I'm enjoying these videos. Just want to make two points. 1) The symbol for the quaternions is H, named after Hamilton. The Q is used for rationals. 2) In describing the topology of SO(3), you should be identifying antipodal points rather than reflections. If you identified reflections you'd end up with a manifold with a boundary.
I like your funny words magic man. Jokes aside, I think I understand what ur saying to be in reference to 50:19. Where the *Z* sub2 is because your identifying polar opposite points, not reflections that Sean accidentally drew. But my understanding breaks down as I don’t really know what ‘manifold with a boundary’ means. I don’t know the implications of the boundary, nor what manifold means here, nor if the sphere is hollow or filled.
16:00 SO(3) - 3 angles - like pitch yaw, and roll! en.wikipedia.org/wiki/Aircraft_principal_axes#Principal_axes Or just think of pointing your camera in space - azimuth (compass) and altitude (above horizon), and third angle tilt camera up away from real up.
You can think of the symmetry group of the triangle as all the ways to relabel the vertices so that the edges connect the same endpoints. So if A is connected to B, it has to stay that way after the relabeling. For a triangle, any way to rearrange the labels will keep the connections the same, but that’s not true once you have a square instead. This way of imagining it can help you compute the symmetry groups of more complicated things like a cube.
Great video! It's surprising how much of particle physics comes down to group theory and topology! (Although I'm sure there's lots of other stuff you haven't gotten to yet)
Congratulations to descend to basics, and especially presenting views that arev obviously your doctrine or understanding. Symmetries (SSYm) are a menace to the many worlds...Thank you
Suddenly I don't understand how SO(2) or U(1) are symmetries anymore. If I multiply by e^iO, I don't have the same value anymore. That's not symmetry. Why am I wrong? Also, when rotating in 3D, don't you regain associativity or something by rotating about the current axis, like x' instead of x?
Well, if you were right, there would be only one symmetry, namely the identity. The point is that if you multiply everything by e^iO you wil get the same geometric configurations.
@@freyc1 yeah looks like I missed the point about the Lagrangian there way back then. The Lagrangian is invariant to those rotations of state, not the states themselves.
great talk - love how when something he is discussing has gotten way more advanced than he probably initially intended to get (based on the demographic he first explained, somewhere in between videos designed for people trained in physics and those designed for non-math super vague videos for the GP you see everywhere) he starts speaking faster and faster and enunciates less lol ..... just like during exams when you are at a point in your answer where you are less certain you start writing smaller and messier. LOVE this whole series though -definitely a level of presentation that is filling a void!!!! I want more!
50:33 It is the antipodal involution (the reflection around the center) that you should divide the 3-sphere to get SO(3) (which also happens to be topologically the same as a 3-dimensional real projective space)
Wait... I'm supposed to understand everything in the normal videos? ... Will there be a test?
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Thank you so much for this series! By the way, tiny typo: around 4:00 you defined N_z and forgot to close the outer bracket and it's making me uneasy :-D
36:00 Cohl Furey has done a ton of research on octonions and their possible relation to phyisics. It's extremely heavy math but may be interesting to some viewers:ru-vid.com/group/PLNxhIPHaOTRZMO1VjJcs7_3dgyJ2qU1yZ
This is for fun (scifi) but i looking up monopoles i found this website describing what engineering with monopoles would look like including how the density and mass of a monopole would interact with matter. www.orionsarm.com/eg-article/48630634d2591
It seems to me a symmetry of moments can be defined by your non-symmetric squigle. Pick a point, draw tangent, add perpendicular to tangents and compute moments of this object. They will come out the same regardless of how you rotate, flip or translate. I guess this is saying the 'shape' is invariant, right?
