Sorry about hiding the earlier version I uploaded today. Caught a last-minute mistake. Really wish UA-cam had a "re-upload" option that maintained the same video URL.
just a few weeks I started reading spinors in spacetime by Penrose, and the beautiful mathematics of writing the celestial sphere as a complex number in the Argand plane. It becomes pretty dense quickly and it is hard to read after the first few chapters because it's not entirely clear what is the significance of the machinery developed
@@samanthaqiu3416 I think I started that book, but didn't make it very far. Not sure I even understood the "celestial sphere" part. I won't be addressing that directly, but hopefully you'll grasp what spinors are following this series.
@@unknownstoneageman81 Hi, I'm a full stack webdeveloper. Your post made me think and gave me an idea. Technically if the correction is small they could change the storage with more precision. Which makes it unnecessary to delete the old video (which is cheap but defragmentation is not) and upload a full new upload. The check for the difference between videos could be on the frontend by saving the original upload in the browser. This way it could save them server cost.
Your tensors for beginners playlist was the thing that finally made tensors click for me years ago and allowed me to dive deep into GR, and for that I will always be grateful. I’m excited to have a similar experience now with spinors! Thanks so much for sharing your knowledge with us!!
Two things I want to ask: 1) is it fair to call a spinor a tensor? I know what you meant, but the fact that under a rotation of 2π they are flipped around means that they don’t transform like tensors. 2) I’m not super familiar with trivectors but weren’t two of the blue arrows on your trivector diagram flipped around backwards? If not, why are both blue vectors on the top and bottom planes pointing the same direction? Thank you!
1) It depends on what you mean by "tensor"... When it comes to the word "vector", we often mean the specific case of a rank-1 tensor. But the more general meaning of "vector" is "an element of a vector space", which means something we can add and scale. Tensors all belong to vector spaces (we can add and scale them), so under this broad definition of "vector", ever tensor is also a vector. When it comes to spinors vs tensors, we normally think of tensors as having rank-0, rank-1, rank-2, etc. Spinors are an extra "generalization" of tensors with rank-1/2. We can use them to make objects of rank-1, rank-3/2, rank-2, rank-5/2, and so on. So spinors are like a generalization of tensors. But if we define "tensors" in a general way as "multilinear maps", then every spinor is also a tensor under this definition, since spinors are multilinear maps. We can give spinors covariant and contravariant spinor indices, similar to what we do with tensors. I'm sorry if this answer was confusing. I can try to give a better one if you're lost. 2) The short answer is that the diagram is wrong, or at least, not too informative. I'll get into more detail when I discuss trivectors, but every multi-vector has an orientation. For vector, the orientation is just the direction it points in. For a bivector, the orientation is either clockwise or counter-clockwise. For a trivector, you can define an orientation by given each of its 6 faces an orientation in a paritcular way. I tried to convey this with arrows, but I think I did it wrong, or at least did it so badly that it's kind of meaningless anyway.
@@eigenchrisaren’t tensors not vectors even by the broad definition because you can’t add a rank-a tensor and a rank-b tensor together? Or is it just that the rank-a tensors belong to the rank-a-tensor vector space and the rank-b tensors belong to the rank-b-tensor vector space?
@@ididagood4335 the latter is certainly true but general tensors form an algebra over a given vector space and can be combined within it. there might be some type of tensor bundle over a manifold but i don’t remember
I'm really grateful that you're putting this together. I come across spinors every now and then and think "WTF, why does nobody explain these properly?" Now all you have to do is rename the playlist "spinors for beginors" :)
And "Tensors for Densors" for tensor playlist? 🤣 Honestly matrices have always been my weakness. My mathematical nemesis, so I doubt I will ever understand either tensors or spinors. Closest video that did it for me was one from SoME2 - with Dirac Belt and with showing "rotations" as going through inside of 2 spheres in straight path. Can't explain it well here, but it made sense, though it wasn't a short video of course.
Unfortunately, I can't tip you, in cause of my current location, but I wish you luck in the series you're making. As MCs math. physics student, I'm already familiar with everything you've said, However, in these 19 minutes I'm feeling my mind cleared a lot, things start to make complete sense, and there are no words for me to describe, how grateful I am for that. You're basically making a solid base for my education, which is kinda flows in air. Sincerely yours
I have been studying Spinors for years. Today is the very first time that I get it. The metaphoric concept of 1/2 spin is what did it for me. I also studied ALL your videos on Tensors. Thanks.
Remarkable Chris. My field is chemical engineering a trillion miles from your field, but I could grasp the ideas, even though there is a vast amount of depth behind each slide. Great channel and videos, you’re a gift to us. Bless you.
I too an a ChE. While on my way to my PhD I chose a minor in Math. As the research on my thesis intensified, my family grew and money became an issue, I learned that I needed only two courses to reach a paper-only master's degree in Math. But I also learned that set theory and number theory would have required me to stay another year beyond my PhD graduation. I opted for a job. Now in my dotage I have been studying about quantum mechanics, especially superstring theory and I trying to understand the concept of spinors
@@throne1797 That’s wonderful, you’re doing really well, really happy for you. We need to stay curious, it’s when we’re not the brain dies and the heart breaks. I can solve a 3x3 rubiks cube in a respectable 3.5 minutes now, I do it 3 times a day, more than I floss😊 Well done to you👏
I just finished all 22 videos -- it took a couple of months! This is an absolutely fantastic lecture series. The quality of the explanation is top notch. It gave me a much deeper appreciation of the math underneath the physics of QFT. Thank you for putting this together!
I literally started reading about spinors today couldn’t have posted this at a better time! I really appreciate your work towards the betterment of math and physics concepts in general and your videos are really helpful!😊
3:12 thanks for your initiative. As a student I was confused too with many physics and math books too. I don't know why professors think it is not necessary to write an understandable and comprehensive books on hard topics. Your videos show that it is possible to explain complex stuff so that one can follow.
