The Biggest Ideas in the Universe | Q&A 14 - Symmetry

hello everyone welcome to the biggest ideas in the universe I'm your host Sean Carroll this is the Q&A video for idea number 14 which was symmetry and we have an interesting grab-bag of different questions that I'm gonna try to get to this week so let's dive in the first one comes right right down to the meaning of what we mean by a group of transformations remember we went through in this imagery video about how a symmetry is a way that you can transform a systems physical system for example in such a way that you've done something to it but the end product is no different like the triangle when you either flip it or rotate it by 120 degrees you get back exactly the same triangle this is kind of an abstract notion when you think about it you're talking about the meaning of a transformation or a group of transformations group both in the figurative sense the casual sense of a set of things but also the mathematical notion of what do you mean by group theory what a group means in that context so let's in particular go right down to the integers this is where the question came up the question was why is group multiplication for the integers group multiplication actually addition and here's a good question right so the point here is that in group theory in general when we have two elements of the group and we combine them together we call that multiplication or I suppose you could ask us the other way around why is what is obviously the operation of addition known as group multiplication in this context so the point here is that we are thinking of the integers in two separate ways and that's why it gets confusing and this is very very common in math this is one of the reasons why both in math and in physics or science more broadly it can be hard to learn things because the experts know what they're talking about and they can use the same words in two different ways and if you're not yet an expert you might get confused about that so the thing about the group idea the idea of a group of transformations is that we can combine them we can transform the triangle one way and then we can 4minute another way and we can then forget about the fact that it was a triangle we can just think about the set of transformations and we can say no matter what you act on you act on it by this transformation and death transformation it's equivalent to some third other transformation remember one of the features of a group was when you combine two transformations together you get a third transformation that is also in the group so we can forget about what we're acting on and just think abstractly about the set of different kinds of transformations and that is a leap of abstraction that can sometimes be hard to get on - so let's explicit us make this brutally explicit by distinguishing between the integers as a set of things and the integers as a group okay these are two different things so let's say the set of numbers the set of integer numbers that's what we're usually familiar with okay and let's call this n sub Z the numbers that are actually the integers okay I'm making this notation up this is not a notation you're gonna find anywhere else but this really means the numbers so this is the set of n such that n is an element of that dot minus 2 minus 1 0 plus 1 plus 2 dot okay the integers and they're numbers there's an infinite number of them there's no notion here in this set of anything like adding them together okay you can add that to it you can say okay here's the integers with the operation of addition on them but these are not transformations this is just a set right now okay and now we can distinguish between that set of objects at infinitely big set of object called the integers and the group of transformations which you might call what do you want to call it adding the integer K let's call it okay so that group we might write as G sub Z okay this is now a set of transformations this is not a set of numbers like the numbers are two means I have two bananas and you know I can add one more banana to that etc this is a set of transformation transformations not of numbers so this is let's call it alpha sub n is the act of adding n to a number and so is the transformation well let me let's write it this way alpha sub n is the transformation such that alpha oops let's call it alpha K is Ida called okay up there alpha sub K means alpha sub K of an integer N equals n plus K a new integer okay so this is a map okay each alpha sub K is a map from the numbers integers to n sub Z the numerical integers to itself for each K you have a map says shift everything over by K okay and then of course it happens to be true that there is because this is a group you can combine things together if you have alpha sub J acting on alpha sub K of some integer n well that's alpha sub J acting on n plus K which is just and put it up here equals n plus K plus J that's just this doing it twice doing this notion of shifting over by an integer twice and then you notice that that is of course the same is doing alpha of J plus K acting on n right alpha J plus K is a different shift it's a shift by the amount J plus K it is not by itself the combination of anything but it is equal numerically to n plus K plus J so what we do then following up on what I just said is we look at this and we say the integer n on which we were acting played no role okay it just went along for the ride it could have been anything at all couldn't even been a real number but we're gonna ignore that right now okay so rather than writing we could write alpha J plus K of N equals alpha J alpha K of n that's true this is saying shifting n by J plus K was the same as shifting in by K and then shifting n by J you knew this we can just write alpha J plus K moves alpha J plus K equals alpha J times alpha K okay now why do you want to call it x because in this particular example where what we're doing is the integers then of course we're just adding the integers themselves together but more generally more abstractly we're going to be taking two transformations and combining them composing them in fact would be the technical term here first you do alpha K then you do alpha J it might be the case that these do not commute for example of in more general circumstances that you can't do one than the other and be expecting it to be the same as the other and then the first one so in the more general group Theory context