The Biggest Ideas in the Universe | 14. Symmetry

hello everyone welcome to the biggest ideas in the universe I'm your host Sean Carroll last week we talked about geometry and topology these were clearly mathematical ideas but I think that most people know these are ideas that have a big role in physics so today since we did math last week we're gonna do math again okay we're gonna talk about symmetry now symmetry is a quintessentially physics topic symmetry is at the heart of a lot of modern physics but the way that we describe symmetries is gonna involve some math and that's basically what today's big idea is how we go about actually describing symmetries if you've ever heard of the standard model of particle physics you may have been told that it's based on a symmetry group su 3 cross su 2 cross u 1 so at the end of this maybe not today but at least today or next week we're going to understand exactly what that means to say that the symmetry group of the Standard Model of particle physics is su 3 cross su 2 cross u 1 those are those letters those symbols su 3 etc correspond to internal symmetries so they're actually transformations they're basically generalized rotations of different fields in the standard model of particle physics so the quantum field theory will come into play we also have what are called external symmetries I guess are they called external symmetries they're not internal symmetries space-time symmetries is what I'm talking about we know that for example you can translate where you are in space you can rotate where you are you can change your reference frame and special relativity all of these are also symmetries so this is why symmetry is so crucial in modern physics but what is it what do we mean this so this is the kind of thing that mathematicians do right they take an idea that you have and you think you understand it at some intuitive level and they try to formalize it make it precise and then talk about its properties in general circumstances far abstracted from any specific application then the physicists come back in and say I bet I can find an application for that so that's what goes on with the study of symmetry so what do we mean let me let's get into what we mean by symmetries by starting with some examples just draw some shapes on the on the iPad here and you can for yourself how symmetric they are so here is a completely non symmetric little blob there you go I don't see any symmetries there at all and look I've been practicing for you folks don't say I never did anything for you I've been learning new things on notability here watch this triangle but boom it becomes all nice now I'm gonna make an equitable triangle there you go hopefully it's neck lateral triangle okay that seems to have more symmetry than the blob like somehow there's some regularity there that we would like to be able to understand better likewise we could draw a square there you go hopefully that's really a square not just a rectangle but again some symmetry there and what if we could also draw a circle this is going to be harder to draw but I can do it here we go there's a circle okay four different shapes and I've arranged them in what I will argue is order of increasing symmetry there's more symmetry as you go from left to right and that kind of has an intuitive sense the circle in particular looks extremely symmetric right it looks the same from every different angle but this is what helps us turn these vague ideas into something formal and manipulatable at the level of equations and theories of physics we want to know exactly what we mean when we say there's more symmetries in the circle than the square more symmetries in the square than the triangle and we kind of already gave it away the idea that you can look at the circle from any different angle is a reflection of the fact that there's some symmetry there the symmetry is a way of saying that there's a transformation we can do to some object to some system that leaves the essential features of the system invariant that's what symmetry is not sure if that's the technical definition well no sorry I am sure that it's not the technical definition but a symmetry is a set of transformations which means in math you terms a set of maps from the thing to itself a set of ways you can manipulate the thing back to itself transformations that leave let's just say leave the system whatever it is invariant what we mean is leave whatever we count as the important features of the system invariant so to make that a little bit more specific think about the triangle we can clearly rotate the triangle by a hundred and twenty degrees right if you rotate it by 120 it goes back to the same triangle you can rotate it again another 120 240 then you rotated a third time and you get 360 degrees it's back to where we started so let's actually make that explicit ok so here's another triangle we're gonna that was not a very a collateral triangle but I'll try to fix it there we go pretty close here's our original triangle I'm gonna duplicate it a couple times and here's my original triangle again duplicate good put it over here and this will all make sense why I'm doing it in this particular way and I'm gonna label the vertices okay ABC I can label them any way the labels are completely arbitrary it's that's up to me I just want to keep track of what I do to the triangle because the whole point of the symmetry is I'm gonna do something it makes it look the same so a B and C are not labels that we physically stick onto the triangle they're just to remember what we've done to it okay so what I mean by this is we could for example rotate clockwise 120 degrees so that's one third of the way around the circle I'm gonna call that operation r plus the notation will make more sense in a second so what our plus means rotate 120 degrees so I get C a B okay that's what I mean by that particular rotation so you see that if I had rotated by 90 degrees I would have gotten the triangle off at some angle 90 degrees whereas if I rotate by 120 degrees I get exactly back to where I started if I were really fancy about it watch what I could do I could take the triangle and I could rotate it and this is why I'm not doing it in real time because it kind of becomes messy and distorted but that's the point it takes the 120 degree rotation to come back where it started that is part of the symmetry we can also rotate backwards so we can do our - I'm not going to clutter it up again but that's a counterclockwise rotation by 120 degrees so that gets us to a b c and we see that these three versions of the triangle are labeled differently so the triangle looks the same in every case but it results from a non-trivial transformation