My professors always used to say that their lectures are easy enough so that even an elementary level of math and science can mke through. In that scale of difficulties, you did explained as if I'm 5. Awesome video!
Your voice and pacing and expressions were made to be able to teach people. Something about it is so soothing yet so expressive of knowledge. It somehow really helps to understand such complicated topics!
Brilliant! I absolutely LOVE your measured, well-researched and qualitative approach to these difficult, abstract, yet deeply fascinating quantitative topics. Can't wait to watch your other videos.
At 7:52, I believe that there's a point of possible confusion regarding the discipline-specific use of the word "rank". In mathematics, a 2x2 matrix that can be "factored" into the (outer) product of a row vector and column vector is a tensor of rank 1, not rank 2. A 2x2 matrix that *cannot* be factored in this way IS a rank-2 tensor. In mathematics, "rank" is defined to be the number of linearly independent columns (or rows) in the matrix. If a 2x2 matrix can be factored the way you have depicted, then the 2 column vectors (or rows vectors) inside of it are not linearly independent, meaning that the column space has only 1 dimension and not 2. Likewise, in mathematics, if a 2-d square matrix having N rows and columns can be "factored" as the (outer) product of a row vector and column vector, each having N components, then it is a rank-1 tensor, and not rank 2, nor rank N. When abstractly representing the set of numbers associated to a tensor object, we often use a variable having a set of subscripts, like A_xyz. In mathematics, the number of such subscripts is called the "degree" and not the "rank". In engineering, and often in physics, the number of such subscripts IS, unfortunately, also called the "rank". Thus, two different uses of the word "rank" can create confusion. Spinors are rarely used in engineering, and appear mostly in physics. Unfortunately, there has not been a consistent use of the word "rank" within physics, and even the word "tensor" can be problematic. In some physics contexts, tensors are considered objects which obey certain transformation rules. In other physics contexts -- quantum mechanics in particular, tensors are treated the same way as they are in mathematics.
Bravo! This is the most clear and concise description I have ever seen that literally takes you from cradle to grave in half a dozen concise steps. If I had only had this video when I took quantum mechanics, my goodness, how many hours of my life it would’ve saved for other more productive learnings.
The graphic at 5:47, along with the comparison of orthogonal state space vectors to physical space, was the best explanation I’ve seen so far. Excellent!
THis is helpful already, connecting dots between different math concepts that I knew are related but could not comprehend fully. Glad you put Clifford Algebra in there. Keep going.
This is fantastic! I spent a summer trying to study Clifford Algebra 15 years ago and gave up because there simply was no lower rungs like this to get on the ladder - even from professors!
🎯 Key Takeaways for quick navigation: 00:00 🧒 Spinors are mathematical objects used in advanced quantum physics, particularly to describe fundamental fermion particles with spin-1/2. 02:10 🌀 Spinors have the property of requiring two full turns (720 degrees) to return to their starting position, unlike vectors that return after a 360-degree rotation. 05:05 icon The relation between the abstract state space and the physical space is projective. Two planes, one at z=0 and one at z=1. The Bloch sphere touches the z0 at [0,0,0] and the z1 at [0,0,1]. A quantum state is a vector from z0 at [0,0,0] to z1 at some point. That point is mapped to the Bloch sphere by projection (the point on the sphere which is a scalar of the vector). 05:08 icon The orthogonal state is the point on the Bloch sphere where the orthogonal vector in the vertical plane hits the Bloch sphere. The ray through this point intercepts the z1 plane at a point which is at radius inverse to the radius of the first point and in the opposite direction. It is the negative of the reflection in the circle. 05:19 icon Taking a circle on z1 centered at [0,0,1] and of radius 1 are points which are orthogonal to their negation (opposite on line through [0,0,1]. 06:01 icon Measurement is a projection from the abstract quantum state to the actual physical state. It is literally a projection so the probability depends on the spread (angle squared) but is then fully determined (although experimentally challenging when the spreads are very small). 07:53 √ Spinors are described as the "square roots" of vectors, and they can be factored into column and row spinors, which are like rank-1/2 tensors. 10:35 🧮 Clifford algebras are used to define spinors in any dimension and involve concepts like bivectors, trivectors, and the wedge product. 13:35 🌀 In particle physics, different particles have various spin values, and spinors are used to describe their transformations under changes of reference frame, involving Lie Groups and Lie Algebras. 16:14 📚 Quantum Field Theory (QFT) utilizes spinors to describe matter particles and their interactions with various fields, such as the electron spinor field interacting with the photon vector field.
Awesome. I received my (undergrad) physics degree a semester ago and this is one of the topics I REALLY struggled with. I'm excited to watch this series!
My physics education ended with tensors and never got to spinors and so when they kept cropping up when reading popular science books and physics articles I tried without much luck to find an overview of them that didn't need several more years of physics and maths than I'd done, which was annoying. This video was exactly what I wanted so thank you very much! I was mildly alarmed at finding "Spinors for Beginners 11" in my search results lol, so I'm glad I decided to see what the first video was like, I'll see how far I get with the rest of the series :)
Thank you, I’ve just started to study QFT and many book get for granted that anyone has already a well established idea on tensor. This video already made me get a grasp of the core principles of this wonderful mathematical objects!
Finally a new good UA-camr channel discover... =). By the way, the first part of the video gave me a flashback from my teenager times (10 years ago), I went with my high school here in Spain to visit our city university and a young recent graduated gave us talk in Physics, when she ends the infantilizated topics, I rise my hand and ask "What's the difference between a boson and a fermion?", She started to sttuter and my teacher just tell to not say nothing alse and friendly to "shut up" and I decided the next days that I will not go college and I spent my first young ages reading Nieztche, smoking weed and working with my father in art. I have not regrets.
