastructure well so we from it is and it also happens to be associated happens to be associated truth that's not quite as obvious OK so rotations are group and we failed to anything about rotation but it's not Beliard This is the fact that a rotation about the X-axis followed by a rotation of the Y axis is not the same as in the changing toward rotation go home and do that if you have done before retained about the X-axis y-axis and do it now voting this followed list have them head yes she wrote a novel sequential 1 after another yes group action is always corresponds to Sykora sequential alone but that is correct the smaller rotations certain cents closers who commuting but the group is definitely and noncommunicable none of Abelian group it's our 1st run a beer groped let's talk about its representation well it going incidentally is over we get to it tonight is a study of color color is a symmetry group and its represented the symmetry Colorado talking about the color of their you sure about the color of quarks red green and blue it's not up and down about 2 state system 3 states and the operations where sure would show Our interesting are the ones which mixed up the different course report Saul much of physics has that pandering Our armed groups of the trees that we need to understand the before we go OK on rotation rotations but How we construct a representation of matrix representation a rotation of the easiest way to do it Mr. take the components of a vector let me basic objects components of an ordinary vector V and as components V X Y and Z for VIP was I can equal 1 2 and 3 Urex warns so for now when you wrote think either that you cannot think of rotating taxis or a rotating physically the vector about some direction of course be components VX and rewind change and they change by being multiplied by a rotation matrix heights of every rotation about any angle about any accidents there is a rotation matrix rotation matrix depends on the angle of rotation and the axis that is a matrix How do you find the new resulting a set of components you multiply the matrix by Jane and you get a quality prime again some Bovard Jake is matrix and vector transformed sector that ladies our rotation matrices narrowed the another definitely anomalies on the properties of rotation matrices of believe we've gone through them but I will go through that again In theory Lane yards just see what that does not words must work out the properties rotation matrices this is another book of course grope open another representation This is a three-dimensional representation of the rotation growth is not only representation of rotation our because many of them but this is the three-dimensional 3 by 3 matrix representation location group it's easy to think about because you just visualize it in terms of the components of a 3 vector of a special vector right FOR rotates me and gives me prime an aura is truly a rotation of record

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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

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prime being the result of rotation that via

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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

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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

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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

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to want you to every year rotation can be represented as a tool by 2 matrix acting on spinners arms have been states let's see what the properties such a matrix this is at the mobile representation Of the rotation group of to by 2 representation is a subtlety about their about this matter get into but uh about sorry may know it to value of representation report to the is gonna represent a rotation all the components of a spinner of the half spin particles when you rotate the direction of space but so what's the 1st thing that's important well 1st of all when you act on state is yet another state status characterized by the sum of the squares of the elements 1 was needed in 4 1 star Alpha 1 Plus Alpha to star Alfred tool 0 because 1 and the same should be true 0 after rotation this should equal resource of equal 1 Bucher also equal powerful 1 Prime Star powerful war trying to cost of about crosses infertility well it's very much a similar pattern the fact that the sum of the squares of the components of a vector should be conserved under rotation the only difference this year with dealing with complex numbers here they have been system is described in terms of complex components here Hey let's count components bomb from young money components does it take the pick a direction toward right I'm really independent fare components of here for for complex numbers and reach a pair of that but will is no we don't care about the the fees of these things the phase of doesn't matter so the vying to buy the Faisan forgetting by overall failures that cut down 3 only really 3 independent numbers describing them as such but well so we require 1 star 4 1 and that's 1 equation so really there are only 2 independent things here 2 independent things even author of 4 complex numbers 1 of these parameters doesn't matter overall phase never affect anything and the other is the leader of the vector of the sum of the squares of a number of them are always difficult 1 sole given any pair of components were given a spare like this there's always some direction that polarized along the axis of the but How do you know what I be rotation where are these rotation operators you well they 1st of all the most important thing the only important thing really is that they have to preserve the lead of any given vector not the Lincoln 3 dimensional space but delayed meaning the total probability that gives rise to some very very similar condition 2 0 0 this over here it gives rise to a condition which is very similar part transposed times are equals 1 I'm not a word about her new co-worker sells it's the condition that you France follows complex conjugate the Collins you His cyclical 1 just an extra will complex conjugate 0 foot conjugated because when we some things we multiplying by complex conjugate this combination he ever transposed complex conjugated is called the permission 100 represented by you dagger wishing that before you dagger dagger because transpose complex conjugated transposed means you flip things about the main diagonal complex conjugate just complex conjugate every element you dagger it is equal to you transforms eroded here redirected again so you

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dagger plans you as you go toward but an insect all

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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.

