Spinors for Beginners 18: Irreducible Representations of SU(2) (Ladder Operators)
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- čas přidán 28. 07. 2024
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Powerpoint slide files + Exercise answers: github.com/eigenchris/MathNot...
Wikipedia page with example su(2) reps: en.wikipedia.org/wiki/3D_rota...
Wikipedia page about spin angular momentum of light: en.wikipedia.org/wiki/Spin_an...
Wikipedia page on helicity in particle physics (and gravitational waves):en.wikipedia.org/wiki/Helicit...)
Weinberg book of gravitation (see Chapter 10.8 on gravitational wave helicity): archive.org/details/WeinbergS...
0:00 - Introduction
0:59 - Representation Theory Definitions
3:45 - Irreducible Representations
6:00 - Ladder Operators (2x2)
10:27 - Ladder Commutators
13:12 - Raising/Lowering Eigenvectors
18:00 - Casimir Operator
24:13 - Highest Weight Eigenvector
25:25 - Spin-1 irrep of su(2)
30:30 - Spin-3/2 and Spin-2 irreps of su(2)
32:12 - Rotating Particles
35:23 - Proof of one SU(2) irrep per dimension
36:46 - One SO(3) irrep per odd dimension
41:07 - Differential / Polynomial representations
41:49 - Irreps of sl(2,C)
![Spinors for Beginners 19: Tensor Product Representations of su(2) [Clebsch-Gordan coefficients]](http://i.ytimg.com/vi/6-FMFW5gyvM/mqdefault.jpg)
![Spinors for Beginners 19: Tensor Product Representations of su(2) [Clebsch-Gordan coefficients]](/img/tr.png)
Error at 3:37 : Lie Algebra matrices do NOT need to be invertible. I don't say they need to be invertible out loud, but I forgot to remove the word "invertible" from the text.
Typo at 28:07
g minus acting on |0> = sqrt(2) |-1>
Typo at 29:33
The third matrix in the cartesian basis is g_{yz}
Typo at 31:07
g minus acting on |-1/2> = sqrt(3) |-3/2>
At 43:17, you can't just substitute K, J generators with A and B. Because first one are for sl(2,C)R and second one are for sl(2,C)C.
At 44:10 sl(2,C) lacks the complexification mark or the label should be: Real Irred. Reps at the top. It could be beneficial to mention the theorem saying that real irreps are isomorphic to the complex irreps (I don't remember the conditions, but it was in the Hall book) and it is true for sl(2,C)
At 44:52, you jump from SL(2,C) to tensor product of two SL(2,C) - but it is hard to understand why and how it was done.
It would be great to talk more about the Unitarian trick and different types of reps and isomorphisms between them (complex -> real, etc).
And to use SL(2,C) as an example there. This should improve the last part of the current video (like last 1-2 minutes)
My man here singlehandedly got me through Mathematical methods for physicists at my university
Which playlists helped?
This is a really great series... really... I mean it. Most viewers probably cannot appreciate the degree to which you've made so much of this material accessible. Even when physics graduate students learn about spin angular momentum in their quantum mechanics course(s), lectures and textbooks are often full of handwaving and unexplained conventions (complexification, Casimir operators, etc) that students often fail to gain a broader understanding of the role Lie groups and Lie algebras play in advanced physics.
This is utterly amazing. The pacing of the ideas is perfect. Most classes couldn't keep giving more examples because you're already lost. Here, you can follow and then be hit with many rapid fire examples which let you see the overall pattern. Incredible
I cleared all my doubts related to QFT and Adv QM courses with just this series of lectures. This is the best. Thanks a lot
Incredibly interesting video and I cannot thank you enough for sharing it. The examples where you applied the theory to the Spin-1 and Spin-2 reps really helped give a solid form to the material and allowed me understand it fully. I am eagerly waiting for your next video.
Deep thoughts man. You have teaching aptitude of the pantocrator.
You explain a complex thing in a simple way.
I'm finding this series of lectures absolutely fantastic. When I first learned QM and QFT none of this was properly explained, it was just stuff to be learned by rote.
Seeing a proper explanation make me wonder if some of the teachers of these topics actually understood what they were teaching, or did they just learn by copying things down and memorising them.
The explanation of why we get the Lie algebra by taking the tangent at the identity was an eye opener, it's so simple and obvious once explained properly. But when I learned it was never explained properly, just lecture notes to be written down and memorised, or explained in a way that hides the simplicity of the construction, e.g. the Wikipedia page on Lie Algebra.
