The Biggest Ideas in the Universe | 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 Idea #14, "Symmetry." Different manifestations of symmetry are at the heart of much of modern physics, so it's worth looking at how we should best characterize it. That's leads us into the mathematics of group theory.

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

[mironmizrahi]

Hi Sean,
I am a big fan of your work and a Patreon supporter of Mindscape. I wanted to express my deep gratitude to you for doing these videos. They fill a massive gap I have had for ages where the 'usual' lectures (by your and many others) have been too high level and a proper physics course too detailed and time consuming. I am sure there are many others like me, who are not laymen but also not able or willing to study physics at a university. Yet we are fascinated by physics and want to learn as much as we can. If anything, I wish there was more math in these videos:). But they have been, nevertheless, something I eagerly await every week. I sincerely hope you will continue until you are done even if the whole Covid situation improves markedly.

[Valdagast]

"4 does not equal 6" was the one thing I really understood in this video.

[DrJeeps]

Dr Carroll is informative, articulate, and engaging. Thank you for doing these sessions!

[Cooldrums]

OK. So I have a BS in NucEng and an MS in EE. This is the first video in the series where I really need a homework set assignment to help me understand the details. I'm starting to think I owe the good professor a tuition payment

[photinodecay]

"It's a reflection of the fact that there's some symmetry there." Indeed :)

[paulc]

Thanks again Prof. Carroll, for another great video.
Although I watch all of them twice, and I consider myself a scientist (Chem & Biol), some of the Maths involved does leave me feeling way out of my depth; sorry to say. But of course, I appreciate that you have a very wide range in the audience, from very basic level maths, up to full math-geek level. But every episode is always interesting and well worth watching.

[pablogh]

Excellent video.
You have given me a new understanding of mathematics, about how it arises etc, and made me see the connections between fundamental concepts that I would not have appreciated before.
For all that, thank you very much!

[vancouverterry]

I can't thank you enough for what your videos are doing for me, Prof Carroll. I am an intellectually-active 70-year-old who needs to dig up and dust off some math and physics I learned and half-learned almost 50 years ago as I need them now in a model I have constructed. For me, your detailed, thoughtful, example-filled videos are tremendously valuable because not only do they solidly develop the fundamental ground, they also are a one-stop-shop for updating one's knowledge of some things, for example entropy or quantum physics. It certainly is great getting the clear views you give about the present understandings, terms, and some of the histories to getting there. You're leaving something of significant value on the internet with your videos, both for new learners and for re-learners, as well as for anyone who wants to verify if their current understanding is still sound. Your videos are truly a bang-on job of spreading out the field in front of people and explaining it in crystal-clear detail as you do it. Many superlatives, Sir! Thank you very much!

[Kafiristanica]

This series is so good, thank you very much Sean

[nujuat]

I love how we’re finally at Nöther’s theorem after 14 episodes starting with conservation laws. It’s full circle.

[bohanxu]

this Feynman way of intuitively understanding neother theorem should be mentioned in every relevant class. it's a shame that this is the first place I heard it (no disrespect to Sean Carroll's wonderful class of course)...given i'm a second year grad student

[paulfrischknecht]

In other words, symmetry is another word for universal quantification or law (of nature) . It basically says that some equation or "invariant" applies for all things from a certain set (a set of some geometrical transformations for example, or all points in spacetime). In other words, it is just a way of saying "no matter what (you do)/where/when... something is always true". For specific applications, you need to define the invariants you are interested in and the set you are quantifying over...

[markweitzman]

I guess this is similar to what Zee talks about in his group theory book that technically its not U(N)=SU(N) X U(1) but rather SU(N)/Z_N x U(1) = U(N) - Group theory in a Nutshell for Physicists p. 253

[ToriKo_]

00:00 Symmetry. And a brief note on the relationship between mathematicians and physicists.

00:04:15 Symmetry and Transformation definition. A Transformation is a map from the object to itself, a set of ways you can manipulate the object to itself.

00:08:43 An short explanation of non-trivial changes. A trivial transformation is, in a certain reference frame, where nothing is done to the triangle.

11:30 Interestingly, the 6 Transformations of the triangle form a Group. I haven’t come across an example of a Group like this. Note, Another Roof is where most of my understanding of Groups comes from. This whole section is very good. What does Abelian mean? It’s when the orders of operations performed matters. What are subgroups? Commutativity? Dihedral Group vs Permutation Group S sub3?