For what i had seen about geometric algebra, It should be able to encode real and imaginary scalars, vectors, quaternions, octonions, spinors, Pauli and Dirac matrices, tensors, Lie and exterior algebras. Yet, I haven't studied It, just relying on these promises. Thrilled to see your perspective on these.
@@eigenchris XylyXylyX also has a video series currently being released on Geometric Algebra as it relates to electrodynamics. Looking forward to your spinor series! You guys are doing great work
Freya Holmer has an awesome talk she gave. I think it was titled something like how do you multiply vectors. It ends up with spinors and geometric algebra. Useful with quaternions and rotating vectors. That was my first introduction to them and I'm keen to learn more. On a side note, her visualizations are second to none. Everyone should watch her two Bezier Curves and Continuity of Spline videos. Top notch.
THIS IS such a good resource!! Thank you so much for sharing you knowledge in such a well paced and well thought out way! We need more of this in physics!
I would love for you to do a breakdown of the math on spinors, like how to derive them or use them in applications, because for someone like me these introduction videos are great but my math skills are terrible, so it would amazing to see a walk through on the math of these topics as well.
Thank you Chris. I’m constantly struggling with various mathematical concepts due to the lack of clarity in some text books. Thanks to your tensor calculus series I was able to understand not only tensors but other topics, since it helped me fill gaps that I had in other topics. Even this introductory video helped me fill gaps related to spinors, exterior algebras, and Clifford algebras. I’m looking forward to watch your spinor series. You deserve a tip 👌🏽…
I've seen a bunch of mathematicians be advocates to stop thinking of vectors as pointy sticks. But I think since this is for 5 year old, they might give you a pass. Awesome video btw!
I'm sure that these have a real application and are a genuine thing but I'm a history and archaeology dual major with anthro and performing arts miners. I have never heard a lecture that struck me as the unhinged ramblings of a monster than this one and I had to listen to an old man slobber over thirteen year olds one semester. Liked and shared with math friends who might not panic when you say something like "quantum fields" or "division"
At 4:21, "a good place" is a singular noun. In English, we use singular verbs with singular nouns. We say the subject and verb agree in number. We do not normally say "A good place are examples from physics." We say "a good place is examples from physics..." Five-year-olds understand this.
Yaaaas I can't wait for the Clifford algebra explanation. I've never cared for matrices or tensors because it seems like they don't carry all the important information, e.g. you obtain your column vector components using a vector basis, and the basis is no longer part of the object. I like Clifford/Geometric Algebras because the objects have transparent meanings and defined relationships that can be reasoned through in a straightforward way. In other words, the object is both the components and basis, and that makes it much easier for me. So I'm psyched to follow this series!
Up/down spin, like the cup trick (t=2:30), is looking under your armpit, compared to looking over your shoulder. One is (looking) upside down, and the other is (looking) right-side up. The spinors as square root of a vector, is 'exactly' the same as adding 'i' (the square root of -1) to the reals to be able to turn positive numbers around to reach negative numbers in two steps (multiply's). There are a similar number of confusions, contradictions and odd things about complex numbers as there are about spinors, but most folks have already internalised the contradictions for complex numbers (brow beaten from basic maths course). It's all about changing "symmetry" from linear shifts to rotational shifts ;-)
This video series are really amazing. So far I have watched all of them because they are so perfect．I am looking for viewing the videos in final phase in the staircase.
Thanks for building this staircase. Looking forward to ascending it. I think you are going to connect a lot of concepts for me and that's always very satisfying.
Congratz on 100k subscribers! Afaik recently you had like ~89k or so. You opened my eyes for some of the math notations that is used in quantum computing and the requirement for reveribility and how it limits some of the possibilities for quantum computing. We all started somewhere, we all, at some point have been... Beginnors!
A matrix is a 2nd rank tensor. A vector is a 1st rank tensor. A scalar is a 0th rank tensor. Chris' statement that a Pauli vector is a ½ rank tensor is wrong, at 8:46. Still a great video.
Very happy to see you make some videos on this - our quantum mechanics prof dropped the word spinor on us just earlier this week without explaining what it was, so the timing here is just perfect! One small note though, at 14:32 I think you forgot to put the angles into the exponent in the left half of the equations as well - as it stands there, the equations only hold for θ=Φ=Ψ=1.
As someone who has been struggling with the concept of spinors for a long time, I find this to be a very nice introduction. Just summing up various ways of looking at the concept that make complete mathematical sense. There are still some minor lacunae, like I don't think it's intuitive what a rotation in phase space is, or how it doubles to a rotation in real space; and the bit on Clifford algebras can use the remark that the algebra elements are kind of like if we treated basis vectors like objects you can multiply, and a basis vector squared is the magnitude squared. At least, that's what I found to be the most straightforward conceptualization of geometric algebra. I'm looking forward to the rest of this serious, and I hope it will bring progress in finally putting this highly complicated topic to rest.
Basis vectors know nothing of multiplication and magnitude, those are facets of the theory within which you are utilising them. Horses for courses sort of thing. For example: I can walk forward/backwards, Left/Right. Up/Down. One step in each of Forward, Left and Up constitute constitutes a basis vector in 3 Dimensional walking. The idea of multiplying left by up is a nonsense in this setting. Different people at different places on the globe at different heights and facing different directions can draw their own arrows and define their own basis vectors by dint of their position and all will be different! An infinity of basis vectors ! And each set of 3 being equally valid to enable walking on Earth, no matter where. My point is that a vector basis is independent of the attributes your field of research adds to them. They are a base class in C++ and are not defined by the classes that you construct from them.
5:37 I get that physical space and abstract space work differently in this instance, but how can you tell apart a from a particle in physical space? There’s a clear difference in the abstract, but what use is it if it doesn’t affect physical space? or is there a situation where it does?
