The Biggest Ideas in the Universe | 14. Symmetry
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- เผยแพร่เมื่อ 7 มิ.ย. 2024
- 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.
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#science #physics #ideas #universe #learning #cosmology #philosophy #math #grouptheory #symmetry - วิทยาศาสตร์และเทคโนโลยี
As noted by "markweitzman's wannabe a theoretical physicist school," I was wrong to say that O(n) is the direct product of SO(n) and Z_2. That's true when n is odd, but when n is even it's a "semidirect product." th-cam.com/video/yCxacFBLHIY/w-d-xo.html&lc=UgxJ8yI8fMwI7n2XMsF4AaABAg&ab_channel=SeanCarroll
Also, at 48:09: U(n) = (SU(n) × U(!)) / Z_n (some people write Z/nZ or Z/n instead of Z_n to avoid the confusion with p-adic integers).
My main man Sean! Philly’s finest.
Soul DFS 😂
If 2pac was alive, he would be a fan of yours. You can’t compose a 2pac without transforming a 1Pac
Hi Sean, an subject unrelated question: what brand / type of notepad/sketch board do you use to make those diagrams. I'm looking for a good one, but there are a lot. And yours is one of the best I've seen in action until now for the purposes I need it. It's hard to find good working examples on the vendors websites...
"4 does not equal 6" was the one thing I really understood in this video.
Hello, I'm mod 2.
"It's a reflection of the fact that there's some symmetry there." Indeed :)
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.
+
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!
Dr Carroll is informative, articulate, and engaging. Thank you for doing these sessions!
I love how we’re finally at Nöther’s theorem after 14 episodes starting with conservation laws. It’s full circle.
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 !!!!!!!!!
So very kind and generous of you, Professor Carroll.
Learning where the SU(2)×SU(3)×U(1) comes from was pretty rad. Thanks science man
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!
I look forward to these with impatience, thank you so much!
I love this series! You do such a great job at explaining complex topics at a down to earth level, Sean. I always enjoy when you take an abstract concept that I "learned" in college and put it in language that actually makes me understand all of the formalism that was taught in the classroom. Can't wait for the next video. :)
This series is just the greatest. Thank you, Dr. Carroll.
Sweet Notability skills, Dr. Carroll! You're the best and this series rules.
you really present these complex concepts fluently and comprehensibly... so much fun to see pieces of understanding creating mega-structural outlines...
ANoether great video!
Oh i m going to love this one. I cannot wait to get off work to check this out. As always, thank you Dr. Caroll :)
Thank you so much Prof.Carroll.
Great physics series.
admire Mr.Sean Carroll every time I listen to his lectures. I hope all of this material will be stored for the future and available to everyone just like today. Truly amazing topics and remarkable person by doing all this work. You can't get it in this form anywhere else at any Educational facility - they don't teach you like that. A great way of doing it Mr.Sean Carroll!
You are the teacher I always missed.. thank you dr Shen Carroll...
This series is so good, thank you very much Sean
Thank you for making these videos! I've never intuitively understood Noether's theorem and principle of action until now, fascinating example
I just want to thank Sean Carroll again for these videos. As they get more and more complicated, I am encouraged to learn more and more. The goal for me is to become a more educated viewer and reader of science journalism so that hopefully those journalists can feel confident in producing videos and articles of more subtle and nuanced areas of active research in physics.
Superb as expected for the normal people. The math is complicated and I had to look some stuff up (trauma doc so no math normally) but you give it in a way that makes sense. PLEASE keep this coming.
Beautiful concepts, appreciate you Sean taking the time to educate us mere mortals on these concepts which have been important to science in the past century.
Euler was just amazing person and his theorems and many corollaries are awesome.
Thank you for these videos Dr. Carroll. I wish that you were my professor when I took quantum mechanics. No knock on who taught my course at BU, but I really love how you explain concepts and I really like your manner. I just love this stuff and I'm learning so much and find myself laughing at your wonderful comments and asides.
Great video! Is it possible to get a copy of your notes?
What teaser, next time ... will come together and tell us something about the fundamental laws of nature.
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.
I really love how you explain complex things with such great clarity 🤗! I do remember many years ago when I did not have access to internet and such great teachers, how impossible was the thought that I would one dat be able to understand these things 🕺
Thanks for your thorough explanation. You filled in a few gaps left over from grad school.
Very excellent lecture l love learning this so much!
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.
As always - exscelent!
Great introduction in 1 hour!
