As a mathematical physicist, I really like this series. Please continue :) Maybe, you are interested in a celebration livestream when we both reach the 100k subscribers? :) Could be fun!
about to take an exam on gauge theories and QFT been struggling to wrap my head around spinor fields and this video has came out almost perfectly timed, tysm !! :)
This is a noteworthy Extreme Super-Synopsis of the exceptionally broad depth of nomenclature for spinors throughout the decades of physics. Congratulations to 'eigenchris' for this attempt ...VERY WELL DONE!...That said, spinor utilization should be exercised with care (and is undoubtedly subject to future modifications)...(Feynman had noteworthy comments about Dirac... and the same could be said for Pauli and Weyl, in a manner as Teller commented about Einstein's post 1919 attempts to advance physics.)
@@eigenchris do you have plans for the next series topic? I would love to know what topics you have in mind. I would love a series in quantum field theory. By the way, my plan for this weekend is to watch your series in relativity haha.
@@GustavoOliveira-gp6nr There probably won't be a next series, honestly. I'll make one-off videos now and then, but I'm pretty burned out from making long series. I will touch on some QFT in videos 21-25.
this is one of the most useful series i have found and is the chirality presented in this video related to the chirality in QFT is how and why (i am a big fan of your work)
Yes, the Chirality shown here is the exact same as the chirality used in QFT. The Dirac Equation I know near the end is one of the main equations used in QFT. It's the equation of motion for spin-1/2 particles and keeps track of both the left-chiral and right-chiral portions of the state.
@ozzymandius666My understanding of QFT is more phenomenological than mathematical, but *any* mechanism by which particles with chirality gain mass, must mix left and right chiral wave functions in order to conserve angular momentum, not just the Higgs mechanism. The only particles for which chiralities don't mix are massless particles and particles with no chirality.
@ozzymandius666 A Higgs coupling is not the only way for a pair of a left and a right hand Weyl fermion to be coupled into a Dirac fermion. My point was that *any* mechanism that generates mass for a particle *must* mix chiralities if more than one spin state exists for that particle (i.e, it isn't a scalar boson).
What a woozie! We are obviously hitting the representational limits of symbols written on a 2D plane. It's so painfully obvious that the only thing making this material so obtuse is the notation. It's just the same simple concepts over and over again but we are forced to spend 99% of our brain power just deciphering the notation.
Holy shit, when you started talking about Dirac spinors and it slowly became clear how antimatter is the time-reflection of matter because of the right dual Weyl spinors, that blew my mind! I feel like everything suddenly makes sense lol
@@eigenchris I might be wrong in seeing a connection there but I feel like that part should play a role, at least vaguely. I have zero knowledge on QFT though, I'm just trying to build a framework of all the math that's used in modern physics and your playlists have been perfect so far. I was always a bit frustrated by how I could tell how intimately connected all these concepts are without being able to figure out how (or why)
15:35 Note: it was quite upsetting that you used L for representing Weyl vector transformation matrices. if you used U insted we all would get the satisfying UWU†
Thanks for the creating these excellent series on tensors and spinors! One thing I noticed on the exercise answer sheet. Last answer (Ltz |+x> ) the lower component should be e^(phi) and not phi/2.
This looks like trying to do spacetime by representing time as a scalar rather than a vector. The Dirac spinors seem more appealing just because they put time on a more even footing with space. In terms of Clifford Algebra, Weyl spinors and Pauli spinors both appear to use Cl(3), though Pauli spinors keep a firm distinction between the different grades while Weyl spinors don't. Dirac spinors appear to use Cl(1,3) while keeping different grades distinct like the Pauli spinors.
You might be interested in Sudgylacmoe's video "A swift introduction to spacetime algebra". It show how Cl(1,3) is related to Cl(3) by replacing pairs of gammas with sigmas.
I learned the hard way (again) that there is a difference between the ^T and the ^\dagger: For the Pauli spinors you use the dagger to give the dual, but for the Weyl spinor the ^T to specify the dual. Then I tried this out with Octave (free Matlab emulation). There the transpose is indicated by a simple hyphen', like A' . But this hyphen actually represents the dagger, that is, if you want A^T (the transposed) for a matrix A containing complex numbers, you have to write conj(A)', or conj(A'). This was a problem when I tried to numerically "confirm" that L^T E L = E = [0 1 ; -1 0] for the 3 boosts and 3 rotations represented by SL(2,C), for some arbitrary angle. I almost thought I found a mistake in your math but no, it was all my own stupidity :-) Chapeau.
Maybe it sounds strange, but your source list at 35:00 is one of my take-aways, . . . . , in addition to all your content of course. :-) I'm teaching myself QFT and have been searching for a good source since some months now. The "no nonsense QFT" book looks exactly like being this input. Thanks for your excellent work. Looking forward to seeing stages "Lie" and "Particle".
I'd be bit cautious with that list. I included it to show how confusing the notation can be, not to suggest these are the best sources. I'm trying to work my way through the "No Nonsense QFT" book, and even though many parts of it are good, I still find QFT very painful. I hope there will be a light at the end of the tunnel, but right now I don't see it.
I too, I'm teachin myself QFT . I have read The no nonsense QFT and Physics from symmetry by the same author and I find it very illuminating but the part about spinors is quite difficult to get through. I like these videos very much because now I can understand spinors a little bit more.
@@barbaramontemurro5515 Yeah, I think he rushes through spinors in the No Nonsense QFT book. I guess it was already 600 pages long and he wanted to cover other things.
So much respect for you extracting a clear summary like this from sources that have differing notations. As much as I like the homage-tradition of things being named after their creators, I really think it has to stop if we're to make progress. I struggle to remember people's names when i meet them in person, never mind through an abstract set of notation.
I think that the left and right chiral spinors are swapped. I've learned from Wikipedia (Lorentz group, Weyl equation) that the Weyl spin vector (or matrix) transforms as 𝑊=[𝑐𝑡+𝑧,𝑥−𝑖𝑦;𝑥+𝑖𝑦,𝑐𝑡−𝑧] → 𝑊′=𝐿_𝑅 𝑊𝐿_𝑅^† and the adjoint Weyl spin vector (parity inverted form of 𝑊) transforms as 𝑊_𝑎𝑑𝑗=[𝑐𝑡−𝑧,−𝑥+𝑖𝑦;−𝑥−𝑖𝑦,𝑐𝑡+𝑧]→𝑊′_𝑎𝑑𝑗=𝐿_𝐿 𝑊_𝑎𝑑𝑗 𝐿_𝐿^†, repectively. The Lorentz transform of the left and right chiral spinors are the member of 𝑆𝐿(2,ℂ), i.e., 𝐿_𝐿,𝐿_𝑅∈𝑆𝐿(2,ℂ), When these spinors are normalized, they satisfies the completeness relation and orthogonality; |𝜓_𝑅 ⟩⟨𝜓_𝑅|+|𝜓_𝐿⟩⟨𝜓_𝐿|=𝐈,⟨𝜓_𝐿│𝜓_𝑅 ⟩=0. The Lorentz transformation of the left chiral Weyl spinor is the conjugate-transposed(dagger) and inverse of that of the right chiral Weyl spinor, i.e., 𝐿_𝐿=(𝐿_𝑅^−1)^†. The explicit forms are given by 𝐿_𝑅=exp(𝑖𝜎 ⃗.𝜃 ⃗/2−𝜎 ⃗.𝜙 ⃗/2) and 𝐿_𝐿=exp(𝑖𝜎 ⃗.𝜃 ⃗/2+𝜎 ⃗.𝜙 ⃗/2) . Thus the spinor (column matrix) obtained by factoring the basic Weyl spinor matrix should be defined as the right chiral spinor.(𝑊=|𝜓_𝑅⟩⟨𝜓_𝑅|,𝑊_𝑎𝑑𝑗=|𝜓_𝐿⟩⟨𝜓_𝐿|)... I'm little confused.
My head is the only thing spinning and this is coming from someone who has done calculus and all the pre calculus algebra. I guess I struggle with this level of abstraction 😂
To Eigenchris I'm very curious about how the Weyl Tranformation translates to pure Clifford algebrra (CA) language. I see clearly how the transformed Pauli Vector V can be expressed in clean CA form with 2 reflections A and B combined to give a rotation, like V' -> (AB)V(AB)-1 But it is unclear how this comes about when dealing with the Weyl Vector W. In video 9, we saw that the Weyl Vector W transforms as something like W' -> SL(C)W(SL(C))-1 but first of all the Weyl Vector is written as W = tI + xsigma_x + ysigma_y + zsigma_z that is, as a linear combination the Identity I and the Pauli Matrices. But note that there is no 'time-reflector', a vector that squares to -1, like the gamma_t in the Gamma matrices. So I think my question is: How is the transformation W' -> SL(C)W(SL(C))-1 related to the language in CA where we reflect along a 'time-reflector' and a 'space-reflector'?