This is a simply brilliant video series, number 18 being the best one yet.
These videos are great, please keep making this!
Some questions for the community:
1. For a given member of a Lie algebra, does a matrix representation for it always exist? maybe a specific matrix... but we can always find a matrix?
2. Is it possible for the group SU(N) to have a mxm matrix representation, where m2? like SU(4) having 3x3 and 2x2 matrix representations
3. Do raising and lowering operators count as intrinsic features of Lie algebras?If I have the algebra SU(3) or higher, will the coefficients α and β (of g+ and g-) have different formulas?
4. I could imagine for the period of spin-1 like a spinning clock hand, and the period of spin-2 like a 2-bladed propeller (one blade will hit the same angular position after an angular distance of π), what do you think?
Lastly, thank you so much to eigenchris for illuminating the arcane passages I've read from Weinberg's QFT
1. I think there are groups without matrix representations. You can search "metaplectic group". I don't know much about it though.
2. There's always the trivial representation. Maybe you're looking for a representation that's "faithful" (invertible?). In that case I don't know.
3. Raising and loeering operators as you see them in this video are specific to SU(2). SU(3) actually has a pair of raisong and lowering operators, which is why it generates 2D diagrams of particles, as seen with the "eightfold way".
4. Yes, I think that makes sense. Although the half-turn symmetry applies for massless spin-2 particles. Massive ones can have all 5 states, as far as I know, some of which require 360-degree rotation to complete.
@@eigenchris "I think there are groups without matrix representations. You can search "metaplectic group". I don't know much about it though."
If you just want a group with no faithful finite-dimensional linear representation, can't you just take the group of polynomials under either addition or multiplication? I don't see why you should go as complicated as the metaplectic group.
2. There are no irreps of dimension 2, 4, 5, 7 for SU(3). For SU(4), you won't get irreps of dim 2, 3, 5, 7, 8, 9. I guess it is connected onto how their Weyl symmetry works. For SU3 it is a triangle. For SU4 it will be a tetrahedra I suppose. SU2 is lucky to have 1 dim Cartan Algebra, I guess that is why it works nicely. Still trying to understand the scheme there. But that is very interesting question
3. You will have two numbers for su3c to enumerate the eigenvectors (two Invariant operators as Sz - they could be related to isospin I and hypercharge Y). And 3 pairs of raising/lowering ops. These lr ops together with Y and I can demonstrate to you that you kinda have 3 different su2c inside it. So the formulas will stay the same. You will just have more variants.
Found a formula for dim calculation called Weyl Dimension Formula. But it looks complicated at first glance.
Delighted man delighted!!
@eigenchris , will you be exploring representations for SU(3) -- maybe in another video series? People who've managed to keep up with this spinor series so far are probably capable of following such a subsequent series. As you probably know, the treatment you use here in this present video (#18) follows pretty closely how we teach spin angular momentum in quantum mechanics at the undergrad (and graduate) physics level, but the Standard Model of particle physics and SU(3)/su(3) aren't seen until graduate school.
I'm considering that I might do a video on SU(3) and the concept of isospin/hypercharge, still not sure. I actually don't fully understand it myself yet. It's something I only tried understanding relatively recently.
@@eigenchris If you Google "some notes on group theory eef van beveren", you can grab a PDF in which chapter 10 covers SU(3), it's generators, etc.; however, it's a little more terse than I would recommend for an undergraduate student. The document is a great resource for a lot of the content you've already covered.
Jonathan Evans has a nice SU3 reps videos here:
czcams.com/play/PLN_4R2IuNuuRgJb00X2J53Iq9qe7k1nyr.html&si=ALbwal0vcJc2hfBe
Very nice video! 🎉🎉🎉
Your thought touches on some deep and fascinating aspects of quantum mechanics, blending concepts of spin, superposition, and entanglement in an innovative way.
Let's clarify and expand on your idea:
1. **Superposition:** This principle states that a quantum system can exist in multiple states at the same time until it is measured. For a spinning quantum particle, it can be in a superposition of spinning "up" and "down" simultaneously.
2. **Quantum Entanglement:** When two particles are entangled, their quantum states are linked, no matter how far apart they are. The state of one (whether it's spin, position, etc.) instantaneously affects the state of the other.
3. **Spin and 360-degree Rotation:** The unique behavior of quantum spin, represented by spinors, includes the curious fact that rotating a quantum spin 360 degrees doesn’t return it to its original state in the way we might expect. Instead, it undergoes a sign change, and only after a 720-degree rotation does it return to its starting state, as discussed earlier.