I'm not sure I totally got all of that but it was amazing, especially the very end :)
Excellent series and very much appreciated. One small quibble about today's video: It is misleading to say that finite groups are "classifiable" (25:30). The finite simple groups have, indeed, been classified but non-simple finite groups (especially those of prime power order) are so numerous, it is hard to imagine ever obtaining a classification, at least up to isomorphism.
This is so helpful. I've watched many lectures and read several textbooks that have group theory in them and everything I saw just assumed that I knew some of these basic groups. I was always wondering why it is called SO2 etc.
Thank you for this
Your teaching explanations I get! Thx.
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.
Note: in _Noether_ , the 't' and the 'h' are not the English digraph for the fricative /ð/ (or /θ/). They are separate sounds. [ˈnøːtɐ] /ø/ is somewhat like "heard".
Looking forward to the upcoming video that covers quaternions and octonions.
U(1) is the rotation in complex plane right. Then how is the SU(n) rotation different from U(1) Rotation?
How are symmetries and conservation applied in non-euclidean spaces? Aren't there aspects of the Riemann curvature tensor that challenge Noether's theorem?
I have a question from a previous video and I apologise for not bringing it up in your earlier video. Why is it that within a black hole, where even light can’t escape, the gravitational effects of whatever is within the black hole can still be observed outside the event horizon? Why aren’t those effects “hidden” from observers outside the event horizon?
Is the e- field at each point in space a single complex value or a set of 4 complex values? I've heard of this 4 component electron field to account for spin and anti-matter, but it doesn't make sense to me for each of those values to be independent and orthogonal. If they were orthogonal, I don't see any reason for positrons and electrons to annihilate. If four values are needed, there must be some bound applied to those values, right? That doesn't seem orthogonal.
How are these values coupled to the photon field? What makes the difference in charge? Is there some way to encode spin or charge in one complex-valued field?
If you had a New York accent I would think you had got Alan Alda to dub over your voice. Good video, I think I need to watch each one of these 14 vids about a thousand times and I might fully catch on.
Sean,I'm not qualified in any scientific joundra but am interested, as I believe that you study time what would a model of the universe look like if time was slowing down. I think you would enjoy that
In the discussion of Noether's theorem, you say that S(AC) is the conserved quantity, but I would think this quantity would depend on the choice of ε and would be infinitesimal anyway. Would the value of the conserved quantity then end up being S(AC)/ε or (for those squeamish about infinitesimals...) d S(AC(ε))/dε (where C(ε) is the starting point after the small shift using the symmetry with parameter ε)?
at the end, you mentioned rotations in space and translation in time as leading to conservation laws. That begs the question on what happens if you perform a rotation in *spacetime* . After all, we're taught that changing inertial reference frames due to velocity is performing just such a rotation.
There is a conserved quantity, which is the spacetime interval and the speed of light. This is the one that I could think of off the top of my head, but I’m not sure if there’s another quantity that’s more fundamental
One common confusion on Quora is the confusion between the symmetry of the eqn. the lack of symmetry of the solution (such as 'our' world). The simplest is how can things change when 'physics' is time independent. You might want to explain that sometime!
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...
Lov'in the details. I enjoy the way you give just enough details to push my knowledge, with going "over the cliff". Nice job. Thanks.
And your said..."I predict that next week...". Yeah, yeah yeah. Let's see :-)
Love the floating head symmetry
I don't understand any sentences that contain the words "Lagrangian" or "Hamiltonian" which makes following this hard. I can't figure out what these words might mean from the videos, from wikipedia or any texts purporting to explain what they might mean. What are these things?
interesting. I have some experience with human anatomy and we have to use special language to describe physical locations of specific body parts in relative environments. I see now why math is needed to describe reality, because of how relative it is.
My anatomy and physiology professor would give test questions like put in order from inferior to Superior: a Angle of Louis, B organ of Corti, C Circle of Willis, d loop of henle. There would be fifty of these on a quiz.
If I was suddenly as smart as Sean Carroll, I’d be very happy! Well, for a while at least…!
Q: These symmetry groups, they were the same ones used to prove Fermat's Last Theorem (In ellipses), right?
Eliptic Curves are not the same thing as ellipses, just as isosceles is not the same as ice cream. (though the ice cream cone cross section is an isosceles triangle)
But yes, same symmetry groups being talked about. Look up "Monster Group"! Check out Numberphile on TH-cam.
I am familiar with the 1D complex plane (real and imaginary), but what does a 2D, 3D, 4D etc. complex plane look like and how many phases are there?
When the symmetry in the triangle was explained, it was 3 rotations around the center and 3 around the axis, what about the 3 symmetries around the edges? We have symmetry around AB, BC and AC.