There's no measurable difference between "positive spin up" and "negative spin up" in the real world. In fact, even more generally, you can not tell the different between the "spin up" state and "spin up multiplied by a complex phase e^iθ" (in the special case of θ=π, this becomes -1). This property is key to why spinors pop up in quantum mechanics. It's the same with the polarized waves example: if a wave is vertically polarized, its negative is also vertically polarized (any arbitrary phase shift is also vertically polarized). If you're familiar with the concept of "electric potential" in electrostatics, you might also know you can add a constant potential everywhere in the universe without changing the electric field, because the electric field only cares about potential DIFFERENCES, so constant offsets in the math are not physically measurable. I'll talk about all this in the series. If you keep studying these ideas, you end up in "gauge theory", which is central to modern particle physics.
Really clarifying as usual. Although I'm not so interested with this topic, I will follow the series just because I love to be led along such a hard path: your exposition makes it interesting and tickles my curiosity...
I have been obsessed with Geometric Algebra (GA), I literally cheered when you mentioned bivector and trivector :D. I wasn't sure if you were talking about GA when you mentioned Clifford Algebra because I'm quite new to the subject.
Yeah, "Geometric Algebra" and "Clifford Algebra" are the same thing. (Some people argue they have slightly different definitions, but they're built on the same core idea.)
@@eigenchris True, but there is a subtle and confusing difference between the Clifford Algebra/Geometric Algebra notation system used by László Tisza of Massachusetts Institute of Technology (available via MIT OpenCourseWare) and the more unified Geometric Algebra notation system developed by David Hestenes that incorporates Clifford Algebra into a notation system for Physics with the geometric product of vectors and use of multivectors. Dr. Hestenes tried to differentiate use of Clifford Algebra notation from his unified notation system for math and physics by appropriating the name Geometric Algebra (which Clifford had used for Clifford Algebra). The point being that the Hestenes Geometric Algebra is more than just Clifford Algebra as uused by Tisza and allows one to work in a unified way with spinors, tensors, vectors and differential forms. geocalc.clas.asu.edu/GA_Primer/GA_Primer/introduction-to-geometric/rotors-and-rotations-in-the.html
Thank you for your kind explanation!!! It's very hepful to me. But I think there is an error at the time 9:04 for the Rotating Pauli Vector and the -i in the rotating matrix should be used instead of i. This is consistent with the rotor in clifford algebra.
Another application of spinors is the Newman-Penrose formalism for general relativity and the Petrov classification of the algebraic classes of the Riemann tensor. Very powerful in my opinion. Spinors allow decomposition of null vectors in relativity (an example of the "square root" of a vector). Then the next stage is "twistors". Oh no another "strange" mathematical abstraction! Regards. Nice videos!
A question about the correspondence between Physical space and State space around 5:00. My intuitive instinct was to try and extend this concept to a third orthogonal state. I can see that obviously we cannot have on the 2D plane more than 2 orthogonal states which are physically 180° apart, but can we have objects which have more orthogonal states at a different spacing? I can't really see how the geometry of the phase space would hold up though.. More to the point, a vector can be an object with a number of different of amplitudes along different orthogonal directions. That is, we have Hilbert spaces where there is a ket encoding the probabilities of for example a ton of energy eigenstates. Why can we not describe spinors like this too?
The 2D circle I show for QM is really a partical view of the 3D Bloch Sphere. And the 2D circle for light polarizations is a partial view of the 3D Poincaré Sphere (the two are mathematically equivalent). I talk about them both in videos 2-5. The word "vector" means different things in math. The abstract meaning is "an element of a vector space". The kets in a Hilbert space are this kind of abstract vector. There's also the more "physical" definition where a vector is a 3D pointed stick. A Pauli spinor is what you get when you decompose this 3D pointed stick into 2 pieces using a special mathematical trick. Both Pauli spinors and 3D poimted-stick vectors can be "kets" in an abstract vector sapce.
I cannot believe you passed on the opportunity to call this “Spinors for Beginors”
Brilliant
You win the a shiny new internet.
7:54
I don’t get it
Sorry about hiding the earlier version I uploaded today. Caught a last-minute mistake. Really wish UA-cam had a "re-upload" option that maintained the same video URL.
I think that would increase server costs for them. Would be appreciated very much if someone who knows well responds
Du bist so guter Dinge
So heiter und rein,
Und wen du ein Fehler begingest,
Konnt's keiner sein.
just a few weeks I started reading spinors in spacetime by Penrose, and the beautiful mathematics of writing the celestial sphere as a complex number in the Argand plane. It becomes pretty dense quickly and it is hard to read after the first few chapters because it's not entirely clear what is the significance of the machinery developed
@@samanthaqiu3416 I think I started that book, but didn't make it very far. Not sure I even understood the "celestial sphere" part. I won't be addressing that directly, but hopefully you'll grasp what spinors are following this series.
@@unknownstoneageman81 Hi, I'm a full stack webdeveloper. Your post made me think and gave me an idea.
Technically if the correction is small they could change the storage with more precision. Which makes it unnecessary to delete the old video (which is cheap but defragmentation is not) and upload a full new upload. The check for the difference between videos could be on the frontend by saving the original upload in the browser.
This way it could save them server cost.
Your tensors for beginners playlist was the thing that finally made tensors click for me years ago and allowed me to dive deep into GR, and for that I will always be grateful. I’m excited to have a similar experience now with spinors! Thanks so much for sharing your knowledge with us!!
Two things I want to ask:
1) is it fair to call a spinor a tensor? I know what you meant, but the fact that under a rotation of 2π they are flipped around means that they don’t transform like tensors.
2) I’m not super familiar with trivectors but weren’t two of the blue arrows on your trivector diagram flipped around backwards? If not, why are both blue vectors on the top and bottom planes pointing the same direction? Thank you!