21:00 Integer Group. What does *Z* sub2 mean? It means a finite group with n elements, where n = 0. Think of a clockface, where 12 = 00 : 00. So *Z* sub2 goes from {0, 1, 2). And only has 2 elements.

26:30 Continuous Groups aka Lie Groups. Wow. I didn’t know that’s what Lie Groups were!!

Radians. Remember, a insightful way to think about Radians is that instead of relating the angle to you via degrees, they relate the angle via the relationship to the circumference. If this is a unit circle, Dia(pi) is the full circle, [ Dia(pi) / 2 ] is 180°, and [ Dia(pi) / 4 ] is 90°. And some of that nice intuitional look is removed once you convert Dia to 2r, and as it’s a unit circle, covert to 2(1), so all in all ur left with 2(pi) = 360°.

00:29:20 Flipping antipodal points on a circle. This is a link to the Topology video that I’ve forgotten

00:34:50 Why there’s 6 degrees of rotation for 4 Dimensional space. Failure of your 3D brain and rotating axis into each other.

36:20 Complex Vectors, Unitary Vectors. This part seems crucial, and requires watching the whole video to get it, which I didn’t, so I should do that.

Man I really regret not watching this one in one go. Definitely lost a lot of momentum in intuition by not doing that.

50:00 Symmetries and Conservation, Noether’s theorem. How laws of conservation are embedded by symmetries.

[truejeffanderson]

If we imagine "i" as more fundamental then "π", then we gain the benefit of measuring in right angles instead of radians. Because e^(i*π) = -1 actually says: e^(i*π) = i^2 or e^(i*π/2) = i, which says that i is equivalent to a right angle turn. And we can say e^(i*radians) = i^right angles.
Measuring in right angles is sensible.
To complete this we just modify our calculators to accept right angle measures for cos() and sin(), since I think the radian input is an arbitrary selection based on the theory that the perimeter distance of the unit circle is significant for angular calculations.
Once that is complete we can say: i^x = cos(x) + i sin(x) ... just like we did with e^ix, but with having to use radians. Maybe π = ln(-1)/i

[davidhand]

It seems like symmetries are all about throwing information away. With the triangle, if you include the labels of the vertices in your object, they aren't symmetric anymore. So when we say there's a symmetry, what we really mean is that there are variables/information that disappear under some form of evaluation. For example, if we have some F(x) = x^2 where x is a real number, and we can only measure or know F, but we are promising anyway that x could be negative, we just would never know it. It gives two solutions for x for every value of F, there is a binary degree of freedom in F.

Knowing that continuous symmetries underlie the gauge fields and their corresponding bosons, is it correct to say that the forces of nature exist not because there are internal degrees of freedom at each point in space, but because those degrees of freedom are immaterial at some level? For example, Psi being "squared" (PsiPsi*, not PsiPsi) when observed disposes of the phase, but we suppose that Psi is, in fact, complex anyway. It has a degree of freedom, but our form of evaluation obscures it.

Therefore, we can relate symmetries to entropy. Entropy is literally a measure of hidden degrees of freedom. For example, we can measure only the P, V, and T of a gas, but there are actually several degrees of freedom _for each molecule, _ an unfathomable number. The domain of states is far larger than the range of our measurement, and that is the entropy of each {P, V, T}. The difference in degrees of freedom for the quantum gauge fields is much, much smaller, but it still seems appropriate to assign each of the fields an entropy, right?

I don't think it matters very much because it would seem that the entropy would be a constant; there are never more or fewer components to each of those fields, and their macrostates (the observed or manifest boson particles) are similar or identical to one another. What we measure is typically reducible to a number of binaries: was there a particle here or in this state or wasn't there? My instinct is that every binary hides the same number of variables. Thus, there would be no gradients of entropy and therefore no work to extract from it. Unless, of course, we consider the expansion of space to be the creation of these new hidden degrees of freedom, in which case entropy would be made to increase through the expansion. That's kind of interesting, maybe.

[ahsanrubel]

You are the teacher I always missed.. thank you dr Shen Carroll...

[reidakted]

When do we start talking about the Premium Model of particle physics? 🤣

[mandarkhadilkar]

Euler was just amazing person and his theorems and many corollaries are awesome.

[nightwng]

Learning where the SU(2)×SU(3)×U(1) comes from was pretty rad. Thanks science man