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
Heisenberg and Noether most scinctlty describe nature's dance. The drive toward simplicity via Symmetry Conservation, vs. the conflicting clumsy uncertain left foot of Heisenberg. The asymmetry is vital even though the drive to simplification, the cancelling process, cleans up most of the slop. The slop left? Us
Things that can be cut or sliced into two symmetrical sides don't count?
So in your discussion on O(2), why is there only one flip symmetry? In the case of the triangle, you had three flip symmetries, and it didn’t matter that you could generate the other two from 1 flip and rotations. Following suit, I would think there would be infinitely many flips in O(2), parameterized by the angle of the axis around which you flipped.
I'm confused by this as well. If you can rotate by any angle and the circle doesn't change it would seem that you could reflect (flip) on any plane that crosses the center of the circle and you would get an unchanged circle. There seems to be an infinite number of those planes.
Upon reflection I realize that the infinite number of planes is just our ability to choose any point on the circle as zero. There is still only reflect or don't reflect, while the operation 1° rotation is different from an infinite number of other rotations.
The way I explain why it's so important to physics: Symmetry and Group Theory is the formalism of "compare and contrast". That is what you want when you go to consolidate and organize all your observations: "This is just like That, in the following ways..." and describing patterns separate from the types features and relationships forming the patterns.
I am going back 30 years in my Engineering Maths days. Higher maths for electromagnetic waves, thermodynamics and applied mathematics.... Optical fibers
Just a nitpick, but with one exception, those pairs of points (at t = 29:30) on the circle are not antipodal.
If a map from the original triangle onto itself counts as a symmetry, why doesn't a 360 degree rotation about the x or y axis count as a symmetry?
what a cliffhanger!
I don't know what it would feel like to live in a complex vector space so I don't know that my experience mismatches that. Argument from lack of imagination is not very convincing and I have tried to give more embodied meaning to such mathematical concepts as imaginary numbers. For example in spacetime metrics difference squared can be negative which would mean that the underlaying amount is complex. So it seems plausible for me that time is imaginary space and distance is imaginary time. There is a difference between not knowing whether that is the case and knowing that is not the case. Afterall I can't just declare that "particles are obviously the nature of reality" just because the ordinary human experience is very particlelike vs wavelike.
Hey Sean! What do you think of chemist Peter Atkins' theory of where the symmetry of the universe and conservation laws came from? His theory: in the beginning, there was nothingness. Now, this nothingness has perfect time symmetry and space (translational and rotational) symmetry. By Noether's Theorem, this implies it has conservation laws of mass-energy and momentum (linear and angular). And thus that explains all conservation laws exhibited by the universe today. Moreover, the nothingness had 0 mass-energy and 0 momentum, and the universe today does too. (If we sum up the positive mass-energy of stuff, it cancels out with 'negative energy' due to gravity. If we sum momentums of galaxies, they sum to 0.)
Thus the theory is that indeed in a physical sense, there is still nothing! So there is no problem of how something came from nothing at all! Out of nothing, nothing still remained! (It just changed 'form', for some reason)
Atkins' theory is elaborated in his book Conjuring The Universe.
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
Does the word Nerd derive from Noether?
For further info on symmetry in physics see my playlist: Symmetry and Group Theory in Physics:
th-cam.com/play/PLrYjnFgP8e0mt4oVaA_FGIdaXkE4-hbSz.html
29:34 flipping a circle does not flip "antipodal" points generally.
29:10 - Fascinating! #Flip ^.^
Are spinor symmetric? My palm is a spinor. Is it symmetric?
My apologies, but I’m going to be a bit dense here.
In the discussion of the symmetries related to the equilateral triangle it was repeatedly stated that the “members of symmetry groups” were the *operations* on the triangle (I.e. the various Rotations, Flips, Identity...)
However, when discussing Z, the integers, it appears “members of the symmetry group” were the *elements* being operated on (I.e the integers themselves).
OK, so what am I missing here?
In the integer group, the elements are the action of shifting over a distance. The thing being operated on was a line.
Ah, thank you. That helps a lot.
So perhaps another way of saying the same thing is that the "symmetry set (of operations)" is the (infinite) set of additions. Right?
if you had labeled the sides of the triangle with arrows and allowed flipping individual arrows, you'd get a bigger symmetry group too
Energy is conserved, but it becomes "more useless" eventually ending up as background heat. (I think). Does this imply there is some underlying asymmetry?
It is because of the 2nd law of Thermodynamics.