Have you watched video #12? I talk all about how Clifford Algebras are used to to rotations and Lorentz boosts. SL(2,C) is just the group Spin(1,3), and I explain how to build it and use it in video #12.
Great video, most useful! May I ask a clarification: do the scalar product in Minkovski space and the inner product (or symplectic form) in Weyl spinor space correspond to each other?
I'm not sure if there's a correspondence. The inner product is on vectors and the symplectic form is on spinors. The inner product of a 4-vector with itself gives the vector's squared norm, but the symplectic product of a Weyl spinor with itself gives zero.
I have a question on your left and right representaion of SL(2,C). 34:42 . In my reference, ( An Introduction to Tensor and Group Theory for Physicist (Nadir Jeevanjee) , Physics from Symmetry ( Jakob Schwichtenberg)) , relation of left handed rep and right handed rep of SL(2,C) is not just complex conjugation but inverse of complex conjugation. I know that there is notational and indexing difference between you and my reference boook , however it is not arise from row and colume vectors, which could be solved by transposed. If I had midunderstood, please recommend a textbook that I can refer to your notaion and indexing
Yeah, actually I'm simplifying at 34:42 and maybe I shouldn't have. Starting from "left", complex conjugation gives you the "right dual" representation, and complex conjugation + inverse gives you the "right" representation.
At 34:38, for the matrizes in the second row, they are both exactly identical for both chiralities (i have been checking for minutes:)). Is there a typo?
hmmm i noticed that while making a weyl vector from the position 4-vector (ct,x,y,z) and pauli matrices (σ0 , σ1, σ2, σ3) we directly sum them together (ct.σ0 + x.σ1 + y.σ2 + z.σ3). I thought if we treat both as 4-vectors, the inner product will include the metric, introducing a minus sign (- ct.σ0 + x.σ1 + y.σ2 + z.σ3)
I believe that the σ0 pauli matrix squares to -1, whereas the other three pauli matrices square to 1. So the metric should be accounted for that way. (Edit: I commented too early before watching the video...)
The σ0 matrix is the identity, so it would square to +1. If you want the equivalent matrices for spacetime, you need the Dirac matrices, also called the gamma matrices γ0, γ1, γ2, γ3. These obey (γ0)^2 = +1 and (γi)^2 = -1 for the other 3. (This is the + - - - signature, which is the opposite of your - + + + signature, but it still works.). Those are the gammas that pop up in the Dirac equation at 43:53. I'll be talking about these when I get to Clifford Algebras. You can watch sugdylacmoe's videos on "A swift introduction to clifford algebra" and "A switft introduction to spacetime algebra" if you want to learn more.
I wrist thought the left and right representation of SL(2,C) is connectedness of this Lie group, but corrected myself as it is connected. is the chirality of SL(2,C) related to it being a double cover for the Lorentz group?
some other things: 1. is it possible that the antisymmetry of the ε matrix for weyl spinor inner product is related to fermions? 2. non-unitarity of the SL(2,C) matrices transforming weyl spinors sounds problematic when we're bringing relativity and quantum mechanics together.... maybe there's more to this 3. can we say a left-handed weyl spinor can never behave like a right-handed one no matter how we choose our coordinate? 4. knowing that left and right weyl spinors have their own transformation matrices, what would happen if we act on a weyl spinor with the wrong matrix? (ex: what if we act on a left-handed weyl spinor with an SL(2,C)-right matrix?) 5. If boost transformation matrices are hermitian, can it be made into a hermitian operator for a physical observable? maybe related to momentum somehow 6. Dirac equation can describe spin-1/2 matter and antimatter in one go with the dirac spinor. maybe there is an equation describing a quantum object with a certain spin using only weyl (left/right) spinor (or dual)? maybe it only selects the left/right handed part or matter/antimatter part no pressure. a dense video this one hahah
1. Possibly, but I don't know how yet. In multi-particle quantum mechanics, the creation and annihilation operators for bosons will follow "canonical commutation relations" and the same operators for fermions will follow "anti-commutation relations". The Weyl spinor inner product looks anti-commutative, but I'm not sure if this is related to the anti-commutibity seen in the fermion creation and annihilation operators. 2. Yes, Born's rule is not invariant under Lorentz transformations. I think this is related to how standard quantum mechanics is incompatible with special relativity. We need quantum field theory (QFT) in order to get relativistic QM. Again, I don't know enough to say why yet. 3. The "mirror symmetry" coordinate transformation I show swaps left and right weyl spinors. This is more commonly called a "parity transformation", which is related to "p-symmetry". 4. You would get results, but the results wouldn't make physical sense. It would be a bit like operating on a 4-component dirac spinor with the 4x4 SO(1,3) matrices that are meant for vectors. The matrix dimensions are compatible, but the result makes no sense. 5. I have no idea. I think each of the 10 spacetime symmetries have operators associated with them. The hamiltonian operator (energy) is associated with time translations. The momentum operator is associated with space translations. Angular momentum operator is associated with spatial rotations. I would guess there is a similar operator associated with spacetime rotations (aka boosts), and also a conservation law for it. Just as we have conservation of energy, momentum, and angular momentum, there is a conservation law for space-time angular momentum. You can look at the 4th row in the table on this page: en.wikipedia.org/wiki/Conservation_law. You can also try reading this on Noether's Theorem, although it might be a bit ahead of your current level: physics.stackexchange.com/questions/12559/what-conservation-law-corresponds-to-lorentz-boosts 6.There is a way the Dirac Equation can "split" into a pair of independent equations for left and right weyl spinors. Although my understanding is that this only works if the particle is massless. Haven't gone through the math. As you can see, I'm still learning and don't have all the answers.
@@eigenchris *3. The "mirror symmetry" coordinate transformation I show swaps left and right weyl spinors.* oh i should've worded this better. Rotation and Lorentz boost transformations can actually be done in a lab by rotating(lol) our detector or moving it with a constant velocity with respect to the particle, and varying how much. I don't see how "applying parity operator" is something that our experiments can do (flipping the universe?), so i thought if we already identified a left weyl particle, there is no lab setup that can register that same object as a right weyl particle... this is what i meant by "no matter how we choose our coordinate" *2. Yes, Born's rule is not invariant under Lorentz transformations. I think this is related to how standard quantum mechanics is incompatible with special relativity.* Yeah i think this is what's going on. One feature of QFT that i remember which sets it apart from just "relativised" QM is violation of particle number conservation; particles can be created/destroyed and what we thought of as one particle can exhibit "virtual cloud of particle-antiparticle pair". If weyl spinor transformation is nonunitary in some cases and born rule really is violated (probabilities can add up to a value other than 1), maybe those missing (or extra) probabilities can be accounted for by more particles being created/destroyed, seeing that ordinary QM is normalized to one particle processes. *4. You would get results, but the results wouldn't make physical sense.* Hmmm... i was hoping that not applying the matching operator to a left(right) weyl spinor would give a weird superposition, like when projecting a harmonic oscillator to a coherent state (maybe you've heard of this) gives a superposition of energy eigenstates, or more simply when applying the Sx operator to an Sz spin eigenstate gives a superposition of spins along the x-axis. oh well... *As you can see, I'm still learning and don't have all the answers.* yeah, same here
@@GeoffryGifari For #3, yes I see your point. There are a few operators in physics that we can't do in real life. In additional to parity reversal P, there's a time reversal operator T which swaps past and future, and a charge conjugation C which swaps matter/anti-matter charges. We can't do any of these in real life, but we can reverse certain variables in our physics equations and see what happens. In real life, I think the Dirac equation tells us that a free spin-1/2 particle will constantly swap between left and right handed states as it travels along. So most particles are a superposition of the two. If you want to just ahead, you can try searching for the solutions to the Dirac equation in the Weyl basis/Chiral basis to see this happen. Though I'm not sure how this works with neutrinos because my understanding is that right-handed neutrinos have never been detected. Again, more for me to learn.
@eigenchris I didn't say you made a mistake. Maybe I missed you saying that unitary matrices aren't always traceless. Remember, not everyone has a math degree. It can be misunderstood by laymen. That's why I wrote a clarification. Don't worry. Your content is great. I've never seen a better explanation in so few words. Will you make one for galois groups as well? Or universal algebra?