it makes sense to call this a product of two transformations because they won't always commute they'll have different properties and so forth here in the integers it is in fact the same thing addition and multiplication so the group product is not multiplying numbers the group product is taking two elements of the group and combining them together one followed by the other in the case of the integers that's the same as adding the integers themselves but that's a special case that's not the general feature and this is also why you will sometimes see people write things like so real people just write Z for either the set of numbers or the set of transformations so let's go back to doing that and they will write two copies of Z as Z plus Z or that's the same thing as Z squared right and there was a question like why in the world then we write that as Z plus Z or maybe a direct sum and the answer is just it's the same thing in this case in this case where it's the integers there is no difference this all this notation means in any of these notations it just means two different copies two ways to shift so this would be for example a symmetry group on two copies of the integers so let's say you add N and M and then you could take one particular value and you would shift it up both in the end direction and the M direction that will be the group Z twice and C squared Z times Z whatever you want to call it okay but really I just wanted to use this as an excuse to really draw out this distinction between a set of things and a set of transformations it's a quick leap of abstraction when you first faced with this but it's a one that it's one that is useful to glom on to okay that was one question here's another question I think this was implicit I don't know if this is explicitly asked but it was it came about remember our first example of a group was on the triangle I know when my triangle disappeared that's too bad here's the triangle there we go see if I can make it look a little bit more regular okay and one of the things we said was that you could flip the triangle around an axis so if we labeled this forget what we labeled it I know as ABC but forget what the original order was but we can flip it around the vertical axis right okay so that was one thing that we could do and some people were a little bit worried you know don't you have to lift at the triangle out of the plane to do that right when you flip it around if the triangle is part of this two-dimensional plane why are you allowed to do that and that this is actually you know a great question because it really brings to life what assumptions are being made and what assumptions are not being made remember if you go back to the geometry video that we did part of the insight that Gauss and Riemann had is that we can and should talk about geometry intrinsically to the object this triangle is clearly embedded in a two-dimensional plane I drew it there and if you were to define what the triangle was you might very well make use of the two-dimensional plane in which it sits embedded but when we get to this point where we're talking about the symmetries of the triangle and more generally the symmetries of other things and particle physics and quantum field theory etc we're not imagining that it's embedded in anything we're imagining it's an intrinsic object somehow for the circle so compare this to the circle for example here's a circle is this gonna become a circle who became an ellipse but I think I can make it more like a circle there we go so the circle one way to define a circle is to say I start with a point and I take the set of all points some fixed distance R from the original point from the origin now clearly that only makes sense if you're embedded in some two-dimensional surface you can generalize by the way this definition two spheres of any numbers of dimension the number of points the whole set of points a fixed distance R from the origin in n dimensions is the N minus 1 dimensional sphere but once we've done that once we made that definition we're now interested in features of the sphere that do not embed depend on that embedding likewise for the triangle up here were interested in features of the triangle that do not depend on the embedding so when we say a flip okay when we say a flip and we visualize it is bringing it out of the try bringing out of the plane and copying it flipping it over well what you really are thinking of is a map from the triangle to a self okay flip in the vertical direction forget what we called this should look at my old notes here but the point is the thing that turns it into this triangle BAC okay it is a map that sends for instance this point gets set to that point okay this point gets set to that point etc and the map doesn't require that we physically lift the triangle up can move it around it just means if you tell me where you are in the original triangle under the map I know where you are on the new triangle okay so that's why flips are just as good reflections or whatever you want to call them when you think about what the symmetries of these spaces or these objects are you should think about them intrinsically you should try your best do not visualize them as being embedded in something else that can be difficult to do but that's the goal to which we strive and one here's a related question I don't even think this is a question to be honest but I think this is something I should emphasize I said it in the original videos but I'm going to say it again which has to do with the non commutativity of these transformations you know the integers up here commute that is to say if I do a shift by N and then shift by M it's the same as doing shift by M and then shift by N the rotations do not commute so rotations and forget about the flips for a minute so so3 this is the set of rotations in three dimensions okay there's a couple questions about this one was why is it three-dimensional which is a good question let's get to that question first why is this a 3-dimensional group so the dimensionality of ASO 3 equals 3 whereas the dimensionality of so2 for example equals 1 does not equal 2 so what this means so3 is rotations in three dimensions of space for so2 it's kind of obvious that it's one-dimensional if you think about taking some vector in two dimensions x and y and you rotate it by an angle theta then there's only one number that you need to give me give me the number theta I tell you by how