on the original triangle so this is part of the symmetry group as we're going to call it of the triangle the set of transformations that we can do the triangle that leave it invariant now of course I could do a rotation twice right I could do our plus twice and that would actually get me to our - right if I did our plus again I'm not going to draw it here because it'll clutter up the diagram but if I rotated this triangle this one right here if i rotated that by another 120 degrees I would get this triangle over here so these are the two rotations that I can do but I'm not done with the triangle because there's things there are things I can do other than rotations right so to see that let's make some more triangles here okay and I'm gonna put these triangles down here not easy to do this but you know we're learning trying to get better at it there you go three more triangles and I'll it'll be clear in a second why there's three more triangles the point is that in addition to rotating I could also flip the triangle right like there's nothing that says that I'm confined to moving in the plane of the iPad or of the board I'm drawing on here right I could flip around some axis so if I flip around this axis for example I would get a B C whereas if I flip around this axis the original triangle I would get C a B and if I flipped around this axis the original triangle I would get B a C and you can notice that every one of these if you look at the vertices and how they're labeled they're all distinct from each other okay no two of these of these six triangles are exactly the same when you consider I mean all the triangles are exactly the same no two of them result from doing nothing to the triangle they are all a non-trivial transformation okay so we can label these let's call this let's call it a flip we have flipped around the vertical axis it's more typical to call it a reflection but we've already used a letter R for rotation so let's call it a flip let's call this F 1 let's call this one f 2 and let's call this one F 3 there you go so I think that's it you can check at home but I think there's nothing else you can do to the triangle that does not get that that gets something different than all of these different possibilities in fact I'm sure of it because in this one particular case the triangles the one case where the set of symmetries of the triangle the geometric shape is just the set of permutations of the three vertices a B and C okay that wouldn't be true in a square or Pentagon or whatever but it's true for the triangle so if I just enumerated ABC ACB etc there are six of them and they correspond to these six versions of the triangle okay so that is those are the symmetries of the triangle how many of them are there you might say well there are five because we got the first rotation the second rotation and the three flips but in fact there's a sixth because we can do nothing we can just transform the triangle onto itself that's an important thing we can do namely you just leave it as it was so I will label that by the fancy number one here the identity operation leave the triangle invariant so I really have six things I can do to the triangle if I include doing nothing as as a thing that I could do so what there's a lot of information here there's a lot of structure going on here this is what makes mathematicians excited about symmetries is that you don't need to simply go ahead and and enumerate all the different possible symmetry transformations you can talk about how they relate to each other and what kind of structure they form so together these six transformations and we'll list them the identity rotation clockwise by 120 rotation counterclockwise 520 flip around one axis flip around the other flip around the third these three form a group that is what the technical definition of the technical term that we attach to this kind of mathematical beast is if a group is a little bit more specific than just say a set it's not just a collection of things a group forms some very specific properties the crucial feature of being a group is that you can take these transformations and combine them and you will always get something that is another transformation that is already in the group okay this is called the closure property of the group the fact that you can bind transformations you never leave the set of operations so let's see that in action a little bit consider if we start up here the original triangle and we do flip one so we go around the vertical axis so we get down to here I claim that I can get to flip two by then doing a rotation so if I go over here what do I need to do I need to get starting from here to get to here I need to move a down to the left see up to the bottom right it looks like I'm going counterclockwise which is in fact the rotation - R - that is what we called it so what this means and equations is you'd write this as f - equals R minus F 1 ok so the order matters here as we will see in a second so you read from right to left first do F 1 then do R plus or minus why why do you why do we do it in that order well because you're supposed to be thinking of these things not as numbers that you're multiplying but as operations that you're performing on the triangle so when we do something like let's just do an example F 1 acting on the triangle here's a triangle that we start with with a b c that is flipping around the vertical axis and so that gets us a oops a and here's the triangle and then we get B and C okay so the reason we read from right to left when we transform when we combine these transformations together is because we're imagining them acting in turn on the triangle itself and the one on the rightmost is the one that acts first and then after that go leftward let me erase all this stuff is just going to get in the way but that's why we go from right to left okay so when you you can check at home whenever you combine two these transformations you stay inside the group that's a crucial property of being what it means to be a group the other crucial properties are there's an identity element okay there's one element that doesn't do anything at all and the other one is that there's always an inverse to any operation so there's always one operation that can be undone by a single element of the group so for example here we would say that our plus is the inverse of our - okay it's our minus to the minus one another way of writing exactly the same thing is that our plus times our - equals the identity element or one our mine is followed by our plus we get you back to where you started and you can check if that's true it is true okay what is not necessarily true I'm sorry there's another technical thing about