@@tomaskubalik1952 In this case for vector spaces, the operation is called the "direct sum". This is what lets you "stack" columns on top of each other. It allows you to add lefts with lefts and add rights with rights.
@@eigenchris Is a Dirac spinor a superposition of Weyl spinors? Is every Dirac spinor a linear combination of left and right Weyl spinors? Is it necessary to multiply the left and right spinor by coefficients when we want to add them?
I wonder if Chris got to the lecture on Clifford Algebras, stumbled on Hestenes, and went down the Geometric Algebra rabbit hole. Hope so. GA is a religious experience for us physics nerds. :)
For all the discussion on Weyl Spinors, wouldn’t it be easier to refer to epsilon as i? It squares to -1. Why is it normally given a different name in this context? Also, does -i (or -epsilon in the lingo) fulfill the equations as well?
Epsilon is more complicated than just "i" despite the fact that they both square to -1. Epsilon maps a pair of spinors to a scalar value using a very particular set of rules that are left invariant under SL(2,C) transformations. "i" doesn't do this in the same way.
@ Hm. I fail to see the fundamental difference. It seems to act precisely like a matrix version of i (or a so3 rotation matrix with the angle pi/2). And the other version for -i seems to fulfill all equations as well. But i’ll try to wrap my mind around it~ thanks for answering as always
8:38 I miss the orthochronous condition Λ_00 > 0 in the SL(2,C) representation - is there a direct connection to the left/right representation? I remember from physics that antimatter = matter traveling back in time and being related to complex conjugation...
I'm still studying QFT so I can only give you a partial answer. With the group O(3), it has 2 "disconnected" parts, which are the det = +1 part and det = -1 part. We can continuously transform WITHIN either of the 2 parts separately, but jumping from one to the other requires a discontiuous jump, which is mirror symmetry, or a "parity reversal". Restricting to SO(3) restricts us to the det = +1 part. The group O(1,3) has 4 disconnected parts, which can be disconntinuously "jumped" between using parity reversal P and time reversal T (we could name the 4 parts: "1, P, T, PT"). Restricting to det=+1 gives SO(1,3), and restricts us to the "1, T" parts, and restricting further to Λ_00 > 0 gives us SO+(1,3) and restricts us to just "1". There is a 3rd discontinuous symmetry operator call "charge conjugation", which swaps positively charged particles and negatively charged particles. If a system obeys CPT symmetry, it means that doing a parity flip P, a time reversal T, and replacing all matter with antimatter using charge conjugation C, it means all particles would appear to obey the same physical laws that we see in our universe. However, CPT symmetry does not always hold. In particular, we know the weak force treats left and right handed particles differently. I can't say much more as I'm still learning, but maybe this will give you a basis to do more searching online.
@@eigenchris I thought "holding" a symmetry means having a Lagrangian that doesn't mix different parts of a representation used to define the state. So like when a mass term is mixing upper and lower part of a Dirac spinor, these two can not be separated into two independent equations to the effect that the particle oscillates between the two (components denoting roughly probability or - in second quantisation - give rise to cross terms with creation and annihilation operators of different particle types that form the basis of the state representation). But maybe I'm just phantasising for I find it hard to interprete the maths. I was hoping that the lower part of the Dirac spinor would be the Λ_00 < 0 part of the Lorentz group or something. Btw, I'm very happy that you study geometric algebra, for I remember suggesting that in a comment under some online lecture you gave to friends/collegues when you wondered about spin. You are much smarter than I am, so I just lean back and watch how you sort it all out :-) Thank you for sharing all of this!!
So with the hints from Chris and GPT, I found that: 1. SL(2,C) acts like the orthochronous, proper Lorentz group, meaning Λ_00>0 i.e, no time reversal T, also no spacial reflection P. 2. This way, the handedness/chirality is not affected, and the two Weyl spinor parts of the Dirac spinor do not get mixed, so it is like two 2x2 matrices on the diagonal of a 4x4 matrix each representing the same boost+rotation on their respective Weyl spinor 3. SL(2,C) acts as a double cover of the orthochronous, proper Lorentz group, so there are always two different 2x2 matrices that do the same. 4. You can have the full Lorentz group by acting with two independent copies of SL(2,C) on a tensor product of two complex 2D vectors, yielding a 4x4 matrix once again. This SL(2,C)×SL(2,C) setup seems to be equivalent to having only one SL(2,C) acting on both Weyl spinors as above, but then optionally composing with P and T operation, where P is just a matrix (γ0), but T involves additional complex conjugation. Here, only P actually swaps the upper and the lower Weyl spinor (with sign changes) and thus chirality. 5. The charge conjugation C acts mathematically as complex conjugation, but this also undoes the composition of P and T with their combined component and sign swapping. CPT = 1 leaves thus all states (tensor product of all representation vectors) in the standard model unchanged. So, at least in the current standard model C = (PT)^-1 = TP which should reflect in the math. 6. I still cannot match spacetime geometric algebra STA, where the Dirac spinor is a full multivector, the left chiral spinor is scalar+bivector (even), and the right chiral is vector+trivector (odd). But P and T seem to flip only signs of some blades, and they do not change grade, so there must be some difference to the complex matrix setup of QFT. Hopefully, we will get there eventually... 7. Regarding symmetry: The Lagrangian that couples left and right chirality will yield vertexes that convert right into left particles, but can still be C invariant when both processes are even likely. So separability (like no mass) is sufficient but not necessary for C invariance and the *overall* conservation of matter/antimatter.
As to 6., Chat-GPT told me this - a bit disappointing, if true. Maybe, it gets clearer over time: The Dirac equation can indeed be transcribed into the framework of spacetime algebra (STA). The catch with chirality in the context of STA is not that it cannot be represented or studied, but that the representation and interpretation of chirality are different from the complex spinor representation used in particle physics. In the complex spinor representation, chirality is described using Weyl spinors and the SL(2, C) × SL(2, C) representation of the Lorentz group. In STA, we can represent Dirac spinors using multivectors and study their behavior under Lorentz transformations and parity operations. However, the geometric interpretation of chirality and its transformation properties are different in STA compared to the complex spinor representation. In STA, a Dirac spinor can be represented as a multivector, which can be decomposed into even and odd parts. The even and odd parts are sometimes interpreted as left- and right-handed chiral components, respectively. However, this interpretation does not have a direct correspondence with the complex spinor representation in terms of transformation properties. As mentioned earlier, one alternative is to use separate spinor spaces to represent left- and right-handed components in STA, where both components are even-grade multivectors (scalar + bivector). This approach allows you to represent and study the left- and right-handed chiral components of a Dirac spinor in the context of STA, but the correspondence with the SL(2, C) × SL(2, C) representation is not direct, and the geometric interpretation will be different. In summary, the "catch" concerning chirality in STA is not that it cannot be represented or studied, but rather that the representation and interpretation are different from the complex spinor representation used in particle physics, which may lead to different transformation properties and geometric insights.
@@franks.6547 I would be careful asking ChatGPT about spinors. I've tried asking it a few things, and while it is sometimes right, it definitely hallucinates often. I think spinors are too obscure a topic for it to answer about reliably.
I didn't quite get left and right representations of spinors. I tried so multiply right spinor to left dual spinor and got Weyl vector, but with inverted space coordinates. Does it mean that right spinor is the same spinor as left spinor, but in a space with inverted x, y, z?
Yeah, I'll be getting there in video 11. I wanted to present a matrix version of everything first, since a lot of physics textbooks take a matrix approach. GA is still in the process of gaining traction with physicists.
The problem with geometric algebra is how the geometric product is ill-defined as a sum of inner and exterior product, because for even grade, that doesn't apply. Also the "gradient" with vector basis is thought as a derivation, but that is false so it is impose by the overdot notation. I think you can do a derivation if you use bivector basis instead.
@@rajinfootonchuriquen you seem to be confused. The geometric product is perfectly well-defined for all grades (and the definition isn't "sum of interior and exterior products", though that happens to be what the geometric product equals for vectors specifically).
@@rajinfootonchuriquen It depends on the dimension considered and the types of bivectors. In 3-d, the outer product of two bivectors may have grade 4. But since the grade over than 3 does not exist in 3-d. In this case the result is just zero. However note that this is not ill-defined.