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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

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I have but the Epsilon cable was street and has a small thing but and be a small thing I what's the determinant determined this time this month this that this time this is quadratic in small numbers it doesn't matter M 1 1 times and to tool is quadratic in small numbers but there is a PC at which is 1 of them To cost In 1 1 times 1 was at its M 1 1 plus and to 2 trace the a trace of every trace of them so the determinant 1 plus a small matrix is our there's basically just a trace of the small matrix the Afghans after they the bottom line is and have to be traced down the bottom line is and have to be traced worth of time so we have we have to construct a permission more matrix which is Chery that's exactly what they said was are they are the trace to buy to her mission makers any traces show by 2 or mission matrix can be written as a sum of Sigma matrices solve these issues here was simply a linear combinations of the signal the particularly a combination you chose will depend on axis of rotation if you because small rotation about the X-axis who puts it next year if want because small rotation about the y-axis to puts it more widely so what you wanna make a small rotation about linear combination of the X and Y axes and a 45 degrees you put a linear combination that's the connection between the city was of rotation there representations of these small rotation OK asked thing is how the combined you know I think I'm a weight of from next week talk about combining representations and just jump the quarks With get off the mathematics a a moment or to have a moment of true love so let's talk about quarks Miller not interested In a span of quarks or we are interested and of course it's been a half but were not but now we're talking about now I'm interested in the color of quarks the cholera quirks is described by a three-state system red green or blue so has single choir that quirk could be in 1 of 3 states just as they could be in 1 of 2 states the quark color could the be red green or blue 0 basilica described state of court by 3 entering the 3 entries will correspond green and blue but not an form now for 2 and a half 3 and the sum of the squares of offers will just b the probability Red quark green quarter blue work end what are the possible symmetries however another thing we can say we just go 1 step further and say we could also think of 83 field the field operator for a rate quark the field operated for green quark the field operated from lukewarm and assemble them will also backed like that's what kind of symmetries Comey imagined among the 3 quirks we can't imagine 1st of all discrete symmetries we can imagine the discrete symmetries which interchange interchange Redding green interchange green and blue or so Bullock and I don't larger family that includes that incidentally these rotation matrices here happened to include the Matrix 1 1 Ms. matrix here it is a special unit carry matrix so among these rotation matrices is a matrix which interchanges up and up-and-down this means the interchange group is a subgroup of the full rotation Pappas 0 Shea a chair the if if if if I think we have to put money in there somewhere is going out with of the pro-Iranian yup now now from such a yacht give determined weren't so high lead commutation of 2 things as a special case is a special case over rotations it's not the fact is there anything more complicated there we could do the other is more complicated thing we can imagine Is unitary opera would not obvious that there is such a cemetery in nature other work there is such a cemetery in nature and in fact the whole point of having color here is that it ends another symmetry of physics important cemeteries symmetry will player important role in the properties of quantum chromodynamics but the symmetry is the multiplication of 3 components red green and blue light special unitary 3 by 3 matrices the same mathematics as appeared here except the mathematics involves 3 by 3 matrices won't fill in all the components but there are few 1 1 about about about about 5 . 3 by 3 matrices special unitary matrices determined warned again special that called special scaup S U 3 of the group as youth but but what charter right now from the start

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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

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group basic symmetry

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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

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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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please visit ash Epstein Stanford down EDU

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### Metadaten

#### Formale Metadaten

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

Serientitel | Lecture Collection | Particle Physics: Standard Model |

Teil | 3 |

Anzahl der Teile | 10 |

Autor | Susskind, Leonard |

Lizenz |
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DOI | 10.5446/15086 |

Herausgeber | Stanford University |

Erscheinungsjahr | 2010 |

Sprache | Englisch |

#### Inhaltliche Metadaten

Fachgebiet | Physik |

Abstract | (January 25, 2010) Leonard Susskind, discusses the rotation of space. |