much I rotate it okay so the question for so3 was well for so3 here I'm gonna draw three axes X Y Z and I give you a vector right here okay and if I want to change that vector to something else well what I will do is I will tell you you know if I drop this down so I can give you coordinates for this vector this is Phi and this vertical vector is Theta I can just change both Phi and theta so if I map it to some other vector over here okay and it's supposed to be a rotation it doesn't look like a real rotation I need to better better-looking rotation here I'll rotate it this way okay so if I can rotate this I could just rotate by a certain amount of Phi and then again by a certain amount of theta and I'm done right it looks like the only requires two dimensions to specify a rotation in three to two numbers okay theta and Phi or Delta Theta and Delta Phi to specify a rotation so the there the answer to that is that in fact when I do so3 rotations reminder so3 I'm not just rotating one vector I'm rotating the whole three-dimensional coordinate system okay so if I want to let's make a different color I want to rotate this somehow I might send it to something looks like this so if this was this X prime Y Prime and Z Prime and this was X Y & Z and so if you go through and well I'll show you an example but you're gonna have to give me three different numbers to specify how all three of these change and that's kind of clear from here because if I rotate this vector to over here okay well I can also rotate around that vector in any way I want while keeping the first thing the first vector that I changed fixed so there's two numbers that I need to change one vector in three-dimensional space to another vector but that I have a whole one more degree of freedom that says I can rotate around that vector that is not specified by those two numbers so clearly I'm going to need three parameters and in this case there are different ways to specify what those three numbers are but you can sort of see from the picture that you're gonna need three different directions to move in if you like you can think about it as a rotation node X rotates around Y rotation around Z and that also helps explain why in this case so3 is non commutative okay which in the case in which the reason why it makes a difference the order in which you do operations I thought it would be fun or useful to actually show that a little bit more explicitly so if you have X Y Z because there's also a little bit of clarity is helpful in what you mean by rotation around X rotation around Y rotation around z so let's rotate that stuff start with our axes here and remember what we're doing is we are rotating all three axes at once and we're gonna start by let's say rotating around the Z so let's call this our Z and when we do all simple simplify my life I'm gonna rotate by 90 degrees in each case okay and one of the points I want to make is we still have X Y Z when we do our rotations we need to be able to specify with respect to what right so what I'm imagining is x y&z are fixed as our original coordinates but we're creating new coordinates we're transforming XYZ into some other set of three vectors defining different coordinate axes so we'll make those different colors so sorry XYZ it's still XYZ but now we're going to rotate 90 degrees around Z so this direction is going to be X prime this direction is y Prime and Z remains unchanged Z prime okay so you see what happened there you see how X moving to where Y was Y move to do minus X and so on and then let's do a rotation around X and this is why it's useful actually work out the example because once you've done that first rotation you might be thinking wait a minute when I rotate around X do I rotate around the original X or do I rotate around X prime and the answer is the simplest way that's the way the most people do it is you rotate around the original X okay so rx ry RZ rotations around the three different axes are all with respect to the originals so now we're gonna take our yellow one and we're gonna rotate that and we're gonna rotate it around your original X 90 degrees so X prime goes up to go where Z prime was so this is now X prime Y prime doesn't change because we're rotating its along the direction of X and that's all rotating along and Z prime goes to be minus y so hopefully you can all see at home how this is happening by rotating around the original X 90 degrees by the way the right hand rule comes in so if you want to say well what is 90 degrees versus minus 90 degrees you put your thumb along the direction of the positive original axis so I'm putting my thumb along the directions direction of plus X and that's how I know to rotate from original Y to original Z that's what it means so this is what I get when I do first our Z and then our X what would happen if I did first our X and then our Z let's verify to ourselves that we would get a different answer so this is X Y Z originally now we're gonna do our X so still have our originals X Y Z we're gonna make this yellow and 90 degrees round X means that X stays the same y is now going up here where Z was this is y Prime and Z goes down here so this is Z Prime and now I'm gonna do a rotation around z comparing to my originals XYZ and I'm gonna get what am I gonna get I am doing this in real time this is not practice so our Z moves around this direction so X prime is gonna slide over where Y is Z prime is gonna slide over to where X's and y prime is not going move let's see what are the chances I did that correctly I think I did that correctly yeah look at that so what you notice is when you compare these two together these are not the same right I did not end up in the same place by doing first a rotation by Z and then a rotation by X followed by rotation by X and then rotation by Z you get different answers so this is why this is just a nice illustration that I think I would be useful to make a little bit more concrete and explicit about how rotation groups can be non abelian can be can fail to commute in some sense okay then we had a question and this was asked by a couple people a couple different ways I'm not exactly sure what the question is so I'm going to talk about the idea of complex dimensions and I think that part of it comes down to the fact that there is always this confusion just about the word dimensions right we think there's two traditional ways to think about dimensions the word dimension and both of them