27:06 I am also always skeptical of this top-down logic of "seeing what maths work", in this case quite arbitrarily turning the Hermitian conjugate into a transpose. Why is that "allowed"? What's the bottom-up logic? What is the intuitive meaning of the object that you get when you take the inner product of two spinors (for real vectors this would be angles, and for complex vectors it's a more abstract angle in complex space, while on the Riemann sphere they become concrete, so can we also visualize the Weyl inner product as visibly encoding distance and angle)?
Unfortunately in this case, I don't have an intuitive reason. The best I can say is that the group SL(2,C) is equivalent to the "symplectic group" Sp(2,C) and the key invariant for symplectic groups involves the symplectic form epsilon that I showed. But I think this is just delaying the real cause to another question. Maybe googling "symplectic group" or "symplectic form" can help you learn more.
@@eigenchris I should have commented after watching because I just got to that point. I have been studying symplectic geometry in the context of Hamiltonian mechanics and thermodynamics as it's one of those topics that you know about but never actually get to, and my interest is now really piqued.
@OrkTv Yeah I'm not sure what the connection is. I think the symplectic group comes up in the case of "anti-symmetry". In Hamiltonian mechanics we often used the symplectic form (wedge product) to measure oriented areas in phase space. Swapping the order of the vectors making the parallelogram will reverse the orientation of the parallelogram, giving you a minus sign. In QM, spin-1/2 particles are fermions and obey anti-commutation relations, so swapping their order gives you a negative sign (also, trying to create 2 fermions in the same state gives you zero under this anti-symmetry, which is expected since spin-1/2 particles obey Pauli's exclusion principle). Not sure if there's any connection beyond that.
20:00 It wasn't obvious to me why C = k₁A* so I detailed it: if A is zero, C can be anything, so let's assume A ≠ 0. AC is real and AA* = |A|² is also real, so AC/(AA*) = C/A* is also real. Call this number k₁.#
Yes. Each Sigma matrix can be written as a pair of gamma matrices. I'd suggest Sudgylacmoe's "a swift introduction to spacetime algebra", but my guess is you've already seen it.
I don't really know what those are, but I think they are a special case of Dirac Spinors that come up in some very specific cases. I didn't include them because they seem very niche and possibly don't describe any particles we know of currently.
@eigenchris they are their own antiparticles, related to sterile neutrinos, and in quantum computing. But yeah, not at the top of the list :-) Thanks again.
I'm not as familiar with those. I know that they are a special case of a 4-component Dirac spinor, for a particle that equals its own anti-particle. I'm not sure what the exact mathematical condition for this is, though.
@@samanthaqiu3416 No, that flips between the left and right chiral representations. That's called "parity conjugation", usually denoted with "P". You want "charge conjugation", which is usually denoted "C".
15:50 "Since we're dealing with complex numbers..." What is this notion predicated on? The SL(2,C) matrices come from Lorentz transformations when these are realized on Weyl vectors. Apparently we can represent this if and only if our special linear matrices admit complex entries. Could this be elaborated a bit? (I really don't like it when, as often happens, people pull complex numbers seemingly out of nowhere. People start talking about "complexified" algebras without proper justification, and in this case I think we're almost there.)
My fist instinct is to say that since Weyl vectors already contain complex numbers, it's reasonable to say that the matrices that transform them will also be complex. We sort of made a similar assumption with the U matrices that transform Pauli vectors. Basically, we start with the most general 2x2 matrices that would transform a 2x2 complex matrix, and then narrow down their properties from there. Another argument could be parameter counting, because there are 6 types of Lorentz transformations and SL(2,C) matrices have 6 free parameters, whereas SL(2,R) matrices have 3 free parameters.
@@eigenchris Makes sense. I guess it would be possible to relate Lorentz matrices and SL(2,C) matrices one-to-one if I sat down, with the complex numbers being introduced because complex numbers are already there in our representation. Thanks as always!
When you make a longer video, please try to vary your vocal tone based on context, emphasis of a significant result and so on, instead of sounding like an unitonal automaton just reading off a script file.
So, why would anyone do modern physics in 3D? We know it is not real. Intervals are real. Distance is an illusion. "locality" based on 4D interval is not the same as naive locality.
@@eigenchrisHmm, okay. But I see a lot of confused stuff out there like the setup of the EPR "paradox" where it is described in 3D but in 4D there are intervals of zero all over it which have bearing on the problem of "locality". And that is often treated like "basic quantum mechanics" and extrapolated to multiverse, with nary a mention of the intervals. What are the guard rails for knowing you have oversimplified by using a 3D model?
Thank you - another great lesson. A little heavyweight this time, but maybe that just goes with the nature of the subject; I'll rewatch later, with the advantage of foreknowledge of where it's going. It probably doesn't help that my Special Relativity is about 35 years old and unused since, so it's very dimly remembered! I've seen mention of some of the groups involved being "double covers": is that the right description for the relationship between SU(2) and SL(2,ℂ)?
Yes. SU(2) double-covers SO(3), and SL(2,C) double-covers SO(1,3). A better name for SU(2) is "Spin(3)". It turns out there's a general way to build Spin(n) groups which double-cover the SO(n) groups, which I will cover when I get to Clifford Algebras. There's also Spin(p,q) groups to double-cover SO(p,q).
![Spinors for Beginners 10: SU(2) double covers SO(3) [ SL(2,C) double covers SO+(1,3) ]](http://i.ytimg.com/vi/Ay482u6b3Xo/mqdefault.jpg)
![Spinors for Beginners 10: SU(2) double covers SO(3) [ SL(2,C) double covers SO+(1,3) ]](/img/tr.png)
As a mathematical physicist, I really like this series. Please continue :)
Maybe, you are interested in a celebration livestream when we both reach the 100k subscribers? :) Could be fun!
I'm hesitant to do anything live as I'd probably be very nervous the whole time. But I might do a pre-recorded Q&A or something like that.
oh i thought you do pure math
@@GeoffryGifari That as well :D
what kind of mathematical physics are you working/have been working with?
@@GeoffryGifari Maybe also something for the Q&A :D
about to take an exam on gauge theories and QFT been struggling to wrap my head around spinor fields and this video has came out almost perfectly timed, tysm !! :)
Glad to hear it!
I feel smarter and dumber at the same time after watching this video. what an experience.
The quantum information scientist would consider your state as point in the blochsphere. Don't worry, confusion is the first stage of learning.
Keep going! These videos should be listed as references on any QFT syllabus.
Framing the math in such a descriptive, visual, and story-like way is necessary for ensuring the physics is understood. Kudos kudos kudos!
This was the most informationally dense video I have ever watched, and I understood every bit of it, thank you!
Pauli, Weyl and Dirac would approve of this video ! Awesome and well done in explaining this in a clear and non confusing way.
This is a noteworthy Extreme Super-Synopsis of the exceptionally broad depth of nomenclature for spinors throughout the decades of physics. Congratulations to 'eigenchris' for this attempt ...VERY WELL DONE!...That said, spinor utilization should be exercised with care (and is undoubtedly subject to future modifications)...(Feynman had noteworthy comments about Dirac... and the same could be said for Pauli and Weyl, in a manner as Teller commented about Einstein's post 1919 attempts to advance physics.)
The quick review of special relativity solidified my understanding of space-time in a big way! Thanks Chris.
I just found this series and I am addicted, it is in the perfrct pace and right to the point and very clear, thanks for creating this series!
Glad you like it!
@@eigenchris do you have plans for the next series topic? I would love to know what topics you have in mind.
I would love a series in quantum field theory.
By the way, my plan for this weekend is to watch your series in relativity haha.
@@GustavoOliveira-gp6nr There probably won't be a next series, honestly. I'll make one-off videos now and then, but I'm pretty burned out from making long series. I will touch on some QFT in videos 21-25.
The part on the 4 different types of weyl spinors was very clear, I remember the headache when I first tried to understand them
Thanks. Yeah, the index notation really drives me nuts with them. It's much easier to start out with matrices in my opinion.
this is one of the most useful series i have found and is the chirality presented in this video related to the chirality in QFT is how and why (i am a big fan of your work)
Yes, the Chirality shown here is the exact same as the chirality used in QFT. The Dirac Equation I know near the end is one of the main equations used in QFT. It's the equation of motion for spin-1/2 particles and keeps track of both the left-chiral and right-chiral portions of the state.
I would imagine it's the same thing, thought I'm not far enough ahead in my QFT studies to understand that yet.
@ozzymandius666My understanding of QFT is more phenomenological than mathematical, but *any* mechanism by which particles with chirality gain mass, must mix left and right chiral wave functions in order to conserve angular momentum, not just the Higgs mechanism. The only particles for which chiralities don't mix are massless particles and particles with no chirality.
