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

The Biggest Ideas in the Universe is a series of videos where I talk informally about some of the fundamental concepts that help us understand our natural world. Exceedingly casual, not overly polished, and meant for absolutely everybody.

This is the Q&A video for Idea #14, "Symmetry." A mixture of abstract thoughts about the integers, what it means for dimensions to be complex, and how we know if a theory has topological defects.

#science #physics #ideas #universe #learning #cosmology #philosophy #math #grouptheory #symmetry

[AthanasiosGiannitsis]

Thanks to this video, first time ever I understood what is the physical interpretation of "symmetry breaking". Feel grateful

[Nathsnirlgrdgg]

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.

[tomhoward]

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.

[lpt]

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!

[ColbyNye]

As always, thank you for continuing this series! Extremely informative and it's much appreciated!

[aresmars]

16:00 SO(3) - 3 angles - like pitch yaw, and roll!
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.

[rtravkin]

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)

[kobevli]

Hey Dr. Carroll !

Thank you for the video, and thank you for all you are doing for the field of physics !!!

[dt]

thanks dr.carroll i dont understand much but i watch to feel smart

[kobevli]

The only thing that would make this video better is if it was x10 longer !!!

Thank you so much Dr. Carroll, this series is very much appreciated.

[icesrd]

Thank you Dr. Carroll. Another excellent Q&A... please keep'm coming!!!!

[kevincleary]

I felt relieved to hear him say in the last minute that he hadn't given us nearly enough information to understand what he said.

[ToriKo_]

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.”

[mgenthbjpafa]

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

[amaarquadri]

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)

[beamfunk]

Love this web series. What presentation tools are used for this series? I am trying to do something similar for my classes next year.

[rbettsx]

Ahhh, thank you so much for answering my question....

[reinerwilhelms-tricarico]

why is it called "spontaneously" broken symmetry?

[nemuritai]

'Photons are real-valued, electrons are complex-valued'. This is is so interesting, I hope the link with charge is elaborated in future videos.

[traruhsynred]

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?