@@43timespersec Or is the second law of thermodynamics a manifestation of some deeper symmetry (or asymmetry in this case). Which are the more fundamental properties, symmetries or the second law of thermodynamics?
Just to add a little "evidence" to my thinking PBS space time th-cam.com/video/xFKgIOX8IRE/w-d-xo.html (from (about) 4 min 00)
Guess the interrelationships of these "ideas" is going to be revelead soon?
Dr Sean, why are Mathe things bad? I love when I can see equations and proofs. It makes so much sense.
It's more that it only makes sense to use mathematical symbols and formalism if there is actually a benefit for doing so.
36:45 COME ON SEAN, DON'T LEAVE US HANGING.
What are the alternatives to real or complex ?
( For "real" problems. I'm assuming you're not talking about the integers.)
Any "Field" can be used. Real and complex numbers are both fields, along with many many more. Fields have lots of nice properties that let us solve most simple equations! You can also use things other than fields (such as "Rings"), but they can be bothersome to do maths with in certain contexts. The study of rings and fields is a part of Abstract Algebra, which is a part of maths that I find endlessly interesting
I don't understand why SO(3) is 3 dimensional... any rotation of the sphere onto itself can be expressed with only 2 [0-2pi) real numbers ( theta, phi.. longitude, latitude)?
@Bob Trenwith Thanks! Yes. Call the point a pole, and the sphere can rotate around it. I did twig, eventually.
Yep... symmetry... got it.... thanks.
Professor Abel?
10:20 looks like a category.
16:00 Abelian groups have more requirements than commutativity. You need closure, associativity, an identity element, and each element require an inverse element.
34:30 lol I wondered if you'd conveniently stop at three dimensions! ;D
I'm not sure if the category comment was meant as a nod to structure you know is there or just an astute observation, but groups can be modeled as categories consisting of a single object (generally thought of as the set of elements) and where every morphism is an isomorphism (typically thought of as the right or left regular actions). This is frequently generalized to "groupoids", or categories with potentially many objects but where every morphism is still an isomorphism.
@@supertraced1391 Ok cool! I didn't know that! I wasn't referring to any particular category, only that it seemed to form a category. Thanks!
Someone’s been watching BG videos lol , when u stop learning u start dying
I dig the electric sheep still in the back
"electric sheep" !!! What are you smoking ? (and where can I get some) . . . . .
These videos send me to sleep every night
Symmetry discussion and Noether's theorem remain a mystery to me ;-((
I burst out laughing: "We live in a real world, did you ever see anyone giving complex coordinates?" Better than Descartes' convoluted proof of reality of existence.
wow, we got a super nerd over here.
** TH-cam viewers frankly lookup Group Theory videos **
Your definition of a radian was actually its reciprocal 1 radian = 360 degrees / 2 π
"for appropriate units" :)
I'll make one in which all equal to zero and win the friggin Nobel.
Quantum Healing and stuff, all go right in the theoreh
I think its worth mentioning that radians is just the distance around the circle if it has a radius of 1 for the angle. That is sometimes more helpful than just 360 which is kind of a random definition the Egyptians apparently picked because it was easy to divide.
@@3dlabs99 Egyptians or Sumerians?
I interpreted the statement as "the number of radians in an angle is (2π/360) * (number of degrees in the angle)", which is correct
I would be working at CERN if Sean Carroll had taught me physics in high school. Unfortunately, I got a public education system slave mill monkey and ended up digging ditches my entire life.
Unless you’re someone who’s main interest is mathematical physics (more power to you) and like getting in the weeds, or are getting some additional info to clarify or improve on academics, then this guy is uninspiring. Now I can of course forgive that, but I can’t help but feel his mathematics and physics worldviews fail to see the forest for the trees, i.e., the more metaphysical meanings behind things like symmetry. He expresses significant interest in it, as if it -and it appears to-has a sort of special quality to the nature of the universe. My assertion is that is Caroll is moved by symmetry for a reason he himself doesn’t consciously understand, which is symmetry’s centrality to a potential metaphysical description of reality.
Not trying to be too critical. I’m of course excited to see many of his videos.
Be sure to see 3blue1brown for "lockdown math" on understanding powers of e.
Love this series! However, your proof of Noether's theorem is not convincing. You can't say that two infinitely close trajectories have exactly the same action (well, maybe you do, but you didn't show that), so you can only say they differ by infinitely small amount dA. But then the whole argument falls apart, and you don't get conservation, you get rate of change dA/dx (partial derivative), which is not infinitely small or zero.