1) It depends on what you mean by "tensor"... When it comes to the word "vector", we often mean the specific case of a rank-1 tensor. But the more general meaning of "vector" is "an element of a vector space", which means something we can add and scale. Tensors all belong to vector spaces (we can add and scale them), so under this broad definition of "vector", ever tensor is also a vector. When it comes to spinors vs tensors, we normally think of tensors as having rank-0, rank-1, rank-2, etc. Spinors are an extra "generalization" of tensors with rank-1/2. We can use them to make objects of rank-1, rank-3/2, rank-2, rank-5/2, and so on. So spinors are like a generalization of tensors. But if we define "tensors" in a general way as "multilinear maps", then every spinor is also a tensor under this definition, since spinors are multilinear maps. We can give spinors covariant and contravariant spinor indices, similar to what we do with tensors. I'm sorry if this answer was confusing. I can try to give a better one if you're lost.
2) The short answer is that the diagram is wrong, or at least, not too informative. I'll get into more detail when I discuss trivectors, but every multi-vector has an orientation. For vector, the orientation is just the direction it points in. For a bivector, the orientation is either clockwise or counter-clockwise. For a trivector, you can define an orientation by given each of its 6 faces an orientation in a paritcular way. I tried to convey this with arrows, but I think I did it wrong, or at least did it so badly that it's kind of meaningless anyway.
@@eigenchrisaren’t tensors not vectors even by the broad definition because you can’t add a rank-a tensor and a rank-b tensor together? Or is it just that the rank-a tensors belong to the rank-a-tensor vector space and the rank-b tensors belong to the rank-b-tensor vector space?
@@ididagood4335 the latter is certainly true but general tensors form an algebra over a given vector space and can be combined within it. there might be some type of tensor bundle over a manifold but i don’t remember
@@ididagood4335 Yeah, usually you can only add tensors of the same rank together. Similar to how you can't reasonably add a 2D vector to a 5D vector.
I'm really grateful that you're putting this together. I come across spinors every now and then and think "WTF, why does nobody explain these properly?" Now all you have to do is rename the playlist "spinors for beginors" :)
And "Tensors for Densors" for tensor playlist? 🤣
Honestly matrices have always been my weakness. My mathematical nemesis, so I doubt I will ever understand either tensors or spinors. Closest video that did it for me was one from SoME2 - with Dirac Belt and with showing "rotations" as going through inside of 2 spheres in straight path. Can't explain it well here, but it made sense, though it wasn't a short video of course.
This series is going to be so good! I'm excited 🤓
Tou should also make a video on spinors
Finally the much awaited series. This channel is like the netflix of mathematical physics. Thanks bro.
Unfortunately, I can't tip you, in cause of my current location, but
I wish you luck in the series you're making.
As MCs math. physics student, I'm already familiar with everything you've said,
However, in these 19 minutes I'm feeling my mind cleared a lot, things start to make complete sense,
and there are no words for me to describe, how grateful I am for that.
You're basically making a solid base for my education, which is kinda flows in air.
Sincerely yours
Oh my god this provided more high-quality explanation than 2 hours of Wikipedia/Google searching. Thanks so much!
I have been studying Spinors for years. Today is the very first time that I get it. The metaphoric concept of 1/2 spin is what did it for me. I also studied ALL your videos on Tensors. Thanks.
Remarkable Chris. My field is chemical engineering a trillion miles from your field, but I could grasp the ideas, even though there is a vast amount of depth behind each slide. Great channel and videos, you’re a gift to us. Bless you.
I too an a ChE. While on my way to my PhD I chose a minor in Math. As the research on my thesis intensified, my family grew and money became an issue, I learned that I needed only two courses to reach a paper-only master's degree in Math. But I also learned that set theory and number theory would have required me to stay another year beyond my PhD graduation. I opted for a job. Now in my dotage I have been studying about quantum mechanics, especially superstring theory and I trying to understand the concept of spinors
@@throne1797 That’s wonderful, you’re doing really well, really happy for you.
We need to stay curious, it’s when we’re not the brain dies and the heart breaks.
I can solve a 3x3 rubiks cube in a respectable 3.5 minutes now, I do it 3 times a day, more than I floss😊
Well done to you👏
As an engineer that like to see “what’s more than I know?” I really appreciated the style. Great job !🎉
Another no nonsense mathematical forest tour de force for physics series!!
Needless to say we are super excited!!
Thanks man...
This can be his masterpiece, eigenchris is explain spinors like they are sweets or candies.
I just finished all 22 videos -- it took a couple of months! This is an absolutely fantastic lecture series. The quality of the explanation is top notch. It gave me a much deeper appreciation of the math underneath the physics of QFT. Thank you for putting this together!
I'm glad you foudn it helpful! Many QFT sources tend to skim over spinors... but they are something you could take a whole course on.
This is the best coverage of spinors and tensors in relation to quantum fields. Explained so *clearly* !
I literally started reading about spinors today couldn’t have posted this at a better time! I really appreciate your work towards the betterment of math and physics concepts in general and your videos are really helpful!😊
3:12 thanks for your initiative.
As a student I was confused too with many physics and math books too.
I don't know why professors think it is not necessary to write an understandable and comprehensive books on hard topics.
Your videos show that it is possible to explain complex stuff so that one can follow.
My professors always used to say that their lectures are easy enough so that even an elementary level of math and science can mke through. In that scale of difficulties, you did explained as if I'm 5. Awesome video!
Also missed an opportunity to name this "Spinors for Beginors"
After going through tensors and relativity, I am so hyped and ready to go through spinors!
Your voice and pacing and expressions were made to be able to teach people. Something about it is so soothing yet so expressive of knowledge. It somehow really helps to understand such complicated topics!
Brilliant! I absolutely LOVE your measured, well-researched and qualitative approach to these difficult, abstract, yet deeply fascinating quantitative topics. Can't wait to watch your other videos.