are not quite applicable here one is like the completely weird science fictiony way like you travel to another dimension okay that is not what we're talking about in any of these lectures a dimension is always a direction you can move in in some abstract space so the other common notion of dimension is the real dimensions of space right one two three up down left right forward backward these are not that either when we talk about the dimensionality of Hilbert space or the dimensionality of a symmetry group what we're talking about is how many numbers do you need to specify that symmetry group or that space what are the number of real valued free parameters that you need to specify exactly where you are in this space and I said real values but but that's this is the point in some cases it is more useful to think about complex valued dimensions okay so what in the world does that mean so very quick reminder of what complex numbers are I know you know this if you've gotten this far already but it's just useful to fix it into your head we define I to be the complex unit the square root of -1 and in fact what we really mean is in some sense we have a two to two real dimensional space called the real values and the imaginary values of whatever we're doing whatever number were specifying and we specify this number to be one and this number to be minus one and then this number to be I and this number to be minus I and then any point in this thing called the complex plane can be thought of as a plus I B so it's very much like a little vector space it is a little vector space to be honest you have two components you have the real part a you have the imaginary part B and you might want to say well okay it's just a two dimensional vector space it's two real numbers that I need to specify one complex number why in the world am i calling this the complex numbers rather than just two real numbers what advantage do I get by thinking of two real numbers as one complex number and the answer is again if you're sort of following along my implicit guide to how to think mathematically the answer is there's a bit more structure on the space called the complex plane then there is in just a two real dimensional vector space so a vector space it's a space of vectors okay collection of vectors a vector space is a set of things that can be added together or multiplied by a number okay so vector space is you know V is set of things little v1 v2 etc such that we have to find an operation v1 plus v2 and we have defined an operation alpha times V V one with the property that alpha times V 1 plus V 2 equals alpha v1 alpha v2 so you can add together and you can scale by numbers that's what makes you a vector space and when you have a vector space you can work in components you could have basis vectors and that sometimes you know there's extra structure you can add to the vector space like what we call a dot product an inner product can you say whether vectors are orthogonal to each other etc but one way or another you can always say you can express these vectors v1 etc as some number a times a unit vector e hat one the put a vector sign over v1 so you remember it's a vector plus B another number times the unit vector e 2 etc and then you would have this picture you know in the one direction you would have a unit vector here he hat one and another unit vector there he had two and there that's a two dimensional vector space if a and B are real numbers elements of the real numbers then we say we have a real vector space so when we say real vector space we don't mean it's an actual vector space we mean that is a vector space where the coefficients the components in some basis are real numbers ok so what I gave you the definition I didn't say where the real numbers are not alpha or beta could be real numbers that could be complex numbers if they're real numbers it's a real vector space so clearly there's a strong family resemblance between a 2 dimensional real vector space and the complex plane they're both in some sense 2 dimensional but there's additional structure on the complex plane and it comes down to this fact I equals the square root of -1 or in other words there is something called I squared which equals oops my equals minus 1 so this equation this relationship is something that doesn't exist down here in just two dimensional vector space land there's no relationship between e 2 and E 1 other than they are all factors in fact they don't even need to be orthogonal you can have vector spaces without an inner product without a one dot e to even being defined as long as they are linearly independent you can still make this vector space but here you have this extra fact there is a way of multiplying numbers in the complex plane that has this relationship that I squared gets you to minus one I don't want to know what to draw there and this also comes along with the idea of complex conjugation that I star equals minus I so there is a map if you like from the complex plane that flips you around the real axis complex conjugation takes you from a plus IB down to a minus IB right that equals a plus IB star so the fact that there is a new product rule defined on the complex plane and associated with that there's this idea of complex conjugation that's something that doesn't exist in the real two-dimensional plane so the question should in some particular cage so do you think about things in terms of two real numbers or one complex number comes down to the question is there a physical meaning to that complex conjugation or to that idea of multiplying I by itself and getting you to minus one okay it turns out that in for things like particle physics when you have all these fields the electron field the quark fields and so forth the Higgs field these are most conveniently thought of as complex valued fields not all of them are the gluon field or the photon field these are real valued fields the gravitons roughly speaking a lot of the bosonic fields a lot of them not all of them but well let's put it this way a lot of the what is the best way to say it the neutral fields the neutral particle fields usually can be thought of as real valued and the complex ones can be thought of as complex sorry the the charge field can be thought of as complex-valued that's roughly the way it works out and why again very roughly like don't tell anyone I'm telling you this but a real-valued field is has the one number the real value okay whatever