associativity how you put parentheses around your group transformations does not matter this is what it means to be a group but group is a little bit more elaborate than a set this is the kind of thing that magicians do they start with a set of things and they put structure on them and they say that structure obeys different conditions the structure here is the way to combine the transformations together it's the multiplication rule if you like that multiplication rule need not be what we call communitive the commutative means you can change the order of operations and it doesn't matter so here it clearly does matter if we took not f1 followed by our - but our - followed by f1 okay well that would start us here the original triangle we go our - brings us here flipping by f1 means that we flip around the vertical axis so we would flip a with see so we get BCA which is not down there it's over here right so that equals f3 look at that so R minus f1 does not equal F 1 R - they do not commute and the lingo we don't need to really know it I mean I need to know it commutativity commuting groups if things commute if the order doesn't matter sometimes that's true are called abelian groups after professor Abel and guess what non commuting groups equals are called non abelian so you can see why mathematicians would fall in love with this you have it this beautiful structure they fit together you can define things you can ask questions like do the different transformations commute or not and so forth so this is the whole subject of group theory and it is the sort of mathematical version of what physicists call symmetry transformations is a fact of the world that we have symmetries in the world and is a mathematical fact that the way of describing them is by talking about groups now there's just a couple other things I can't resist telling you because this is an infinitely interesting subject remember the point of the group is that different transformations complying together will always give you another group element ok so we can have subgroups we can have subsets of the original group which are themselves groups so for example if you have the identity element and the two rotations are plus + or minus they form a subgroup so that's the subgroup of these the top 3 things right here if you rotate and rotate backwards you're never gonna flip a flip is a whole separate thing ok so this is a perfectly good subgroup if you combine our + and our - you will get each other in fact there's a what you would do if you were being very careful about this is you would write out the multiplication table so we already said that r + x our - is just the identity but you would also write out things like r + squared which is guess what our - as you can go back and figure it out okay I'm gonna erase that because not done writing down subgroups but that's the kind of thing you would do so this is a subgroup good but what about just the identity and our plus that is not a subgroup I'm going to make this red okay big red X they're not a subgroup because our plus x itself is not in that collection of things it gives you our - there you go likewise if you have just the identity and one of the flips pick any one f1 f2 f3 is this a subgroup okay well you can go back and look I could start here in the original triangle I could flip it by f1 if I flip it again I just get back to where I started so indeed this is a perfectly good subgroup f1 squared because f1 squared equals the identity that's all you really need to know whereas if you try to complicate your life what if you said one f1 and f2 okay so you have two different flips what is where does that get you well so you start here here is F here's nothing here's f1 I flipped around the vertical axis if I do f2 that's gonna flip around the axis going this way so it's gonna flip a and B so I'm gonna get B at the top a at the bottom left C at the bottom right which gets me here which is the arm - element which is not in the group that I just wrote down so this is not a good subgroup so you can see how there's sort of an interesting set of possibilities what is a subgroup what is not a subgroup one last little fact you can talk about take two non-trivial group elements say R + + f1 it turns out that all six of the elements in the group can be written as some combination of these two so these are called the generators it's a choice of what the generators are there's other choices the generators of the group transformation from these you can build up the entire group okay I mean I can go on there's other things we can say about all these things but you see why groups are so important groups are the way to mathematically represent the idea of symmetry transformations okay just to be one-one fact I really should tell you here what a these groups all have names okay and I kind of don't want to like dwell on the names this group is called d3 the dihedral group of an object with three corners so as you might guess there's something called d4 which is the symmetry group of the square d5 which is the symmetry group of the pentagon etc and this happens to equal s 3 which is the permutation group of three elements ABC right so ACB BAC etc are all different symmetry transformations of those three elements in general as we said the dihedral group is not going to be equal to the symmetric group on n elements because you have a Pentagon you're not gonna be able to do all the different permutations of its vertices those are just the words who cares what the words are what the symbols are with the notation is what we care about is the essence of it but when it comes to these little examples there is another couple of examples where knowing the terminology will help so there's a group called the integers actually it's not called the integers let me let's let's be careful z this fancy Z is how we denote the integers and it equals you know dot dot minus 1 0 1 2 okay you know what the integers are when I say the integers are group what's going on there it means I can add them together if I take 0 plus 1 I get 1 if I take 1 plus 2 I get 3 etc so if I think of the integers as a shift of a real line so let's imagine that I have some real line I literally take the integers or any set of points equally spaced and I call each element of the integers a shift by that much that is a symmetry transformation on this set of points ok but notice that I do in fact fulfill all the requirements the integers if I take any two integers and add I get a third integer there is something called the inverse of any integer namely - that integer and there's also my identity elements at number zero okay but we we are we have baggage about the integers we think that not only