At 7:52, I believe that there's a point of possible confusion regarding the discipline-specific use of the word "rank". In mathematics, a 2x2 matrix that can be "factored" into the (outer) product of a row vector and column vector is a tensor of rank 1, not rank 2. A 2x2 matrix that *cannot* be factored in this way IS a rank-2 tensor. In mathematics, "rank" is defined to be the number of linearly independent columns (or rows) in the matrix. If a 2x2 matrix can be factored the way you have depicted, then the 2 column vectors (or rows vectors) inside of it are not linearly independent, meaning that the column space has only 1 dimension and not 2. Likewise, in mathematics, if a 2-d square matrix having N rows and columns can be "factored" as the (outer) product of a row vector and column vector, each having N components, then it is a rank-1 tensor, and not rank 2, nor rank N. When abstractly representing the set of numbers associated to a tensor object, we often use a variable having a set of subscripts, like A_xyz. In mathematics, the number of such subscripts is called the "degree" and not the "rank". In engineering, and often in physics, the number of such subscripts IS, unfortunately, also called the "rank". Thus, two different uses of the word "rank" can create confusion. Spinors are rarely used in engineering, and appear mostly in physics. Unfortunately, there has not been a consistent use of the word "rank" within physics, and even the word "tensor" can be problematic. In some physics contexts, tensors are considered objects which obey certain transformation rules. In other physics contexts -- quantum mechanics in particular, tensors are treated the same way as they are in mathematics.
You're correct. I am using the word "rank" to mean "thr number of tensor indices", not "the number of linearly independent columns".
Bravo! This is the most clear and concise description I have ever seen that literally takes you from cradle to grave in half a dozen concise steps. If I had only had this video when I took quantum mechanics, my goodness, how many hours of my life it would’ve saved for other more productive learnings.
The graphic at 5:47, along with the comparison of orthogonal state space vectors to physical space, was the best explanation I’ve seen so far. Excellent!
Thanks. I was happy when I figured that out.
THis is helpful already, connecting dots between different math concepts that I knew are related but could not comprehend fully. Glad you put Clifford Algebra in there. Keep going.
This is fantastic! I spent a summer trying to study Clifford Algebra 15 years ago and gave up because there simply was no lower rungs like this to get on the ladder - even from professors!
🎯 Key Takeaways for quick navigation:
00:00 🧒 Spinors are mathematical objects used in advanced quantum physics, particularly to describe fundamental fermion particles with spin-1/2.
02:10 🌀 Spinors have the property of requiring two full turns (720 degrees) to return to their starting position, unlike vectors that return after a 360-degree rotation.
05:05 icon The relation between the abstract state space and the physical space is projective. Two planes, one at z=0 and one at z=1. The Bloch sphere touches the z0 at [0,0,0] and the z1 at [0,0,1]. A quantum state is a vector from z0 at [0,0,0] to z1 at some point. That point is mapped to the Bloch sphere by projection (the point on the sphere which is a scalar of the vector).
05:08 icon The orthogonal state is the point on the Bloch sphere where the orthogonal vector in the vertical plane hits the Bloch sphere. The ray through this point intercepts the z1 plane at a point which is at radius inverse to the radius of the first point and in the opposite direction. It is the negative of the reflection in the circle.
05:19 icon Taking a circle on z1 centered at [0,0,1] and of radius 1 are points which are orthogonal to their negation (opposite on line through [0,0,1].
06:01 icon Measurement is a projection from the abstract quantum state to the actual physical state. It is literally a projection so the probability depends on the spread (angle squared) but is then fully determined (although experimentally challenging when the spreads are very small).
07:53 √ Spinors are described as the "square roots" of vectors, and they can be factored into column and row spinors, which are like rank-1/2 tensors.
10:35 🧮 Clifford algebras are used to define spinors in any dimension and involve concepts like bivectors, trivectors, and the wedge product.
13:35 🌀 In particle physics, different particles have various spin values, and spinors are used to describe their transformations under changes of reference frame, involving Lie Groups and Lie Algebras.
16:14 📚 Quantum Field Theory (QFT) utilizes spinors to describe matter particles and their interactions with various fields, such as the electron spinor field interacting with the photon vector field.
Awesome. I received my (undergrad) physics degree a semester ago and this is one of the topics I REALLY struggled with. I'm excited to watch this series!
My physics education ended with tensors and never got to spinors and so when they kept cropping up when reading popular science books and physics articles I tried without much luck to find an overview of them that didn't need several more years of physics and maths than I'd done, which was annoying. This video was exactly what I wanted so thank you very much! I was mildly alarmed at finding "Spinors for Beginners 11" in my search results lol, so I'm glad I decided to see what the first video was like, I'll see how far I get with the rest of the series :)
Thank you, I’ve just started to study QFT and many book get for granted that anyone has already a well established idea on tensor. This video already made me get a grasp of the core principles of this wonderful mathematical objects!
The most important channel on UA-cam.
Finally a new good UA-camr channel discover... =). By the way, the first part of the video gave me a flashback from my teenager times (10 years ago), I went with my high school here in Spain to visit our city university and a young recent graduated gave us talk in Physics, when she ends the infantilizated topics, I rise my hand and ask "What's the difference between a boson and a fermion?", She started to sttuter and my teacher just tell to not say nothing alse and friendly to "shut up" and I decided the next days that I will not go college and I spent my first young ages reading Nieztche, smoking weed and working with my father in art. I have not regrets.
loved the overview, understood a frightening amount from dipping my toes in lie algebra previously. Will watch all the videos!
For what i had seen about geometric algebra, It should be able to encode real and imaginary scalars, vectors, quaternions, octonions, spinors, Pauli and Dirac matrices, tensors, Lie and exterior algebras. Yet, I haven't studied It, just relying on these promises.
Thrilled to see your perspective on these.
Geometric algebra is neat. I recommend the series "Plane-based Geometric Algebra" by Bivector on UA-cam.
Another channel to try for Geometric Algebgra is Sudgylacmoe (it will probably be at least 3 months before I get to Geometric Algebras in my video).