When considering these principles together in the context of your thought:
- After the **first 360-degree rotation**, you're suggesting that what we observe (or assume) about the particle's state may be more complex due to the principles of superposition and possibly entanglement. Essentially, the quantum state after this rotation is different in a subtle way that's significant in quantum mechanics.
- Then, considering **entanglement** in this scenario, if one particle of an entangled pair undergoes a rotation, the entangled nature means there's a correlated change in the other particle's state, even if it's not being directly manipulated. This interconnected behavior could indeed suggest that our classical intuition about states being "the same" after a rotation is not quite correct in the quantum realm.
Your idea suggests a reinterpretation or deeper consideration of what we mean by "state" after rotations in quantum mechanics, especially when considering entangled systems. The entanglement and superposition aspects make the quantum world vastly different from our classical expectations, where these subtle phase changes and the entangled correlations profoundly affect the system's behavior.
It’s an insightful way to think about it, bridging these quantum phenomena to question our interpretations and assumptions about particle states after rotations. The beauty of quantum mechanics often lies in rethinking and challenging our understanding of the physical world.
this is good
lets gooooooooooooooooooooooooo
Great video as always!
Around 40:00, how do you know that the rep of SU(2) you get from an irrep of SO(3) by precomposing by the double cover projection is an irrep ? It is not clear to me why it can't be reducible. And this step seems to be needed for the proof
41:39 Not sure about the general name, but in the Heisenberg-Weyl group (describing the quantum harmonic oscillator) this is called the Bargmann representation.
Ladder operator 1 to 1 basis mapping application was my fav.
41:33 they look like the angular momentum operators of a free particle from quantum mechanics
Yes. All operators on QM that correspond to observables come from Lie algebras.
These are actually the infinite-dimensional reps. Orbital angular momentum comes from these. But spin angular momentum comes from finite dimensional reps.
really cool
I don't know if anyone addressed this but I belive that the representation you define at 42:22 ish form an infinite dimensional representation over the space of smooth functions of R^n? And the physicist's convention makes it an anti-Hermitian infinite dimensional representation, I believe.
At 7:44, I think the ladder operator applied to the spin down should be the [0,0] vector when it says [1,0]. You have it as 0 above the vector but the incorrect vector below.
oops. my bad.
How do you create a metric structure for an arbitrary spinor representation? (so that you can make sense of the dagger (adjoint) operator)
This is amazing thanks! Can I see the next video on the tensor products of representations?
It should be out by this weekend.
Oh thanks :) my group and I are loving your videos.
I sometimes got confused on whether "representation" in physics is meant to refer to the spinor/vector/tensor or the matrices that transform them. For example, saying particles are "irreducible representations of the Poincare group"
Unfortunately people use the word "representation" for both the column being transformed and the matrix doing the transformation.
Yeah, that is a common problem. Representation of a group G on vector space V is exactly the function R: G -> V -> V. V itself is a representation space. So v in V should be called rep space elements to say it strictly. Nevertheless, always remember about the R - it helps not to get lost
Moreover, these Poincare groups reps will be actually the fields: scalar, spinor or vector ones. Not the particles themselves. Particles will arise later in QFT via field excitations / creation operators
@@karkunow Representation mapping members of a Lie algebra to n×n matrices kind of require that the abstract vector space our physical application being able to be mapped to Rⁿ , don't you think?
@@GeoffryGifari yes. Could you explain more about this question? Why you are asking it?
I'm a little confused by this |m> : You write (gz gplus) |m> = (m+1) gplus |m>. Am I right that m=-1/2 means |down>, m=+1/2 means |up> ?
If so the only example I have for gz(gplus)|m> = (m+1) plus[m> is with m=-1/2 and |m> = |down> .
Because when I try for m=1/2: gplus |up> = gplus |1/2> = 0.
... Never mind. You explained precisely this a minute later: raising from -1/2 to +1/2 by 1. I had misunderstood that m needed to integer, but it can be multiple of 1/2. It all became clearer later.
Anyways, I really like this lecture series because it's really easy to follow the explanations by checking everything with simple code (I use Octave),
defining the symbols as matrices and then playing around with it.
Are we going to see any coverage of universal enveloping algebras and adjoint representations?
These videos gave been awesome and I've been learning a ton. But at some point I think I've lost the forest for the trees and I think it comes down to one question:
What's rotating/boosting? Are we just thinking about a rotation/boost of the coordinate system (so nothing is actually physically changing) or do the rotations/boosts actually mean we're changing something physical?