can we add them but we can also multiply them two times three equals six okay so you might say well maybe there's another thing I can do a different kind of transformation in the integers that I could do to make a group namely multiplying them not dividing them or not adding them false that doesn't work I'll give you three seconds to think about why it doesn't work it doesn't work because you need an inverse to define a group so if I have you know plus one as my transformation then I have minus one is my inverse transformation whereas if I do multiplication I can multiply by zero and I can't undo that if I take an integer like six and then multiply it by 0 I get 0 and there's no way to multiply anything by zero to get back to six so it's the integers under addition that are a group not the integers under multiplication so when we say in this kind of context just the integers we very well we generally mean under addition the integers have an infinite number of elements in the group there are infinite number groups and there's also finite number of groups like this group up here the triangle group or the dihedral group has six elements the integers have an infinite number but there's a version of the integers only as a finite number namely the integers modulo n and what that means is we define Zn we define addition on the integers but only up to n so if you get above n whatever n is you pick it so there's Z - there's Z 3 Z 4 different versions of this group you just say you know 1 plus 1 equals 2 but if you're doing addition modulo 2 you say the 2 is the same as 0 okay so Zn is just the set 0 1 2 dot n minus 1 once you get up to n you wrap back around so Zn is very much like a circle let's make it a circle where you've written n points okay so this is Z six and there's an element there's another element etc okay you come back to where you started so these groups here this one is isomorphic the same as Z three whereas this group here is the same as Z two okay so these are abelian groups if you notice I can you know that doesn't matter what order I do our plus or a minus in doesn't matter if I only have two elements in the group it better be a billion there's no way they could not commute with each other but so these are two very very important discrete groups okay discrete sets of transformations now you notice there are an infinite number of different groups called the integers modulo and one for every single value of N greater than 2 or 2 or 2 or greater every single integer so you could have Z a billion or whatever as in terms of finite groups it turns out there classifiable there's only a finite number of ways to be a finite group there an infinite number of finite groups like Zn but there's that this is only one way to be a finite group and there's only a finite number of ways and it was only like in 2004 they finally classified this there's both these continuous families of finite groups and there's also what are called these four addict groups there's some weird ones there's something called the monster group which has something like 10 to the 53 different elements and so of course mathematicians just love this stuff physicists like the integers and and the integers modulo n as groups they like the dihedral group or the symmetry group of the sphere the of the triangle but what they love are continuous groups ok so that's where physics becomes really exciting continuous groups are just what they sound like they're groups where there is some continuous transformation a transformation that is that is indexed if you want by a real number or by some smooth parameters and continuous parameter not buying into integer or just some number continuous groups also called lis groups after professor Li who studied them classified the earliest ones okay what's an example of a continuous group well remember we have the circle right remember we have ups that's not a very good circle there can I fix it is it fixable yes look at that kind of like a circle now a circle you can just rotate by any angle right any angle between 0 and 360 degrees or we're gonna have to update our units here sorry about that but we're gonna have to go to radians instead of degrees so I'm gonna have to teach about radians if you don't already know radians is just degrees times 2 PI over 360 so a right angle equals PI over 2 radians all the way around the circle is 2 pi radians although this will become clear why this is important in a second it's just units just a choice of units okay but physicists mathematicians are always going to be using radians rather than degrees I've been shielding you from that to some extent but now we're gonna have to go there so anyway any rotation of the circle by between 0 and 2pi radians is a symmetry transformation so there's a continuous group an infinite number of different possible angles by which you can do that rotation and in fact to some extent the basic most importantly groups are in fact generalizations of this idea of a rotation group ok in particle physics as we'll get to in a second we have SU 3 cross su 2 cross u 1 in the standard model yeah and these are all examples of continuous groups that are sort of generalized rotations in some sense so let's do this a little bit more systematically let's look at that continuous group which is called the symmetry group of the sphere funny sorry of the circle circle is of course a sphere is a 1 sphere it's a one dimensional sphere but I will call it the circle so what are the symmetries of the circle well very similar to the analysis for the triangle any rotation what we have the identity we have any rotation by any angle theta with theta being an element of 0 to 2pi oops so including 0 not including 2 pi because you can't include 2 pi ik is already there for 0 ok in fact we shouldn't even clued the identity element because that any element is there but it's a rotation by 0 ok that's what it is and there's another one there's another can you guess what that is yes you can a flip around some axis so you can write some access and you can imagine just flipping antipodal points points on opposite sides ok we kind of talked if we kind of implied the existence of something like this transformation when we did the identification of antipodal points in the lord topology discussion last time so together these are called o2 that is the letter O not the number 0 2 for a two-dimensional plane where the circle is living and oh for orthogonal this is the orthogonal group or a thug's and mole and in general o n is the set of rotations oof once again a bad handwriting day sorry about that rotations and reflections in n dimensions so the circle is a one-dimensional object itself but if I erase this bit