it's doing it might be a vector so it has components but they're just all real valued the complex valued fields have two different things that can go on there real part in there imaginary part and that roughly corresponds to being positively charged or being negatively charged okay being a particle or being an anti particle that's why you need a little bit more freedom a little bit more information for a charged field and the complex numbers are just right for that okay so if I have complex value complex vectors I would write again you know this is exactly the same formula v1 equals a he hat one plus B he had to with both a and B elements of the complex numbers okay that's what it means to have a complex valued vector a vector over the complex numbers a complex vector space so it's not that the vectors change but there's more information in them because you can meet both the real part and the imaginary part of each particular coefficient and then of course you had this extra operation you can take the complex conjugate etc so it turns out and this is just a little bit of extra information you don't really need to know but it's fun background knowledge this idea that you generalize the real numbers by adding this unit called I which is in some sense orthogonal to the real numbers and satisfies this equation y squared equals -1 so they're related to the real numbers in a very specific way can be generalized not arbitrarily but in a very very specific number of ways and this is you know you certainly don't know there's how many going to explain to you what it means but this is the this is the topic of normed division algebras to be honest I would not be able to give you this precise mathematical definition of this without looking it up but I know what they are there's only a couple of them a few of them they are the real numbers real numbers so that's so a is a real number let's say today is a real number okay see complex numbers are things of the form a plus bi b with I squared equals minus 1 okay so in the complex numbers there's the real direction and the imaginary direction so a single complex number is like two real numbers then there are also many of you will have heard of this the quaternions I think this is a simple for the quaternions and for reasons why we're not gonna get into you might thankfully you have a plus IB so you add is another dimension maybe shed two more dimensions rather than just one the answer is no you have to go all the way up to four so you have three different dimensions a plus I B plus J C plus K D where ABC and D are real and I J and K are these different kinds of directions you can move in in the quarter neonics space so one quaternion is for real numbers worth of information and just like I squared equals minus one likewise I squared equals minus one and that also equals J squared and that also equals K squared and that equals I times J times K okay so these this set of rules this is a product table right instead of a multiplication table for I J and K and you know that I I squared equals minus 1 K squared equals minus 1 I times J times K equals minus 1 so you might want to say well like what is I J what is I times J right I would like to figure that out do I need to give you more information well the answer is no you know that I a K equals minus 1 so let me multiply on the right by K okay okay and then k squared equals minus 1 so this is minus IJ and this is just minus K so therefore IJ equals K there you go you can figure it out so anyway these rules give you the attorney ins and they are they are a generalization with complex numbers so the point being you could have not only real valued vector spaces complex valued vector spaces you can have quaternion ik pal U vector spaces and finally you can have octo neon ik yes I'm supposed to tell you what the symbol is right I think it's just to know I don't know this these never actually come up in most physics Oh octo means and there you have you start running out of letters right so instead of writing ijk you write down one which is still the number one and then you write down e 1 e 2 dot up to e 7 so there's hope that looks like a 7 so there are eight different directions you can go in eight different real valued numbers equal one octo nyan and then there's a rule for you know what is e1 squared etc there's a whole multiplication number multiplication tables you can multiply any of the YZ by each other and get what the answer is the question you should be asking is how do I know what kind of vector space I have or what I should have what is the mathematical structure I should be using when I'm doing my physics problem and the answer is it depends on what the physics problem is so you know the fact that Schrodinger's equation applies to wave functions that are complex right the wave function is a quintessential element of a complex vector space Hilbert space how do you know well it turns out that's what you need to describe the physical phenomena so there are people who you know very sensibly say look there are these elegant mathematical structures quaternions octo nians and so forth I mean it not and so forth these are the only normed division algebra there you go I think that that's the right answer you could have other kinds of mathematical structures that do generalize even these vaults and then the question you ask is well what is the physical situation you're trying to describe there are physics-based reasons why the Higgs boson is a complex valued field and the photon is a real valued field the Higgs boson that's a terrible example isn't it because the Higgs boson is of course the Higgs boson which results from spontaneous symmetry breaking after the fact is a real-valued field that was a terrible example the for there are four components to the Higgs field before spontaneous symmetry breaking and those are complex numbers I haven't told you about that yet so you'd have no idea what I'm talking about so therefore you're confused but don't worry we will talk about that eventually better example the electron ok the electron is a complex valued field the photon is a real valued field why because electrons have electric charge an electric charge is either plus 1 or minus 1 photons do not have electric charge photons are neutral ok so the question is not you know here's the math let's fit it into the physics the question is what is the physics and now let's look for the math that conventionally conveniently