@@eigenchris XylyXylyX also has a video series currently being released on Geometric Algebra as it relates to electrodynamics. Looking forward to your spinor series! You guys are doing great work
Freya Holmer has an awesome talk she gave. I think it was titled something like how do you multiply vectors. It ends up with spinors and geometric algebra. Useful with quaternions and rotating vectors. That was my first introduction to them and I'm keen to learn more.
On a side note, her visualizations are second to none. Everyone should watch her two Bezier Curves and Continuity of Spline videos. Top notch.
@@cbbbbbbbbbbbbreally good summary, indeed. Reminds of an article of Matt Ferraro called "what is the inverse of a vector?"
THIS IS such a good resource!! Thank you so much for sharing you knowledge in such a well paced and well thought out way! We need more of this in physics!
I was just about to read some QFT stuff on my own and you kinda saved me there. Thank you so much . I will be eagerly waiting for the next videos.
I’m so excited for the rest of this series!
After years of searching in you tube this the first time I am begining to understand the topic of Spinors
How lucky I am to meet you while undergrad. Your videos helped me a lot. Thank you!
You deserve the Nobel prize for education!
I would love for you to do a breakdown of the math on spinors, like how to derive them or use them in applications, because for someone like me these introduction videos are great but my math skills are terrible, so it would amazing to see a walk through on the math of these topics as well.
Thank you so very much EigenChris!!! Very informative & helpful!!!
Thank you Chris. I’m constantly struggling with various mathematical concepts due to the lack of clarity in some text books. Thanks to your tensor calculus series I was able to understand not only tensors but other topics, since it helped me fill gaps that I had in other topics. Even this introductory video helped me fill gaps related to spinors, exterior algebras, and Clifford algebras.
I’m looking forward to watch your spinor series. You deserve a tip 👌🏽…
I've seen a bunch of mathematicians be advocates to stop thinking of vectors as pointy sticks. But I think since this is for 5 year old, they might give you a pass.
Awesome video btw!
I'm sure that these have a real application and are a genuine thing but I'm a history and archaeology dual major with anthro and performing arts miners. I have never heard a lecture that struck me as the unhinged ramblings of a monster than this one and I had to listen to an old man slobber over thirteen year olds one semester. Liked and shared with math friends who might not panic when you say something like "quantum fields" or "division"
I’m so excited, I really really can’t wait to see how you tackle on teaching this field and I can’t wait to learn.
At 4:21, "a good place" is a singular noun.
In English, we use singular verbs with singular nouns. We say the subject and verb agree in number.
We do not normally say "A good place are examples from physics." We say "a good place is examples from physics..."
Five-year-olds understand this.
Glad to hear there are people who know this stuff!
Thank you sir, for this outstanding contribution to mankind (not even exaggerating, it’s fantastic !)
15:09 missing 1 's in the 3D rotation matrices on the neutral axes.
I am beyond grateful man! Fantastic initiative.
this is so exciting! Finally, I would be able to wrap my head around this topic
Yaaaas I can't wait for the Clifford algebra explanation. I've never cared for matrices or tensors because it seems like they don't carry all the important information, e.g. you obtain your column vector components using a vector basis, and the basis is no longer part of the object. I like Clifford/Geometric Algebras because the objects have transparent meanings and defined relationships that can be reasoned through in a straightforward way. In other words, the object is both the components and basis, and that makes it much easier for me. So I'm psyched to follow this series!
The best introduction to spinors I've seen is through the topic of geometric algebra, and you can explain using pictures.
Up/down spin, like the cup trick (t=2:30), is looking under your armpit, compared to looking over your shoulder. One is (looking) upside down, and the other is (looking) right-side up.
The spinors as square root of a vector, is 'exactly' the same as adding 'i' (the square root of -1) to the reals to be able to turn positive numbers around to reach negative numbers in two steps (multiply's).
There are a similar number of confusions, contradictions and odd things about complex numbers as there are about spinors, but most folks have already internalised the contradictions for complex numbers (brow beaten from basic maths course).
It's all about changing "symmetry" from linear shifts to rotational shifts ;-)
This seems like it's gonna be a great series; I loved your tensor series.
I'm very looking forward to this series. Always a hard topic for a physicist, due to its deep mathematical roots.
This video series are really amazing. So far I have watched all of them because they are so perfect．I am looking for viewing the videos in final phase in the staircase.
Looking forward to the start of another great series after going through both tensor playlists 👍
Thanks to this video, all that mathematical stuff is finally clearing up and everything is falling into place! This is great!
Your explanations are outstanding and extraordinary. May God bless you!
6:20 HOLY SHIT I GET IT NOW THANK YOU SO MUCH YOU LEGEND
Thanks for building this staircase. Looking forward to ascending it. I think you are going to connect a lot of concepts for me and that's always very satisfying.
YESSSSSSS YESSSS YESSSSSSSSSSSSSSS christmas is early this year THANK YOU EIGENCHRIS
Excellent and very lucid presentation.
Thanks, Chris. do you realise that you are a super-star !!!
Thanks for your work in describing the layout of the path to understanding spinors.
im so Happy you decided top start a News series! i absolutely Loved your relativity one!
I'm so beyond excited for this series
WOW, very concise and clear explanation!
So excited about this! The perfect Christmas gift 🎄
Chris that was a great intro! I’m excited about your next videos on this topic.
I am super hyped to see more of this video series!
Congratz on 100k subscribers! Afaik recently you had like ~89k or so. You opened my eyes for some of the math notations that is used in quantum computing and the requirement for reveribility and how it limits some of the possibilities for quantum computing.
We all started somewhere, we all, at some point have been... Beginnors!
14:53, the exponentiated 3x3 matrices are missing some 1's.
Yup. My bad.
Excellent. Physics needs such interpretation of mathematics.👍
Amazing video. Cannot wait for the video on the Lie algebra perspective!!
Wow, this clears so much up already... Thank you so much. Excited for your series!
Great new series! Can't wait for the next video.