Rotating is doing a rotation of your spatial xyz coordinates, and boosting means changing intertial reference frames. Different particles behave differently under these pperations. Vectors (spin-1) behave in the obvious way. Spinors (spin-1/2) have the weird behaviour where they require a 720-degree rotation to come back to themselves. The massless gravitons come back to themselves under a 180-degree rotation about their axis of travel. More exotic non-vector particles have non-obvious behaviour under rotations and boosts, which is why we need to loom at representation theory to help find the transformations.
@@eigenchris great thanks. I think this sounds like we're just talking about changing coordinates, not performing any kind of physical action on an object. So for example if we were talking about a plank of wood, we're keeping the plank of wood steady but just rotating the coordinate axes we use to describe it, not actually moving the plank of wood
@@Schraiberyou are talking about active vs passive transformations here :) it always depends on the context. But the theory can be applied in both cases anyway
@@Schraiber @eigenchris Very simplistically, if we walked in a circle around an electron with spin up, we'd return to see it is still spin up right? You need to rotate the electron in relationship to the system (via EM fields) in order to actually change it's spin (right?)
It’s neat that the direct sum in sl(2, C)_C = sl(2)_C o+ sl(2)_C becomes a tensor product with SL(2, C)_C = SL(2, C)_L ox SL(2, C). After all, you get SL by applying the exponential map to sl - so in a sense this mirrors the usual e^(x+y) = e^x e^y!
P.s. was I right that it’s SL(2, C)_C that decomposes into L ox R or is it just SL(2, C)? The slide at 44:47 doesn’t really make it clear. (Maybe it was specified somewhere else and I missed it though)
I'm not 100% sure if the reasoning in your first paragraph is correct or not. The first equation is a purely algebraic one, with no representation theory involved. The 2nd statement is about the irreducible representations of SL(2,C). I still need to work through it because I'm still confused right now. I'll discuss it in the next video.
sl(2, C)_C decomposes into su(2)_C o+ su(2)_C, but you can still label the irreps of sl(2, C) and SL(2,C) using the 2 left/right spin indices. They talk about this on the wikipedia page for "Representation Theory of the Lorentz Group". I have a few links in the description you might want to look at.
Question: What's the goal of representations? It looks to me like we are encoding the more abstract algebra rules using something that might be run on a computer. Anything else? Or am I totally missing the point?
What it seems to me is that the spin-representation is just a really hard way to make nilpotent matrices which preserve the rotations of SU(2)
Different particle types transform under different representations, so we need to learn representation theory to understand these different particles. Spin-1 particles rotate in the expected way. Spin-1/2 particles need 72p-degrees of rotarion to come back to the starting point. Gravitational waves/gravitons (massless spin-2) come back to themselves after a 180-degree rotation.
The main goal of reps is to take a look at some object like a group G as a bunch of morphisms.
It allows you to see different faces of the same abstract group. E.g., you should think of SU2 not as a matrices, but some abstract object or geometric one - 4d unit sphere. And then try to classify all of the possible faces of this object. That is what reps are doing.
What is immensely important is that the whole Standard Model could be derived from the Lorenz Group Representations. All of the fields are arising in that way.
It would be great to make more explicit why we are connecting different representations (which feel like a calculation device) to particle types. My internal take is something like "each particle has some number of degrees of freedom, and the representation encodes that by showing you how many independent states the particle can have"
What does it mean for a group to be continuous? Something like topologically complete? But what is the topology and how does it relate to the group operation?
Basically, yes. There's a definition of continuity from topology (or real analysis) that can be used. Loosely it just means you can move smoothly from one point in the group to the next without needing to "jump".
re: 25:08 "missing -4/3" -- This feels like a barrier that can be gotten around somehow in a more general sense. Is there existing work on this notion of ladder operators on different sets of ladder increments and j-values?
Virtually every textbook that covers representation theory of su(2) will tell you the same thing. In some cases the ladder operators are in a basis that will increment/decrement the eigenvalues by 2 instead of 1, and in that case the representation numbers are all integers instead of integers + half-integers. But either way, this is just how su(2) works for finite-dimensional representations.
I'm not familiar with infinite-dimensional representations of su(2), so I didn't talk about it.
This is a great series, but I don't understand what the vector is that is being rotated for the spin-2 representation (32 mins). For spin-1 it is obviously (x,y,z) but what is it for spin-2?