1:15 Explicating Group multiplication = addition by distinguishing the Integers as a Group from Integers as a set of whole numbers. And a note on why it can be hard to learn things, as experts can use the same words for different things, and you won’t notice. 10:10 Why are allowed to flip the triangle? Beautifully, Sean tells us this is a good question. Because it’s encouraging us to explicitly discuss tacit assumptions. We can flip the triangle, because really the Flip is just a *map* from the triangle to itself but flipped, and so we don’t actually make use of any other dimensions. ‘Map’ is doing a lot of heavy lifting there but it makes sense. Remember, SO(n) is the Set of Rotations in n dimensions. SO(3) is the Set of Rotations in 3 dimensions, and the dimensionality of SO(3): Dim(SO(3)), is 3. Dim(SO(2)) = 1. Dim(SO(4) = 6. 22:40 Talking about Complex Dimensions. Rutvik’s video on why the square root of -1 is unreasonable effective is super helpful here. There is a description of oscillation or rotation baked into Complex numbers that’s helpful for describing some physical phenomena, for example the electron is part of a Complex valued field that has equal opposite charge. 39:30 Topology, Spontaneous Symmetry Breaking and Cosmic Strings. Didn’t grasp this part. I don’t really know what the vacuum manifold is. “If the Fundamental Group of the Vacuum Manifold is Topologically non-trivial, the Field Theory can give you Cosmic Strings. “ 46:30 “Groups have Topologies!” You can have topological invariance, for example Homotopy [pi sub1 is an example of that], Cohomology, etc for Groups. Also just because you know what a Group acts on, [for example SO(3) the Set of Rotations in 3 dimensions] doesn’t mean you know the Topology or Geometry of that Group [for SO(3) that is S ^3 / *Z* sub2, a (hollow?) hemisphere]. 50:50 a good definition of a Topological Defects. “A configuration of the Field which we know must contain dimensionally configured energy simply because of the Topology of the Vacuum Manifold.” I don’t think I understand what Breaking Symmetry means. 1:02:00 “Topology is important for particle physics because it can predict the existence of real physical things according to different theoretical hypothesise, eg hypothesising Grand Unification.”
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The vacuum manifold was explained in the previous video (14 not-Q&A) :-) But I also don't understand something, even after checking Wikipedia: what would a cosmic string "look like"? What would happen if I poked it?
@ I’ve been watching the whole playlist in order, but despite that I just *feel* like I don’t *really* know what a vacuum manifold is. Like maybe I can say some words that “sound right” - I could deepfake an understanding - but I don’t really understand what’s going on there. But to have an ignorant stab at your question, I guess you wouldn’t be able to ‘see’ the cosmic string. Because it might be something that is truly one dimensional, and so from any angle you look at it, it’d be too ‘small’ to see. But the deeper part of your question is pointing to the fact that I’m not really sure what a cosmic string would be made up of, and so I don’t know how light would interact with it as compared to the normal vacuum of space, and so I don’t know how a human’s visual systems would perceive it.
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@@ToriKo_ Thanks for the attempt! I put "look like" into quotes because I meant not just what it would look like to human eyes, but how it would interact with stuff around it in general. I just don't know what to imagine - "topological defect" makes some sense but does not give me any intuition. Other than the gravitational effects cosmic strings would have, I wonder if anyone knows :-D
It would be great if there were means to jump over the current answer to a question and skip to the next (in case I am rather save with some of the questions).
Is it coincidence that, if you had rotated the first around x' (rather than x), you'd end up with the second? I mean, does rotation in transformed coordinates relate to rotating in the original space in reverse order?
Hi Prof. Carroll. Thanks again for another great Q & A. And you are dead right - you gave way too little information near the end, for us in the Non-maths group, to follow. These are HARD going for some of us. But I keep coming back for more. And BTW, does the Higgs Boson have Anti-particle ? Thanks again.
@@Valdagast Quarks, being fermions with electric charge, are complex-valued. You distinguish between Electric charge and Color charge. We're referring to Electric charge here when discussing real or complex-valued fields. A quarks anti-particle is the same quark with the opposite charge and ant-color, e.g. A Red Up quark' has Electric charge = 2/3. It's anti-particle is ananti-Up quark with Electric charge = -2/3 and anti-Red Color charge.