@ozzymandius666 A Higgs coupling is not the only way for a pair of a left and a right hand Weyl fermion to be coupled into a Dirac fermion. My point was that *any* mechanism that generates mass for a particle *must* mix chiralities if more than one spin state exists for that particle (i.e, it isn't a scalar boson).
Awesome video!!! Thank you so much for all the effort you put in to these videos. They are an incredible resource for my learning.
i was using this in weyl spinors in the context of Supersymmetry and I had no idea where it came from. Cool video
It is definitely what I want to know about spin representation. thank for your video
What a woozie! We are obviously hitting the representational limits of symbols written on a 2D plane. It's so painfully obvious that the only thing making this material so obtuse is the notation. It's just the same simple concepts over and over again but we are forced to spend 99% of our brain power just deciphering the notation.
I find the spinor-index/Van der Waeden notation pretty painful. I was hoping to make it less painful here. I'm not sure if I succeeded.
This video was so good and helpful! Really helped me understand better. Looking forward to the next video in the series!
Holy shit, when you started talking about Dirac spinors and it slowly became clear how antimatter is the time-reflection of matter because of the right dual Weyl spinors, that blew my mind! I feel like everything suddenly makes sense lol
It's interesting that you say that, because I'm not sure I understand anti-matter yet, or the time-reversal of Dirac spinors.
@@eigenchris I might be wrong in seeing a connection there but I feel like that part should play a role, at least vaguely. I have zero knowledge on QFT though, I'm just trying to build a framework of all the math that's used in modern physics and your playlists have been perfect so far. I was always a bit frustrated by how I could tell how intimately connected all these concepts are without being able to figure out how (or why)
just realized that so many abstract stuff in theoretical physics were brought in by the duo weyl and wigner
The table of indices conventions at 35:00 is very much appreciated
15:35 Note: it was quite upsetting that you used L for representing Weyl vector transformation matrices. if you used U insted we all would get the satisfying UWU†
Some people are upset I didn't call the series "Spinors for Beginors", but this is a much worse mistake on my part.
Please check Spinors for Beginners 17 at timestamp 31:08 for some good news.
@@eigenchris this is wonderful thank you so much for this and for this series
Thanks for the creating these excellent series on tensors and spinors! One thing I noticed on the exercise answer sheet. Last answer (Ltz |+x> ) the lower component should be e^(phi) and not phi/2.
True. Thanks. I'll get around to correcting that this weekend.
This looks like trying to do spacetime by representing time as a scalar rather than a vector. The Dirac spinors seem more appealing just because they put time on a more even footing with space.
In terms of Clifford Algebra, Weyl spinors and Pauli spinors both appear to use Cl(3), though Pauli spinors keep a firm distinction between the different grades while Weyl spinors don't. Dirac spinors appear to use Cl(1,3) while keeping different grades distinct like the Pauli spinors.
You might be interested in Sudgylacmoe's video "A swift introduction to spacetime algebra". It show how Cl(1,3) is related to Cl(3) by replacing pairs of gammas with sigmas.
@@eigenchris Oh I'm subscribed to him and have seen every one of his GA videos (multiple times for the Swift Introductions). 😉
I reached that point where I stopped following the math. Need to back up a bit. But another brilliant series of videos from eigenchris.
you are the GOAT, tnks alot. i used to use the landau book (4) and i dont understand alot of things, but your videos made all clear(:
I learned the hard way (again) that there is a difference between the ^T and the ^\dagger: For the Pauli spinors you use the dagger to give the dual, but for the Weyl spinor the ^T to specify the dual. Then I tried this out with Octave (free Matlab emulation). There the transpose is indicated by a simple hyphen', like A' . But this hyphen actually represents the dagger, that is, if you want A^T (the transposed) for a matrix A containing complex numbers, you have to write conj(A)', or conj(A'). This was a problem when I tried to numerically "confirm" that L^T E L = E = [0 1 ; -1 0] for the 3 boosts and 3 rotations represented by SL(2,C), for some arbitrary angle. I almost thought I found a mistake in your math but no, it was all my own stupidity :-) Chapeau.
Another wonderful video! Thank you. Easy to understand.
Maybe it sounds strange, but your source list at 35:00 is one of my take-aways, . . . . , in addition to all your content of course. :-)
I'm teaching myself QFT and have been searching for a good source since some months now. The "no nonsense QFT" book looks exactly like being this input.
Thanks for your excellent work. Looking forward to seeing stages "Lie" and "Particle".
I'd be bit cautious with that list. I included it to show how confusing the notation can be, not to suggest these are the best sources.
I'm trying to work my way through the "No Nonsense QFT" book, and even though many parts of it are good, I still find QFT very painful. I hope there will be a light at the end of the tunnel, but right now I don't see it.
@@eigenchris I know that this was not your intention - but I'll give the book a try. ;-)
Your list is a jewel! Maybe there are better books, but they all suffer by their own van-der-waerden notation. Thank you for the systematic overview!
I too, I'm teachin myself QFT . I have read The no nonsense QFT and Physics from symmetry by the same author and I find it very illuminating but the part about spinors is quite difficult to get through.
I like these videos very much because now I can understand spinors a little bit more.
@@barbaramontemurro5515 Yeah, I think he rushes through spinors in the No Nonsense QFT book. I guess it was already 600 pages long and he wanted to cover other things.
good grief, all the different conventions...my respect for trying to be reasonably comprehensive with the table.
Yeah it definitely drove me a little crazy. Writing things as arrays instead of using summation conventions definitely makes learning it easier.
You explain very well! Thank you
Please upload a playlist on particle physics.
So much respect for you extracting a clear summary like this from sources that have differing notations.
As much as I like the homage-tradition of things being named after their creators, I really think it has to stop if we're to make progress. I struggle to remember people's names when i meet them in person, never mind through an abstract set of notation.
Epsilon is also known as Levi-Cevita symbol btw
Damn, didn't realize that at first, nice
Great content as always (I assume, im still waiting in the premier XD)
I think that the left and right chiral spinors are swapped. I've learned from Wikipedia (Lorentz group, Weyl equation) that
the Weyl spin vector (or matrix) transforms as 𝑊=[𝑐𝑡+𝑧,𝑥−𝑖𝑦;𝑥+𝑖𝑦,𝑐𝑡−𝑧] → 𝑊′=𝐿_𝑅 𝑊𝐿_𝑅^† and
the adjoint Weyl spin vector (parity inverted form of 𝑊) transforms as 𝑊_𝑎𝑑𝑗=[𝑐𝑡−𝑧,−𝑥+𝑖𝑦;−𝑥−𝑖𝑦,𝑐𝑡+𝑧]→𝑊′_𝑎𝑑𝑗=𝐿_𝐿 𝑊_𝑎𝑑𝑗 𝐿_𝐿^†, repectively. The Lorentz transform of the left and right chiral spinors are the member of 𝑆𝐿(2,ℂ), i.e., 𝐿_𝐿,𝐿_𝑅∈𝑆𝐿(2,ℂ),
When these spinors are normalized, they satisfies the completeness relation and orthogonality; |𝜓_𝑅 ⟩⟨𝜓_𝑅|+|𝜓_𝐿⟩⟨𝜓_𝐿|=𝐈,⟨𝜓_𝐿│𝜓_𝑅 ⟩=0. The Lorentz transformation of the left chiral Weyl spinor is the conjugate-transposed(dagger) and inverse of that of the right chiral Weyl spinor, i.e., 𝐿_𝐿=(𝐿_𝑅^−1)^†. The explicit forms are given by 𝐿_𝑅=exp(𝑖𝜎 ⃗.𝜃 ⃗/2−𝜎 ⃗.𝜙 ⃗/2) and 𝐿_𝐿=exp(𝑖𝜎 ⃗.𝜃 ⃗/2+𝜎 ⃗.𝜙 ⃗/2) .
Thus the spinor (column matrix) obtained by factoring the basic Weyl spinor matrix should be defined as the right chiral spinor.(𝑊=|𝜓_𝑅⟩⟨𝜓_𝑅|,𝑊_𝑎𝑑𝑗=|𝜓_𝐿⟩⟨𝜓_𝐿|)... I'm little confused.
My head is the only thing spinning and this is coming from someone who has done calculus and all the pre calculus algebra. I guess I struggle with this level of abstraction 😂
To Eigenchris
I'm very curious about how the Weyl Tranformation translates to pure Clifford
algebrra (CA) language.