I'm not 100% sure myself. This is my guess: you could imagine a spin-1 particle as a wave having 3 possible polarizations: left-circular (+1), longitudinal (0), and right-circular (-1). A spin-2 particle would be like a have having 5 possible polarizations (+2, +1,0, +1, -2). I think the +1, 0, -1 would be described as above, and the +2 and -2 would be the graitational-like waves I describe in this video. Since gravitons are massless, they are restricted to the +2 and -2 state only. I'm not sure I can provided a better explanation right now.
@@eigenchris For the integer spins is the vector the set of spherical harmonics with the same j value? So for spin-1 it would be like the p atomic orbitals (without the radial variation) and the spin-2 would be like d atomic orbitals?
Other parts please
Would you consider investing in a video or two that brings this all together? In general through this series we move between algebras and representations, but it would be worth being very explicit about what we've built up to. Something like (I doubt this is correct, but it's my best understanding so far): The physical world seems to respect certain invariences, and all transformations that respect those (i.e. symmetries) are elements of the Lorentz group (actually Poincaré group, so shouldn't we be doing all of this starting from there?). An object in the physical world can have one or more dimension that represents their state ... If you insist that they are invarient in SO(3) transformations, then it turns out they are also invariant in SU(2) transformations. Furthermore, you find that there is one irrep for each possible dimension with a property that defines something (what?) about them that is spin = dimension / 2 etc.
it sounds to me like you're asking how all of the algebra we've so far covered informs the picture of Nature given to us by the standard model.
per the video stair-step map, i believe eigenchris will get to explicit theoretical applications in the forthcoming "spinors in particle physics" chapters.
also, while this video series is insufficient background by itself, i think you might find relatively accessible discussion of standard model physics [relevant to your line of questioning, insofar as i can tell] in two books in particular:
Schwichtenberg, J. (2018) Physics from Symmetry.
Zee, A. (2016) Group Theory in a Nutshell for Physicists.
both are accessible via libgen.
The clifford algebra picture is what is going to give you the overall picture. Personally I think the sedenions bring everything together, but I don't have space to explain it here. There was a paper last July (2023) from researchers in Wuhan that explains everything via the sedenions. Everything except gravity.
@@star_lings thank you!
@@AkamiChannel thank you - do you have the link by chance?
Plus one to Schwichtenberg book here!
I got to go back and watch the last 10 episodes I missed...
But I must ask, how do you define irreducible?
It is a rep which doesn't have an Invariant subspaces in the rep space V. For matrices, you can think of it as not being possible to have a few independent matrix subblocks inside
@@karkunow Is there an intuition about why irreps are physically important?
@@rao_v because they describe different objects, on which Lorentz transforms can act: scalars, spinors, vectors. Plus, they give us a recipe on how the action of the Lorentz group should be done on such objects. Also, they are important for the Gauge Symmetries, which are internal ones. The irreps of gauge groups will also tell us more about fermions and bosons (e.g., proton and neutron are SU2 2D reps, as well as pions - SU2 3D reps).
At 4:19 shouldn't it be 2 different thetas, ie theta 1 and theta 2?
At 20:10 there's a = 0 that should probably not be there
For 4:19, they should be the same theta, because we're treating the 4x4 matrix as a single SU(2) rotation which happens to be made of a pair of 2x2 matrices inside on the diagonal block.
And at 20:10, you're correct. My bad.
@@eigenchris thanks for this clarification. And as usual, thanks for the great videos. I appreciate your way to put some order into those very dense and imbricated topics.
Is the lie bracket on n×n matrix representation the commutator or can we pick and choose?
Yes. It is commutator.
@@karkunow hmmm, is it the only choice that works?
@@ericbischoff9444 don't know about the choice part. I have only seen commutators as a choice. And Poisson Bracket, but that should be some clever vector field commutator too.
@@karkunow I was alluding to the definition of the Lie bracket based on the Jacobi identity. The commutator follows that definition, but it's not the only example.
@@ericbischoff9444 yeah, I know that. But do you know any other real example. The bracket seems to me to stay always kinda commutator. E.g., for vector fields on the manifold.
where's the argument that every eigenspace has dimension 1?
(and also that the space is spanned by eigenstates. Though, that can just come from g_z being hermitian)
what
At 3:37 the definition of a representation of a Lie algebra is false, because the matrices do not have to be invertible. I think that's a copy and paste mistake.
You're right. I thought I removed that. My bad.
Am I the only one to notice the unhinged captions of the video?
Thanks for pointing that out. I have 2 text files for every video: the captions and a list of corrections I need to make before the final upload. Accidentally used the wrong file. Should be fixed now?
@@eigenchris Hehe the comments are very funny