helpfully describes it so in noting that we're gonna do for the rest of these videos our quaternions wrote onehans gonna be very helpful they could be in other examples but we're not going to get that far I mean I guess I never said this out loud but the philosophy of these videos the biggest ideas in the universe is not to really be very speculative you know I'm not talking about my favorite ideas in the world I'm trying to talk about the ideas that have established themselves as really really important so the videos aren't here to speculate about crazy ideas that I'm fond of many which I have nor are they to sort of chase the most recent news or the most recent wrinkle about you know particles being discovered and stuff like that I'm really trying to talk in these videos about the ideas that we know are very very helpful trying to explain the universe so su 3 across su 2 cross u 1 is very very helpful in trying to explain the universe otoni ins who knows maybe they are maybe they're not but that they do not qualify yet as one of the biggest ideas in the universe by our standards here okay let me talk about one more thing one final thing which actually goes right in to the fumble I just made talking about the Higgs boson which is the idea of spontaneous symmetry breaking now we'll talk about this a little bit better later on actually we'll be talking about this I don't even know we haven't talked about it yet but we we mentioned it in the last Q&A video because we talked about topological defects remember imagine that you have two scalar fields Phi 1 and Phi 2 with a potential V of Phi 1 and Phi 2 equaling minus mu squared Phi 1 squared plus Phi 2 squared plus lambda v 1 to the fourth plus Phi 2 to the fourth I think that works but anyway what I want is to have a so-called Mexican hat potential Phi 1 Phi 2 and then V of Phi looks something like this I'm drawing it here so that the minimum is not at zero but that okay it doesn't really matter I you just add a constant to it plus a constant okay and then what you notice for a potential like this where there is a Phi squared piece that is a negative coefficient and Phi to the fourth piece that is a positive coefficient the minimum of the potential is on top of a hill sorry the center of the potential is the top of a hill but the minimum of the potential is down at the brim of the Hat so there's not a unique single point that is the brim that is the minimum of the potential there's a whole vacuum manifold that's what we called this the vacuum manifold the space of possible states of lowest energy and what I said was reason why I mentioned this in the last Q&A video because the last Q&A was about geometry and topology and I said look the vacuum manifold in this case is a circle s1 right it has the property that PI 1 of s1 equals Z and PI 1 is the fundamental group it's the way that we can map circles into the vacuum manifold and the fact that we can map circles in a topologically non-trivial way into the vacuum manifold gives rise physically to cosmic strings because if you think here's space here's 2d space and let's imagine that in this region of space actually let me draw let me draw it this way we drove up with better here a little bit more careful here's space and here's the vacuum manifold of my scalar field theory the space of all field configurations phi1 and phi2 which minimize the energy and the reason why there are cosmic strings is field space here right here is secretly I'm not to draw the potential but this is phi1 and phi2 and the vacuum manifold has the property if this is the value v little v don't ask me why it's usually called little V Phi 1 and Phi 2 combine Phi 1 squared plus Phi 2 squared equals V that's the vacuum manifold the set of all possible values of Phi 1 and Phi 2 so they sit on that circle down there now we want to do is say like let's imagine I have a region of space which is actually this circle okay not the interior of the circle let me draw a circle in space and imagine that at every value of that circle the fields Phi 1 and Phi 2 are in their vacuum and volt okay so I have a point a base point and I map it to the vacuum manifold and if the whole circle gets mapped to just one point in the vacuum manifold and everything is fine everything is just topologically trivial nothing interesting going on but it can be to this vacuum manifold that this region of space gets mapped with a winding number it wraps topologically non-trivial e around the vacuum manifold and what that means is that somewhere inside here you're gonna have to climb out of the vacuum manifold that's the usefulness of topology in spontaneous symmetry breaking it can be the case that if your vacuum manifold is non-trivial their configurations in space which imply that you can't be in the vacuum manifold everywhere you cannot smoothly continue the field values while remaining in the vacuum manifold because they cannot be unwound there you go and therefore you say that here there is some energy right there and this is just two dimensions of space so I can just do it I can extend in the third dimension if this is Z direct direction then this region of space which climbs out of the vacuum manifold and therefore has energy because it's sitting up here at the top of the potential is gonna look one-dimensional it's going to look like a string so long story short if the fundamental group of the vacuum manifold is topologically non-trivial the field theory can give you cosmic strings what does this have to do with symmetries well look I mean look at this potential that I drew here I'm not I'm trying to hide the equation I'm not exactly sure the equations right but you can look at the potential and there's clearly a symmetry here right there's an so2 symmetry there's an so2 symmetry that rotates phi1 phi2 thought of as a 2-dimensional vector in a 2-dimensional space 2 Phi 1 prime Phi 2 prime right so if you think of it this way you've been drawing it like this Phi 1 Phi 2 and I can do a little rotation to make this Phi 1 prime Phi 2 Prime okay I can rotate it by a little bit and just as clearly so this is a this is a symmetry of the theory this is not just a that's nothing to do with spontaneous symmetry breaking yet this is what the symmetry is of this theory if I have Phi 1 and Phi 2 it doesn't matter what direction I call Phi 1 and what direction I qualify to they're both there okay and I can rotate them the physics is remaining unchanged so