I see clearly how the transformed Pauli Vector V can be expressed in clean CA form with 2 reflections A and B
combined to give a rotation, like V' -> (AB)V(AB)-1
But it is unclear how this comes about when dealing with the Weyl Vector W.
In video 9, we saw that the Weyl Vector W transforms as something like
W' -> SL(C)W(SL(C))-1
but first of all the Weyl Vector is written as
W = tI + xsigma_x + ysigma_y + zsigma_z
that is, as a linear combination the Identity I and the Pauli Matrices.
But note that there is no 'time-reflector', a vector that squares to -1, like the gamma_t
in the Gamma matrices.
So I think my question is: How is the transformation W' -> SL(C)W(SL(C))-1 related to the
language in CA where we reflect along a 'time-reflector' and a 'space-reflector'?
Have you watched video #12? I talk all about how Clifford Algebras are used to to rotations and Lorentz boosts. SL(2,C) is just the group Spin(1,3), and I explain how to build it and use it in video #12.
Great video, most useful!
May I ask a clarification: do the scalar product in Minkovski space and the inner product (or symplectic form) in Weyl spinor space correspond to each other?
I'm not sure if there's a correspondence. The inner product is on vectors and the symplectic form is on spinors. The inner product of a 4-vector with itself gives the vector's squared norm, but the symplectic product of a Weyl spinor with itself gives zero.
I have a question on your left and right representaion of SL(2,C). 34:42 . In my reference, ( An Introduction to Tensor and Group Theory for Physicist (Nadir Jeevanjee) , Physics from Symmetry ( Jakob Schwichtenberg)) , relation of left handed rep and right handed rep of SL(2,C) is not just complex conjugation but inverse of complex conjugation. I know that there is notational and indexing difference between you and my reference boook , however it is not arise from row and colume vectors, which could be solved by transposed. If I had midunderstood, please recommend a textbook that I can refer to your notaion and indexing
Yeah, actually I'm simplifying at 34:42 and maybe I shouldn't have. Starting from "left", complex conjugation gives you the "right dual" representation, and complex conjugation + inverse gives you the "right" representation.
@@eigenchris thanks for your anser and it is really helpful to me understanding left and right spin representation toptic .
@@junghoonkwon3949 complex conjugate + inverse of rotation in xz plane for SL(2,C) is not same, have you tried that?
At 10:09, I think the element in the upper right corner of the boost matrix in tx and ty-plane should be 0.
Yeah, those are typos. My bad.
Can you start another series of topology for beginners
At 34:38, for the matrizes in the second row, they are both exactly identical for both chiralities (i have been checking for minutes:)). Is there a typo?
hmmm i noticed that while making a weyl vector from the position 4-vector (ct,x,y,z) and pauli matrices (σ0 , σ1, σ2, σ3) we directly sum them together (ct.σ0 + x.σ1 + y.σ2 + z.σ3). I thought if we treat both as 4-vectors, the inner product will include the metric, introducing a minus sign (- ct.σ0 + x.σ1 + y.σ2 + z.σ3)
I believe that the σ0 pauli matrix squares to -1, whereas the other three pauli matrices square to 1. So the metric should be accounted for that way. (Edit: I commented too early before watching the video...)
The σ0 matrix is the identity, so it would square to +1. If you want the equivalent matrices for spacetime, you need the Dirac matrices, also called the gamma matrices γ0, γ1, γ2, γ3. These obey (γ0)^2 = +1 and (γi)^2 = -1 for the other 3. (This is the + - - - signature, which is the opposite of your - + + + signature, but it still works.). Those are the gammas that pop up in the Dirac equation at 43:53. I'll be talking about these when I get to Clifford Algebras.
You can watch sugdylacmoe's videos on "A swift introduction to clifford algebra" and "A switft introduction to spacetime algebra" if you want to learn more.
@@eigenchris thanks! i'll look into it
I hope you will explain Majorana spinors too.
I wrist thought the left and right representation of SL(2,C) is connectedness of this Lie group, but corrected myself as it is connected. is the chirality of SL(2,C) related to it being a double cover for the Lorentz group?
some other things:
1. is it possible that the antisymmetry of the ε matrix for weyl spinor inner product is related to fermions?
2. non-unitarity of the SL(2,C) matrices transforming weyl spinors sounds problematic when we're bringing relativity and quantum mechanics together.... maybe there's more to this
3. can we say a left-handed weyl spinor can never behave like a right-handed one no matter how we choose our coordinate?
4. knowing that left and right weyl spinors have their own transformation matrices, what would happen if we act on a weyl spinor with the wrong matrix? (ex: what if we act on a left-handed weyl spinor with an SL(2,C)-right matrix?)
5. If boost transformation matrices are hermitian, can it be made into a hermitian operator for a physical observable? maybe related to momentum somehow
6. Dirac equation can describe spin-1/2 matter and antimatter in one go with the dirac spinor. maybe there is an equation describing a quantum object with a certain spin using only weyl (left/right) spinor (or dual)? maybe it only selects the left/right handed part or matter/antimatter part
no pressure. a dense video this one hahah
1. Possibly, but I don't know how yet. In multi-particle quantum mechanics, the creation and annihilation operators for bosons will follow "canonical commutation relations" and the same operators for fermions will follow "anti-commutation relations". The Weyl spinor inner product looks anti-commutative, but I'm not sure if this is related to the anti-commutibity seen in the fermion creation and annihilation operators.
2. Yes, Born's rule is not invariant under Lorentz transformations. I think this is related to how standard quantum mechanics is incompatible with special relativity. We need quantum field theory (QFT) in order to get relativistic QM. Again, I don't know enough to say why yet.
3. The "mirror symmetry" coordinate transformation I show swaps left and right weyl spinors. This is more commonly called a "parity transformation", which is related to "p-symmetry".
4. You would get results, but the results wouldn't make physical sense. It would be a bit like operating on a 4-component dirac spinor with the 4x4 SO(1,3) matrices that are meant for vectors. The matrix dimensions are compatible, but the result makes no sense.
5. I have no idea. I think each of the 10 spacetime symmetries have operators associated with them. The hamiltonian operator (energy) is associated with time translations. The momentum operator is associated with space translations. Angular momentum operator is associated with spatial rotations. I would guess there is a similar operator associated with spacetime rotations (aka boosts), and also a conservation law for it. Just as we have conservation of energy, momentum, and angular momentum, there is a conservation law for space-time angular momentum. You can look at the 4th row in the table on this page: en.wikipedia.org/wiki/Conservation_law. You can also try reading this on Noether's Theorem, although it might be a bit ahead of your current level: physics.stackexchange.com/questions/12559/what-conservation-law-corresponds-to-lorentz-boosts
6.There is a way the Dirac Equation can "split" into a pair of independent equations for left and right weyl spinors. Although my understanding is that this only works if the particle is massless. Haven't gone through the math.
As you can see, I'm still learning and don't have all the answers.
@@eigenchris *3. The "mirror symmetry" coordinate transformation I show swaps left and right weyl spinors.*
oh i should've worded this better. Rotation and Lorentz boost transformations can actually be done in a lab by rotating(lol) our detector or moving it with a constant velocity with respect to the particle, and varying how much. I don't see how "applying parity operator" is something that our experiments can do (flipping the universe?), so i thought if we already identified a left weyl particle, there is no lab setup that can register that same object as a right weyl particle... this is what i meant by "no matter how we choose our coordinate"
*2. Yes, Born's rule is not invariant under Lorentz transformations. I think this is related to how standard quantum mechanics is incompatible with special relativity.*
Yeah i think this is what's going on. One feature of QFT that i remember which sets it apart from just "relativised" QM is violation of particle number conservation; particles can be created/destroyed and what we thought of as one particle can exhibit "virtual cloud of particle-antiparticle pair". If weyl spinor transformation is nonunitary in some cases and born rule really is violated (probabilities can add up to a value other than 1), maybe those missing (or extra) probabilities can be accounted for by more particles being created/destroyed, seeing that ordinary QM is normalized to one particle processes.
*4. You would get results, but the results wouldn't make physical sense.*
Hmmm... i was hoping that not applying the matching operator to a left(right) weyl spinor would give a weird superposition, like when projecting a harmonic oscillator to a coherent state (maybe you've heard of this) gives a superposition of energy eigenstates, or more simply when applying the Sx operator to an Sz spin eigenstate gives a superposition of spins along the x-axis. oh well...