inside here just to make my life easier the symmetries of the circle are the same as the symmetries of rotations around a point in the plane so the point the plane itself is two-dimensional right it has two dimensional vector space you rotate around the center that is the orthogonal group in two dimensions ok so in this case and in fact in every case what's gonna happen is oh and the entire set of symmetries of this n-dimensional space the generalization of a rotation is going to be rotations so sorry let me just be a little bit more explicit about it it's going to be something which we call the special orthogonal group so this is just rotations times z2 reflections or flips this is always going to be true so just like the triangle you could separate out the rotations and the flips you can just do that in general in any number of dimensions for an orthogonal orthogonal group for the continuous symmetry group in n dimensions so generally that's what we do we don't usually talk about Oh n we could but the point is once you get to the math of it and the physics of it if you can separate out a group into two different kinds of things and then combine them together later on that is almost always a sensible thing to do there will be things that are invariant under just s Oh N and not oops that's z2 sorry sorry it's just a two element group not an infinite number anyway there will be things not necessarily this plane but there will be things that are invariant under s o n but not the z2 flips so whenever you can sort of separate out these groups into separate pieces that's different kinds of transformations that's generally a good thing to do okay so then this generalizes right we have for example so3 or rotations in three dimensions it's not a generalization this is an example in the 3d so think of you know a three dimensional space parametrized by three vectors and you can rotate it you can rotate it to be wherever you want right so if this was if we started with Basie's X Y Z and you rotate and you get something X Prime why prime Z Prime and every single possible way to rotate that is an element of so3 so even though there's an infinite number of such rotations what you can do now is not talk about the number of transformations but the dimensionality of the space of transformations so oh - let's think about it let's see let's see if you can guess so2 is the rotations in the plane okay this is so2 how many dimensions knot is the thing we're rotating but is the group how many dimensions is the group so2 well the answer is it's just one dimensional there's only one thing we can do so we're rotating two dimensions into each other but there's only one way to do it and therefore so2 is a one dimensional group okay in fact the group itself is just the circle topologically so3 it turns out there are three ways to do it and basically you can think about that as you can rotate around this axis around that axis or around that axis so this is a three dimensional group and that sometimes confuses people into thinking that somehow the number of rotations should equal the number of ways you can the number of dimensions in the thing you're rotating but that's not the case so if you have s o n that is rotations in n-dimensional space but the dimensionality of s om works out to be one-half n n minus one okay so if you plug in three you're gonna get back to where you started right one-half times three times two is three but if you plug in four 1/2 times four times three is six so 4 does not equal six in general once you get beyond three-dimensional space the number of rotations is bigger than the number of dimensions you're rotating you might wonder how that can possibly be true isn't it always true that you just be able to rotate around an axis but that's not true this is just because your ability to visualize in dimensions greater than three is failing you when you rotate in four dimensions you don't rotate around an axis you rotate around a plane so a rotation moves one axis into the other leaving all the other axes invariant so that's the question you should be asking yourself how many different ways how many different ways are there to leave all the other axes invariant and the answer is this so all that is just to say s o n the group of rotations and n dimensions is not n dimensional it is a bigger number of dimensions ok now I'm going to complicate your life a little bit ok the the rotations is a pretty obvious thing it is a non abelian group you know other than so2 where it's just rotating in the circle once you start rotating in three dimensions the order matters if you rotate a little bit around z-axis and then a little bit around the x axis you can convince yourself you're gonna end up with a different orientation than if you rotate a little bit around the x axis and then a little bit around the z axis it's a non abelian group most groups are not commutative or not abelian also in physics many things turn out to be complex numbers right we don't use real numbers all the time we sometimes use complex numbers and vector spaces can be either real by which we mean we take basis vectors and take linear combinations of them with real numbered coefficients or they can be complex and there's other alternatives also but they can certainly be complex let's put it that way which means that our vectors are sets of combinations of basis vectors but the coefficients of those basis vectors are complex numbers not just real numbers so there's the version of rotation groups that applies to complex vector spaces and we call those unitary groups for reasons which I'm not gonna reveal so the unitary groups um are symmetries of complex and dimensional spaces just like oh n were the symmetries of real and dimensional spaces UN are the symmetries of complex and dimensional spaces so just to be a little bit more explicit about that vector stuff I just said you know a real vector means you might have seen things like this you have a vector and you'd say you know a times the unit vector in the X Direction plus B times unit vector in the Y direction if you're two dimensions if you're in three then you might have C times Z and implicitly a B and C are all real numbers okay let me just I don't need to be quite that Matthew I just said the correct thing but you don't need to be Matthew unless it is necessary right so a B and C are real there you go whereas in a complex vector space same exact notation okay V equals a and because I don't want because space the three-dimensional space we live in is a real vector space okay we locate where we are using real numbers not complex numbers I hope you've noticed this in your life so I'm not gonna write x y&z let me just write some generic