we know that so2 itself is a one-dimensional group so so2 it turns out well let me say more generally groups have topologies that's like escalation which should be the punchline of this what is the topology of this group well in this case what I mean by having a topology I mean that for all the topological invariants we talked about like the homotopic groups like PI 1 etc the dimensionality there's an answer for this group what that is so I can of course can talk about PI 1 of the circle how can I wrap one circle around another one and we've talked about you know wrapping circles around punctured planes or spheres or things like that I can do the same thing with a group so I can talk about PI 1 of so2 and it equals maps from the circle into so2 top of the set of topologically equivalent Maps the set of maps that can be deformed into each other so you probably have guessed this already but look the action of so2 is just a rotation here by theta by a single thing and theta can go from 0 to 2 pi so topologically it turns out that topologically so2 group of rotations in two dimensions is equal to the circle that's exactly what it is the set of all rotations in two dimensions can be parameterized by points on a circle how much if you rotate by there you go so therefore PI 1 of so2 equals Z it's the integers and you might now be thinking well oh good so the symmetry group gave rise through the vacuum manifold and that's why we know there are cosmic strings not quite that simple let me let me drive home how complicated this can get because it can get complicated pretty quickly here so what about you know another subject what about so3 well you might think you know we have three dimensions we're gonna rotate them so just like in two dimensions when we rotate it around a circle maybe in three dimensions we rotate around some kind of sphere right the sphere is left invariant by rotations but that's not true that's why we had the previous discussion about the dimensionality a 2-dimensional sphere is two-dimensional but so3 is three dimensional so3 cannot be the same as a two-dimensional sphere and has nothing to do with a 3-dimensional sphere right it seemed it seems to have nothing to do with it so there's a long story short but let me the point is the real point that I wanted to get to is just because I tell you what a group does what it acts on doesn't mean you can tell me what it's topology or it's geometry is right away it turns out that I won't go into details here but that so3 is a three dimensional sphere with an identification by a z2 which is to see there's a single discrete symmetry mapping different sides of the sphere so it's very hard to draw a 3-dimensional sphere imagine that instead of being two-dimensional which it really is this were a three-dimensional sphere okay so if this were a three-dimensional sphere this thing that I said divided by z2 I mean if I take this point on the sphere identify it as the same point as one on the other side of the Equator okay bike D this point identified up there with that point at this point right here as they didn't find it at that point right there so it's really just the top half of the sphere in some sense so this is all just a lesson trying to tell you well you know if you're if you're watching this far in then you're here for some of the fun things that I'm not going to explain in complete detail the point is that the topology of groups is very intricate and it becomes very complicated and the punchline of all this is topological defects when do we get a topological defect like topological defect is a cosmic string a domain wall magnetic monopole etc it is a configuration of the field which we know has to contain energy in some kind of dimensional configuration simply because of the topology of the vacuum manifold that's what makes it a topological defect topological defects happen when the vacuum manifold has non-trivial topology how do we know what the vacuum at a fold is here the symmetry group so2 was a circle and the vacuum manifold was a circle that is what will happen if you break the symmetry entirely but sometimes you break the symmetry partway and not the entire way right and then you have to think harder so the answer is topological defects you have a symmetry group and again I know perfectly well I'm not giving you enough information here to completely absorb what's being said but maybe it is a little hint or inspiration to go further so imagine I have some theory with a symmetry group G and I break it spontaneously and what that means is whoops spontaneously to a subgroup H subgroup of G so a smaller group embedded in the bigger group what does that mean that means I have some physical configuration which is rotated by the symmetry group but it's not left invariant by it so for example very very simple example let's imagine we have so3 okay so we have X Y Z this is real three-dimensional space but let's imagine it's just more abstract than that it could be real three-dimensional space but let's imagine some space of fields or something like that and let's imagine that just like over here the field would take a nonzero value it would point in some direction right so here this field if it were living right there the field is not invariant the field value living in the vacuum manifold changes when you do a symmetry transformation right the set of all field values doesn't change it maps into itself but the specific field value that we have in that particular location of space does change under the field configuration that's when you get spontaneous symmetry breaking so let's imagine you have a theory that is invariant under all so3 three dimensional rotations ok but let's imagine there's some object in the theory whether it is a vector field or some other set of fields which takes a value pointing along the z direction so in other words in empty space ok let's imagine there's a vector field that picks out a direction everywhere in space so then the underlying theory is invariant under the entire rotation group but that rotation would change the direction in which Z the this vector call it V is pointing in at least some rotations would change the direction if you rotated around the x axis or if you rotate around the y axis then V would change but rotations around the z axis leave V unchanged so there's an unbroken sub group which is just rotations around the z axis and that is so2 so in this case the original symmetry group G is