*As you can see, I'm still learning and don't have all the answers.*
yeah, same here
@@GeoffryGifari For #3, yes I see your point. There are a few operators in physics that we can't do in real life. In additional to parity reversal P, there's a time reversal operator T which swaps past and future, and a charge conjugation C which swaps matter/anti-matter charges. We can't do any of these in real life, but we can reverse certain variables in our physics equations and see what happens.
In real life, I think the Dirac equation tells us that a free spin-1/2 particle will constantly swap between left and right handed states as it travels along. So most particles are a superposition of the two. If you want to just ahead, you can try searching for the solutions to the Dirac equation in the Weyl basis/Chiral basis to see this happen. Though I'm not sure how this works with neutrinos because my understanding is that right-handed neutrinos have never been detected. Again, more for me to learn.
Unitary matrices don't need to be traceless.
This is just a clarification.
Did I say they needed to be traceless? Sigma matrices are tracless, but not unitary matrices.
@eigenchris I didn't say you made a mistake.
Maybe I missed you saying that unitary matrices aren't always traceless.
Remember, not everyone has a math degree. It can be misunderstood by laymen. That's why I wrote a clarification.
Don't worry. Your content is great. I've never seen a better explanation in so few words.
Will you make one for galois groups as well? Or universal algebra?
@@devalapar7878 Sorry. Usually when people make comments like this it's pointong out a mistake I may have made. My bad.
भगवान आपका भला करें ।
Is the Dirac spinor composed of the left and right spinor or the left spinor and right dual spinor? I would say that the first option is correct.
Yes, the first option: left and right.
@@eigenchris Did Dirac perform a direct sum when he combined right spinors with left spinors? Thanks a lot
@@tomaskubalik1952 In this case for vector spaces, the operation is called the "direct sum". This is what lets you "stack" columns on top of each other. It allows you to add lefts with lefts and add rights with rights.
@@eigenchris Is a Dirac spinor a superposition of Weyl spinors? Is every Dirac spinor a linear combination of left and right Weyl spinors? Is it necessary to multiply the left and right spinor by coefficients when we want to add them?
What a masterpiece !
Love this thank you
I wonder if Chris got to the lecture on Clifford Algebras, stumbled on Hestenes, and went down the Geometric Algebra rabbit hole. Hope so. GA is a religious experience for us physics nerds. :)
Yeah, I've learned the basics of GA. Videos 11-15 will be on them (although I'll be calling them "Clifford Algebras", which is the same thing).
@@eigenchris Awesome. Can't wait. 💯🎉
For all the discussion on Weyl Spinors, wouldn’t it be easier to refer to epsilon as i? It squares to -1. Why is it normally given a different name in this context? Also, does -i (or -epsilon in the lingo) fulfill the equations as well?
Epsilon is more complicated than just "i" despite the fact that they both square to -1. Epsilon maps a pair of spinors to a scalar value using a very particular set of rules that are left invariant under SL(2,C) transformations. "i" doesn't do this in the same way.
@ Hm. I fail to see the fundamental difference. It seems to act precisely like a matrix version of i (or a so3 rotation matrix with the angle pi/2). And the other version for -i seems to fulfill all equations as well. But i’ll try to wrap my mind around it~ thanks for answering as always
Will you teach gauge theories
I don't have plans to teach them fully. More just mention them.
At 23:15 I think there's a missing "i" in the exponent on 3rd line
then on 4th line |B|² might be missing some exponent part
Oops. You're right.
Thank You so very much.
8:38 I miss the orthochronous condition Λ_00 > 0 in the SL(2,C) representation - is there a direct connection to the left/right representation? I remember from physics that antimatter = matter traveling back in time and being related to complex conjugation...
I'm still studying QFT so I can only give you a partial answer. With the group O(3), it has 2 "disconnected" parts, which are the det = +1 part and det = -1 part. We can continuously transform WITHIN either of the 2 parts separately, but jumping from one to the other requires a discontiuous jump, which is mirror symmetry, or a "parity reversal". Restricting to SO(3) restricts us to the det = +1 part.
The group O(1,3) has 4 disconnected parts, which can be disconntinuously "jumped" between using parity reversal P and time reversal T (we could name the 4 parts: "1, P, T, PT"). Restricting to det=+1 gives SO(1,3), and restricts us to the "1, T" parts, and restricting further to Λ_00 > 0 gives us SO+(1,3) and restricts us to just "1".
There is a 3rd discontinuous symmetry operator call "charge conjugation", which swaps positively charged particles and negatively charged particles.
If a system obeys CPT symmetry, it means that doing a parity flip P, a time reversal T, and replacing all matter with antimatter using charge conjugation C, it means all particles would appear to obey the same physical laws that we see in our universe.
However, CPT symmetry does not always hold. In particular, we know the weak force treats left and right handed particles differently.
I can't say much more as I'm still learning, but maybe this will give you a basis to do more searching online.
@@eigenchris I thought "holding" a symmetry means having a Lagrangian that doesn't mix different parts of a representation used to define the state. So like when a mass term is mixing upper and lower part of a Dirac spinor, these two can not be separated into two independent equations to the effect that the particle oscillates between the two (components denoting roughly probability or - in second quantisation - give rise to cross terms with creation and annihilation operators of different particle types that form the basis of the state representation). But maybe I'm just phantasising for I find it hard to interprete the maths. I was hoping that the lower part of the Dirac spinor would be the Λ_00 < 0 part of the Lorentz group or something.
Btw, I'm very happy that you study geometric algebra, for I remember suggesting that in a comment under some online lecture you gave to friends/collegues when you wondered about spin. You are much smarter than I am, so I just lean back and watch how you sort it all out :-) Thank you for sharing all of this!!
So with the hints from Chris and GPT, I found that:
1. SL(2,C) acts like the orthochronous, proper Lorentz group, meaning Λ_00>0 i.e, no time reversal T, also no spacial reflection P.
2. This way, the handedness/chirality is not affected, and the two Weyl spinor parts of the Dirac spinor do not get mixed, so it is like two 2x2 matrices on the diagonal of a 4x4 matrix each representing the same boost+rotation on their respective Weyl spinor
3. SL(2,C) acts as a double cover of the orthochronous, proper Lorentz group, so there are always two different 2x2 matrices that do the same.
4. You can have the full Lorentz group by acting with two independent copies of SL(2,C) on a tensor product of two complex 2D vectors, yielding a 4x4 matrix once again. This SL(2,C)×SL(2,C) setup seems to be equivalent to having only one SL(2,C) acting on both Weyl spinors as above, but then optionally composing with P and T operation, where P is just a matrix (γ0), but T involves additional complex conjugation. Here, only P actually swaps the upper and the lower Weyl spinor (with sign changes) and thus chirality.
5. The charge conjugation C acts mathematically as complex conjugation, but this also undoes the composition of P and T with their combined component and sign swapping. CPT = 1 leaves thus all states (tensor product of all representation vectors) in the standard model unchanged. So, at least in the current standard model C = (PT)^-1 = TP which should reflect in the math.
6. I still cannot match spacetime geometric algebra STA, where the Dirac spinor is a full multivector, the left chiral spinor is scalar+bivector (even), and the right chiral is vector+trivector (odd). But P and T seem to flip only signs of some blades, and they do not change grade, so there must be some difference to the complex matrix setup of QFT. Hopefully, we will get there eventually...
7. Regarding symmetry: The Lagrangian that couples left and right chirality will yield vertexes that convert right into left particles, but can still be C invariant when both processes are even likely. So separability (like no mass) is sufficient but not necessary for C invariance and the *overall* conservation of matter/antimatter.
As to 6., Chat-GPT told me this - a bit disappointing, if true. Maybe, it gets clearer over time:
The Dirac equation can indeed be transcribed into the framework of spacetime algebra (STA). The catch with chirality in the context of STA is not that it cannot be represented or studied, but that the representation and interpretation of chirality are different from the complex spinor representation used in particle physics.
In the complex spinor representation, chirality is described using Weyl spinors and the SL(2, C) × SL(2, C) representation of the Lorentz group. In STA, we can represent Dirac spinors using multivectors and study their behavior under Lorentz transformations and parity operations. However, the geometric interpretation of chirality and its transformation properties are different in STA compared to the complex spinor representation.
In STA, a Dirac spinor can be represented as a multivector, which can be decomposed into even and odd parts. The even and odd parts are sometimes interpreted as left- and right-handed chiral components, respectively. However, this interpretation does not have a direct correspondence with the complex spinor representation in terms of transformation properties.