unit vector e1 plus be e 2 plus C III or plus e 4 plus C 5 you can have any number of dimensions in your vector space you like where a B and C are complex and of course a complex number ie there's a real part plus I times an imaginary part where I equals the square root of -1 okay and I've been careful here you know when we did oh and versus s oh and oh and are all the symmetries of the n-dimensional vector space s Oh n is just the ones that do not change the orientation do not flip around some axis okay you might think that since this is the same set up but with complex numbers rather than real numbers it's the same thing you might think that UN is just su n a special unitary group and dimensions plus flips of orientation and in fact you already know there's something called su 3 and there's something called su 2 so you're waiting for me to say su n is just u n plus some flips but that turns out not to be true and the reason why is because if you if you think about what is a one-dimensional group sorry what are the transformations of a one dimensional vector space okay 1d space well let's look at real 1d space that's just the number line okay it's just the real numbers there's zero at the middle what can you do to it well you can translate it but we're not counting those okay you by translate I mean you literally shift over that's certainly absolutely something you're allowed to do but we're looking at rotations or the the generalization of rotations in 1d space and there's the only thing you can do in a one-dimensional space to rotate it is actually to flip it right is actually to exchange points on opposite sides with the same valve now is it pretty bad I can do better to exchange points on opposite sides with the same value so you know if this is minus two plus two they get flipped back and forth that is something you can do that's all you can do so that is in fact oh one if you want and that's just Z 2 it's just a group with two elements either you don't do something or you flip it so many possibilities but what about 1d complex space the complex numbers right well one complex number is kind of like two real numbers right you have the real part and you have the imaginary part and any complex number call it Omega it can be written as the real part plus the imaginary part or it can also be written as the length which we can call are times some angle theta e to the I theta so R is the length and theta is the angle in the complex plain so in other words when if you write Omega as let's say a plus IB okay so a is this real part B is this imaginary part you can write that equally well as are times e to the I theta now E I gotta do a little bit of math here sorry about that e is the Euler constant 2.718 dot that the Euler constant it's special for many reasons it's a special number in mathematics it's an infinite number there's no simple way to write it down using a finite number of digits there's no way to write it down using a finite number of digits it has the wonderful feature that I was tell you this and then I'll erase it there's a function e to the X okay this particular number raised to the power X I can take the derivative of that function with respect to X ask what is the slope of that function if I think that function looks like it's exponential growth right exponential growth something we all know about these days e to the X versus X it turns out that this is the unique function other than just zero everywhere with the property that its derivative is equal to itself if it were 2 to the X or 10 to the X it would not quite have that property that that's another little pre factor out there so e is a very special number good letting you letting you in on that and even better when you raise it to an imaginary number that's what we're doing here e to the I theta so e to the I theta theta is a real number theta is just this angle here e to the I theta equals the cosine of theta plus I times the loop s-- don't need that I times the sine of theta and here in order for this formula to be true theta has to be measured in radians okay in going from 0 to 2 pi so if you plot let's see if I can do this probably not do this in real time if you plot the cosine of theta it looks something like this whereas if you plot the sine of theta it would look something like this okay that's sine theta and this is cosine theta so again this formula happens to work out it's a miraculous wonderful formula that is true if theta is measured in radians so it repeats at theta equals two pi this is theta equals zero right there and why is it so interesting well remember when we did complex numbers squared so when you have a complex number and you take it square what you really do is you take its complex conjugate and then you multiply it so by itself so e to the I theta squared equals the complex conjugate let's see how we can write this e to the I theta star times e to the I theta e to the I theta star I mean the complex conjugate just says change all the I's to - sighs so this is e to the minus I theta times e to the I theta which equals e to the minus I theta plus I theta which equals e to the 0 which equals 1 so that's why this is such a nice little thing this formula e to the I theta gives us a vector in the complex plane of unit length okay that's why it's convenient to write any complex number as a length R times what we call the phase or the angle e to the I theta why am i doing this why am I going through all this effort well we said up here in one dimensional space the orthogonal group in one dimensional space is just flips it's just z2 it's either you flip it or you don't those are the only two things to do in one dimensional complex space you have really two dimensions to work with there's another thing you can do you can rotate okay so the group u1 is actually a multiplication of a complex number by e to the I theta so if you have some complex number so here's the real part here's the imaginary part and here is the complex number call it Omega okay if you multiply by theta it keeps the length by e to the I theta it keeps the length constant let's zoom in here the length stays the same but you rotate by an amount given by theta so this is Omega this is e to the I theta times Omega so the set of all rotations in the complex plane is multiplications by e to the I theta because even though I 2 pi equals 1 right we can write that we did that e to the 0 equals 1 that's also e to the 2 pi I because you go back to where you started by that little picture up there so anyway the point being that in the orthogonal group case in the real vector space case oh n is som just rotations plus flips whereas that's not quite true in the unitary group case UN in