so3 rotations in three dimensions the unbroken group is so2 these rotations that leave V invariant and then who is a long way to go much longer than I expected a long way to go to get to the punchline which is that the vacuum manifold is the group that you start with divided by the symmetry group that is unbroken the unbroken group okay this is called a quotient space and what this means is I take the in the full set of transformations in the original group but then I ask can they be related to each other by a transformation in the unbroken sub group and if they can I count them as one element of this vacuum manifold this quotient space called a quotient space for obvious reasons and it's very much like this example we gave here if you have a three sphere and you identify opposite points that's quotient anout by z2 here you would have so3 mod so2 that would be the vacuum manifold for this theory okay and in fact you can actually do the math there's a some computations involved or some more math for you to learn it turns out that topologically this space so3 mod so2 well what what's the simplest thing it could be how lucky could we get right of so3 is three dimensional group it has this weird topology we set up here it's as three points okay it's still three dimensional right so2 is one dimensional group right and it turns out that you can you can sort of check that this makes sense but the dimensionality of a quotient group G divided by H is the dimensionality of G minus the dimensionality of H because this is saying how many directions are there to go in G how many directions are there to go in H how many are left over okay so if you like this this makes more sense if you write the dimensionality of G the whole group is the dimensionality of H the number of symmetry groups left unbroken by this field living in empty space plus what is left over so so3 is three dimensional so2 is one dimensional this quotient space this vacuum manifold is going to be two dimensional what's the simplest thing it could be a two dimensional sphere and indeed it is a two dimensional sphere that's not that many two dimensional compact closed manifolds that it could be so nice what this says is if you start with so3 you break it by some vector down to so2 then you have a vacuum manifold which is two dimensional that's the sphere remember when we said up here could you think of so3 as a two dimensional sphere no you can't but if you break it to so2 then the vacuum manifold is indeed that exact two dimensional sphere it is the set of points that you could transform V into by an so3 transformation that's the vacuum manifold and so it makes perfect sense that the vacuum manifold is a two dimensional sphere just look at this picture look at this what V is doing and think about everywhere I could take it by rotating in someplace that fills up a single 2-dimensional sphere and that is why you know PI one of let's just tell you these facts PI 1 of s 2 is trivial you can take a circle map into a sphere you can always contract it down to a point but pipe tune of a sphere is the integers okay if you take a sphere and map it to asphere a two sphere method to a two sphere you can wind around once or twice or 0 times or -1 times or whatever so there are no cosmic strings in this particular theory that I have of so3 breaking down two so2 but there are mana poles one dimensional topological defects and in fact the same thing the same kind of logic works for grand unification theories in particle physics why do grand unified theories guts as they're called predict mana poles well because they imagine that you start with some symmetry group G let's just say it's G I'm not gonna tell you what it is well okay it could be like su 5 or something like that but there's lots of things that could be some simple group not let's just let's imagine it's not the product of two groups because then it wouldn't be really unified and we're trying to grandly unify everything let's imagine it's a simple group and then H the unbroken symmetry group the whole point is you imagine breaking the grand unification group down to the symmetry group of the standard model so we know what the unbroken group is at SVSU three across su 2 cross u one by the way giving more details here there's really a I'm gonna write this and then I'm going to unright it there's a secret that they never tell you but it's actually the symmetry of the standard model is this modulo z6 there's a z6 discreet subgroup that gets modded out in the standard model but nobody cares and it's not actually relevant for what we're saying right here so let's ignore that right now the point is that for any simple group g mu what happened there for any simple group g pi/2 the set of topologically distinct Maps from a two sphere into G / su 3 cross su 2 cross u 1 is the integers Z no matter what you do if you unify the gauge group of the standard model the symmetry group of the standard model into a single simple lis transformation group then when you break it down to the standard model there will be a non-trivial archly non-trivial vacuum manifold that will allow for the existence of magnetic monopoles that's why we know the grand unified theories predict magnetic monopoles and we haven't seen any magnetic monopoles so either grand unification is wrong or there's some mechanism that gets rid of them like the inflationary universe scenario or something like that again I have not given you nearly enough information to understand what I've been saying for the last few minutes but I want to give you a hint because it's the Q&A video right yeah in the regular videos you're supposed to understand everything but in the Q&A videos we can let our hair down a little bit and just let you in on some of the things that people think about when they're doing this for a living so this is the one of the one of the reasons why - Paula G is important in particle physics because it can predict the existence of real physical things according to different theoretical hypotheses your hypothesis is there's grand unification you know just on the basis of topology then you must predict the potential existence of magnetic monopoles one of the ways we'll see a few others how topology and symmetry are really important in understanding modern quantum field theory

Channel: Sean Carroll

Views: 27,913

Rating: 4.9225807 out of 5

Keywords:

Id: 7m-USntpQh4

Length: 62min 32sec (3752 seconds)

Published: Sun Jun 28 2020