As mentioned earlier, one alternative is to use separate spinor spaces to represent left- and right-handed components in STA, where both components are even-grade multivectors (scalar + bivector). This approach allows you to represent and study the left- and right-handed chiral components of a Dirac spinor in the context of STA, but the correspondence with the SL(2, C) × SL(2, C) representation is not direct, and the geometric interpretation will be different.
In summary, the "catch" concerning chirality in STA is not that it cannot be represented or studied, but rather that the representation and interpretation are different from the complex spinor representation used in particle physics, which may lead to different transformation properties and geometric insights.
@@franks.6547 I would be careful asking ChatGPT about spinors. I've tried asking it a few things, and while it is sometimes right, it definitely hallucinates often. I think spinors are too obscure a topic for it to answer about reliably.
16:52 why can't we say the same about SU(2) matrices that r=1, thus overall 2 constraints?
I didn't quite get left and right representations of spinors. I tried so multiply right spinor to left dual spinor and got Weyl vector, but with inverted space coordinates. Does it mean that right spinor is the same spinor as left spinor, but in a space with inverted x, y, z?
Yes. Inverting the x,y,z axes (called a "parity transformation", often denoted with "P"), will swap left and right spinors.
Personal opinion time: this subject is a hundred times better with geometric algebra.
Yeah, I'll be getting there in video 11. I wanted to present a matrix version of everything first, since a lot of physics textbooks take a matrix approach. GA is still in the process of gaining traction with physicists.
The problem with geometric algebra is how the geometric product is ill-defined as a sum of inner and exterior product, because for even grade, that doesn't apply. Also the "gradient" with vector basis is thought as a derivation, but that is false so it is impose by the overdot notation. I think you can do a derivation if you use bivector basis instead.
@@rajinfootonchuriquen you seem to be confused. The geometric product is perfectly well-defined for all grades (and the definition isn't "sum of interior and exterior products", though that happens to be what the geometric product equals for vectors specifically).
@@MatthewDickau what is the exterior product of two bivectors?
@@rajinfootonchuriquen
It depends on the dimension considered and the types of bivectors.
In 3-d, the outer product of two bivectors may have grade 4. But since the grade over than 3 does not exist in 3-d. In this case the result is just zero. However note that this is not ill-defined.
27:07 That's a completely new version of what it means to be an inner product. Could use some detailing.
It's really a "wedge product", or "symplectic form". It comes up in symplectic geometry.
27:06 I am also always skeptical of this top-down logic of "seeing what maths work", in this case quite arbitrarily turning the Hermitian conjugate into a transpose. Why is that "allowed"? What's the bottom-up logic? What is the intuitive meaning of the object that you get when you take the inner product of two spinors (for real vectors this would be angles, and for complex vectors it's a more abstract angle in complex space, while on the Riemann sphere they become concrete, so can we also visualize the Weyl inner product as visibly encoding distance and angle)?
Unfortunately in this case, I don't have an intuitive reason. The best I can say is that the group SL(2,C) is equivalent to the "symplectic group" Sp(2,C) and the key invariant for symplectic groups involves the symplectic form epsilon that I showed. But I think this is just delaying the real cause to another question. Maybe googling "symplectic group" or "symplectic form" can help you learn more.
@@eigenchris I should have commented after watching because I just got to that point. I have been studying symplectic geometry in the context of Hamiltonian mechanics and thermodynamics as it's one of those topics that you know about but never actually get to, and my interest is now really piqued.
@OrkTv Yeah I'm not sure what the connection is. I think the symplectic group comes up in the case of "anti-symmetry". In Hamiltonian mechanics we often used the symplectic form (wedge product) to measure oriented areas in phase space. Swapping the order of the vectors making the parallelogram will reverse the orientation of the parallelogram, giving you a minus sign. In QM, spin-1/2 particles are fermions and obey anti-commutation relations, so swapping their order gives you a negative sign (also, trying to create 2 fermions in the same state gives you zero under this anti-symmetry, which is expected since spin-1/2 particles obey Pauli's exclusion principle). Not sure if there's any connection beyond that.
@@hyperduality2838 That's nonsense
20:00 It wasn't obvious to me why C = k₁A* so I detailed it: if A is zero, C can be anything, so let's assume A ≠ 0. AC is real and AA* = |A|² is also real, so AC/(AA*) = C/A* is also real. Call this number k₁.#
when is the video on clifford algebra coming?
Sometime in July. I'm working on it now.
10:46 is this like a spacetime split in spacetime algebra
Yes. Each Sigma matrix can be written as a pair of gamma matrices. I'd suggest Sudgylacmoe's "a swift introduction to spacetime algebra", but my guess is you've already seen it.
What about Majorana spinors? Good stuff
I don't really know what those are, but I think they are a special case of Dirac Spinors that come up in some very specific cases. I didn't include them because they seem very niche and possibly don't describe any particles we know of currently.
@eigenchris they are their own antiparticles, related to sterile neutrinos, and in quantum computing. But yeah, not at the top of the list :-) Thanks again.
should name it spinors for beginnors
I only read it as such every time, the big funny.
Awesome!
Wonderful !!!
Q: How do Majorana spinors fits in this entire opera?
I'm not as familiar with those. I know that they are a special case of a 4-component Dirac spinor, for a particle that equals its own anti-particle. I'm not sure what the exact mathematical condition for this is, though.
@@eigenchris is complex conjugation of the SL(2,C) the way to map between particles and anti-particles?
@@samanthaqiu3416 No, that flips between the left and right chiral representations. That's called "parity conjugation", usually denoted with "P". You want "charge conjugation", which is usually denoted "C".
15:50 "Since we're dealing with complex numbers..." What is this notion predicated on? The SL(2,C) matrices come from Lorentz transformations when these are realized on Weyl vectors. Apparently we can represent this if and only if our special linear matrices admit complex entries. Could this be elaborated a bit? (I really don't like it when, as often happens, people pull complex numbers seemingly out of nowhere. People start talking about "complexified" algebras without proper justification, and in this case I think we're almost there.)
My fist instinct is to say that since Weyl vectors already contain complex numbers, it's reasonable to say that the matrices that transform them will also be complex. We sort of made a similar assumption with the U matrices that transform Pauli vectors. Basically, we start with the most general 2x2 matrices that would transform a 2x2 complex matrix, and then narrow down their properties from there.
Another argument could be parameter counting, because there are 6 types of Lorentz transformations and SL(2,C) matrices have 6 free parameters, whereas SL(2,R) matrices have 3 free parameters.
@@eigenchris Makes sense. I guess it would be possible to relate Lorentz matrices and SL(2,C) matrices one-to-one if I sat down, with the complex numbers being introduced because complex numbers are already there in our representation. Thanks as always!
Nice
Color coding spinors is really helpful. Notation, notation. notation! $%@#!
whoever deslikes this must be clinically insane
When you make a longer video, please try to vary your vocal tone based on context, emphasis of a significant result and so on, instead of sounding like an unitonal automaton just reading off a script file.
😀
yep I'm lost
Sorry to hear. Is there anything I can help clear up?
Great series but +---
So, why would anyone do modern physics in 3D? We know it is not real. Intervals are real. Distance is an illusion. "locality" based on 4D interval is not the same as naive locality.
Most modern particle physics does use 4D spacetime. However, basic quantum mechanics and some condensed matter systems don't require relativity.
@@eigenchrisHmm, okay. But I see a lot of confused stuff out there like the setup of the EPR "paradox" where it is described in 3D but in 4D there are intervals of zero all over it which have bearing on the problem of "locality". And that is often treated like "basic quantum mechanics" and extrapolated to multiverse, with nary a mention of the intervals. What are the guard rails for knowing you have oversimplified by using a 3D model?
Great videos but your voice is so monotone… makes it much harder to follow imo
second lol
Make more joke videos 😭😭😭
Thank you - another great lesson. A little heavyweight this time, but maybe that just goes with the nature of the subject; I'll rewatch later, with the advantage of foreknowledge of where it's going. It probably doesn't help that my Special Relativity is about 35 years old and unused since, so it's very dimly remembered!
I've seen mention of some of the groups involved being "double covers": is that the right description for the relationship between SU(2) and SL(2,ℂ)?
Yes. SU(2) double-covers SO(3), and SL(2,C) double-covers SO(1,3). A better name for SU(2) is "Spin(3)". It turns out there's a general way to build Spin(n) groups which double-cover the SO(n) groups, which I will cover when I get to Clifford Algebras. There's also Spin(p,q) groups to double-cover SO(p,q).