general is su n which are just generalized rotations as we expected times u 1 which is multiplication by a phase so multiplication by e to the I theta and this is just called the phase so anyway all this is to get to the fact that this is a general rule ok it's not the same quite the same thing as the orthogonal group case there's a continuous set of transformations that we can take out of U n to leave us with su n ok and s here stands for a special which is just there's a fancy way of saying what that means but it means not just multiplying by a face it means everything other than multiplying by a phase ha ok and as before we can calculate the dimensionality of su n as it is not going to be n it is in fact N squared let me get it right yeah I was gonna say N squared minus 1 and correct I should just said it should have believed myself there you go it goes up very quickly the minus one is basically the U 1 okay the dimensionality of U 1 equals 1 because it's just that one rotation so the dimensionality of U n is going to be N squared that's how many things you can do to n complex numbers all right where are we you n this is the rotations complex vector spaces very important because all of the matter fields of the standard model all of the electrons and the quarks and neutrinos those are all complex valued fields so the version that we have to use to operate on them if there's some symmetry between them will be su n naught s ohem having said that there is your secret fact I'm not going to right now but so3 turns out to be almost quite but almost the same group as su 2 they're both three dimensional groups we'll get in a little bit to why they're a little bit different not a little bit I mean not today someday someday down the road we'll show you why they're different the topology is a little bit different is the answer but the basic set of transformations looks very very similar both three-dimensional groups okay good so what I wanted to get to is the fact that these UN's or su ends really these this su n the real rotation here this is where we get the su 2 cos su 3 of the standard model the U one is another symmetry of the standard model and these are give us all the forces of nature roughly speaking the U one is going to give us electromagnetism the electromagnetism earth is a reflection of an underlying u 1 symmetry of the complex valued electron field and positron field the su 2 gets us the weak interactions the su 3 gets us the strong interactions don't have time to do any of that today also there's other secrets that will be revealed but don't worry I predict that next week we will get to exactly that so I mostly wanted to get to you the idea of a group and also the specific groups that we'll be dealing with z ZN s o n & you and these are our biggest most famous groups I can't however leave you without one more fact about symmetries it is indeed the first fact about symmetries that many people know in physics which is the connection between symmetries and conservation laws conservation was the very first topic the very first idea that we had in this series so it's nice to see the old friends coming back and this relationship is summarized in nerds theorem I mean earther famous mathematician of early twentieth century and she said basically well it's a one-way street let's put it this way symmetries imply the existence of conservation laws it's more subtle than that most conservation laws in nature do come from symmetries there's a few tiny exceptions but typically if you find a conservation law you should search for the symmetry underlying it there are subtleties dealing with what kind of symmetry it is again there's a nervous first theorem and second theorem we're not going to any of those things the basic point is that this that when you have a symmetry there is a conserved quantity and it turns out to be hard to explain why actually Richard Fineman came up with what is probably the most transparent explanation of why northers theorem is true which I'll try to reproduce at least a little bit casually here it all relies on the action principle remember the action principle of least action says that there's this thing that you can calculate by integrating over time something called the Lagrangian and the Lagrangian depends on the position of some let's just say a single particle let's make our life easy it could be fields right and we'll be but make our lives easy single particle as a position and a velocity and it could also depend on time but let's say it doesn't okay so this is the action this is the Lagrangian in field theory we would also write the Lagrangian as an integral over space the lagrangean city okay we can do that and the point is that if you consider all the possible pads the particle or the fields could take what you would find is the actual physical path that obeys the equations of motion minimizes or at least extreme eise's the action okay so it's like kind of like not exactly like but kind of like if you considered all the trajectories of your system and you plotted the action it turns out that the action would be a minimum at the actual physical trajectory so this is all possible trajectories like a particle sitting there without any forces acting on it but just moving around in all sorts of directions violating Newton's laws that's a possible trajectory that we can contemplate that conceivable maybe a conceivable trajectory would be better the physical ones are the ones that actually obey the laws of physics okay so nervous theorem says that if you have a physical system that is described by an action like this and by the way not all systems are but that's what we're gonna consider right now most systems in quantum field theory anyway certainly are so what it says is let's imagine a symmetry is described by some shift of all the values so we have X of T the position as a function of time and V of T the velocity is a function of time and you shift it to something parametrized by a parameter let's call it Epsilon okay so to some new value X sub epsilon of T and V sub epsilon of T so what you're saying is you know let's say that you are in a world where rotational invariance is obeyed okay you see we're in outer space the earth does not below you you can rotate your coordinate system however you want and the shift might be I just changed my coordinates by rotating a little bit or in quantum chromodynamics the theory of the strong interactions QCD maybe you change red quarks into blue quarks by a little bit and this parameter epsilon tells you how much you change and you're gonna say well if it's like SU three I had many different transformations I can do yes that's true what we're saying here is pick one okay and move it in one direction by an amount Epsilon so what here is fineman's argument for doing this for deriving nervous theorem so here again we have trajectories space and here now we have time so the actual physical trajectory may be looks like this there you go this is X of T and from that you could get V of T okay and call this point a where you begin at Point a before B where you finish and he says okay now let's imagine doing this transformation so I shift by Epsilon okay to Point C right there and then I do the same kind of trajectory but I've shifted everything by Epsilon shifted everything by some amount by some transformation rotation translation some sort of internal rotation whatever it is okay and I will label this point D okay so here's what here's the argument there's a bunch of things you can sort of say about this picture let me move over the picture so I can say things about it okay one is one is that the original action s of a going to be let's just write it as a P okay the original path this path right here that's some value and it better be equal to the action of the path going from C to D because that path is just related to the original one by asymmetry that is the assumption that we're making there's a symmetry relating these two paths okay but at another fact the D that's the sort of the obvious fact there's another fact that is a little bit more subtle but look we're saying here that the action is minimized on the real path which means if you do an infinitesimal by little tiny bit of epsilon then the action won't change okay so the total action on a path that is displaced from the original one by just a little bit is equal to the action of the original path to the two first order to the smallest approximation okay this is why it's subtle this way it doesn't sound completely kosher but you can make it completely kosher trust me on this so what that means is if this if this little displacement epsilon is infinitesimal then the action associated with a b is also equal to the action associated with a c d b that is to say think of this little v as an actual trajectory you'll be moving faster than the speed of light or something like that but that's okay it's just a conceivable trajectory think of the trajectory is going from a moving over to C on that trajectory then following the physical trajectory up to D then going backward okay because that is only an infinitesimal change away from the original one the action will be the same again to a good approximation and you can make it all rigorous if you want to therefore what we can say is it had better be the case that the action going from A to C that little tiny trajectory better be equal to minus the action of going from D to B just from what we said you know we didn't say that the action on this path was zero going from A to C we didn't need to say that was zero we just need to say that when you go a to C then up to D then back to B the whole thing had to equal the original action so there's a more general way for that to be true maybe it's zero but it can be some nonzero value as long as it's compensated for when we go back from D to B finally if you look at what the formula for the action it here is it's an integral over time okay so going from D to B is the opposite in time of going from B to D going the opposite direction so si C equals minus s DP that means it equals plus s BD if you had gone in this direction you would get the same answer you get sorry you would get minus what you got by going from D to B so look what we found look at that this is kind of fun si see the action along this little path right here is equal to SB D the action of that little path right there and indeed it didn't matter where we picked our original or or stopping points all of these little paths if you calculate the action on them you would get the same answer the quantity would be the same so we say therefore s a si or its generalizations later in time is conserved it's the same at every moment in time so the action along the path that you get from wherever you are and going by this amount by an infinitesimal shift gives you a conserved quantity okay that's nervous Theorem the existence of a symmetry gives you a conserved quantity it's sort of not at all surprising in some sense what a symmetry says is that there's a direction in the space of all possible things that can happen where it doesn't matter right you can shift in this direction it doesn't matter doesn't matter where you do your experiment how you set up your orientation of your axes or anything like that so the fact that it doesn't matter sort of morally corresponds to the fact that some number is constant it would not change if you did shift in that particular direction this is just the formalization of that so that's nervous theorem so it turns out this is kind of a big deal so for example spacetime symmetries we have so translations translations are that's a terrible way of spelling translations I'm just go start from scratch translations are just shifts right shifts in either space or time translations in space give rise to observe conserved quantity called momentum makes sense how many ways are there do a translation in space well there's XY and Z there are three of them and there are three components of momentum that works translations in time well how many of them are there for that there's only one and guess what there's one conserved quantity we call it the energy rotations how many of them are those well we have three axes in space they're all real right so the number of transformations is so3 the dimensionality of so3 is three so there are three conserved quantities and we call those angular momentum and then we can go on to the internal symmetries like the u 1 symmetry that I told you we haven't really talked about in detail yet but we will of electromagnetism is what gives rise to electric charge being conserved and so forth so this fact that norther discovered near this theorem right there which is that this quantity is conserved is not just a mathematical curiosity it's crucially important in all of physics it gives rise to things that we care about like energy conservation charge conservation stuff like that ok that's all I wanted to say about symmetry what we're gonna need to do in the next video the next idea is get some payoff oK we've done two videos in a row on math math is great but we're trying to understand the physical universe so next time geometry and symmetry are going to come together to tell us something about the fundamental laws of nature

Channel: Sean Carroll

Views: 68,028

Rating: 4.9257293 out of 5

Keywords:

Id: yCxacFBLHIY

Length: 63min 37sec (3817 seconds)

Published: Tue Jun 23 2020