A matrix is a 2nd rank tensor. A vector is a 1st rank tensor. A scalar is a 0th
rank tensor. Chris' statement that a Pauli vector is a ½ rank tensor is wrong, at 8:46.
Still a great video.
The slide says a Pauli Vector is a rank-1 tensor, and calls Pauli Spinors "rank-½" tensors.
Very happy to see you make some videos on this - our quantum mechanics prof dropped the word spinor on us just earlier this week without explaining what it was, so the timing here is just perfect!
One small note though, at 14:32 I think you forgot to put the angles into the exponent in the left half of the equations as well - as it stands there, the equations only hold for θ=Φ=Ψ=1.
Yup, my bad. Hope you find this series useful!
As someone who has been struggling with the concept of spinors for a long time, I find this to be a very nice introduction. Just summing up various ways of looking at the concept that make complete mathematical sense. There are still some minor lacunae, like I don't think it's intuitive what a rotation in phase space is, or how it doubles to a rotation in real space; and the bit on Clifford algebras can use the remark that the algebra elements are kind of like if we treated basis vectors like objects you can multiply, and a basis vector squared is the magnitude squared. At least, that's what I found to be the most straightforward conceptualization of geometric algebra. I'm looking forward to the rest of this serious, and I hope it will bring progress in finally putting this highly complicated topic to rest.
Basis vectors know nothing of multiplication and magnitude, those are facets of the theory within which you are utilising them. Horses for courses sort of thing. For example: I can walk forward/backwards, Left/Right. Up/Down. One step in each of Forward, Left and Up constitute constitutes a basis vector in 3 Dimensional walking. The idea of multiplying left by up is a nonsense in this setting. Different people at different places on the globe at different heights and facing different directions can draw their own arrows and define their own basis vectors by dint of their position and all will be different! An infinity of basis vectors ! And each set of 3 being equally valid to enable walking on Earth, no matter where. My point is that a vector basis is independent of the attributes your field of research adds to them. They are a base class in C++ and are not defined by the classes that you construct from them.
look forward to your whole spinor series
I can't wait for the rest of this playlist!
I got it! So a spinner is the first domino that is played in a game of dominoes. Thanks 😊
Wow! Thanks for the clear and easy to understand explanations!
5:37 I get that physical space and abstract space work differently in this instance, but how can you tell apart a from a particle in physical space? There’s a clear difference in the abstract, but what use is it if it doesn’t affect physical space? or is there a situation where it does?
There's no measurable difference between "positive spin up" and "negative spin up" in the real world. In fact, even more generally, you can not tell the different between the "spin up" state and "spin up multiplied by a complex phase e^iθ" (in the special case of θ=π, this becomes -1). This property is key to why spinors pop up in quantum mechanics. It's the same with the polarized waves example: if a wave is vertically polarized, its negative is also vertically polarized (any arbitrary phase shift is also vertically polarized). If you're familiar with the concept of "electric potential" in electrostatics, you might also know you can add a constant potential everywhere in the universe without changing the electric field, because the electric field only cares about potential DIFFERENCES, so constant offsets in the math are not physically measurable. I'll talk about all this in the series. If you keep studying these ideas, you end up in "gauge theory", which is central to modern particle physics.
Awesome! Looking forward to this, especially the geometric algebra which is a super-power I'm struggling to understand.
Really clarifying as usual. Although I'm not so interested with this topic, I will follow the series just because I love to be led along such a hard path: your exposition makes it interesting and tickles my curiosity...
Best channel on youtube, i followed your series in tensor calculus and relativity. Definitely will follow this series.
Gotta love the real deal, compared to the typical UA-cam "science" videos.
This will be so helpful for my upcoming QFT courses!
It seems that this series will be a bomb.. Please continue..
I have been obsessed with Geometric Algebra (GA), I literally cheered when you mentioned bivector and trivector :D.
I wasn't sure if you were talking about GA when you mentioned Clifford Algebra because I'm quite new to the subject.
Yeah, "Geometric Algebra" and "Clifford Algebra" are the same thing. (Some people argue they have slightly different definitions, but they're built on the same core idea.)
@@eigenchris True, but there is a subtle and confusing difference between the Clifford Algebra/Geometric Algebra notation system used by László Tisza of Massachusetts Institute of Technology (available via MIT OpenCourseWare) and the more unified Geometric Algebra notation system developed by David Hestenes that incorporates Clifford Algebra into a notation system for Physics with the geometric product of vectors and use of multivectors. Dr. Hestenes tried to differentiate use of Clifford Algebra notation from his unified notation system for math and physics by appropriating the name Geometric Algebra (which Clifford had used for Clifford Algebra). The point being that the Hestenes Geometric Algebra is more than just Clifford Algebra as uused by Tisza and allows one to work in a unified way with spinors, tensors, vectors and differential forms.
geocalc.clas.asu.edu/GA_Primer/GA_Primer/introduction-to-geometric/rotors-and-rotations-in-the.html
Thank you for your kind explanation!!! It's very hepful to me. But I think there is an error at the time 9:04 for the Rotating Pauli Vector and the -i in the rotating matrix should be used instead of i. This is consistent with the rotor in clifford algebra.
Yeah, I think you're correct. I got the direction of rotation reversed.
Another application of spinors is the Newman-Penrose formalism for general relativity and the Petrov classification of the algebraic classes of the Riemann tensor. Very powerful in my opinion. Spinors allow decomposition of null vectors in relativity (an example of the "square root" of a vector). Then the next stage is "twistors". Oh no another "strange" mathematical abstraction! Regards. Nice videos!
I'm not familiar with twistors other than the name. Is there a good introduction to them somewhere that you're aware of?
Nice timing, just started looking at these myself.
A question about the correspondence between Physical space and State space around 5:00.
My intuitive instinct was to try and extend this concept to a third orthogonal state. I can see that obviously we cannot have on the 2D plane more than 2 orthogonal states which are physically 180° apart, but can we have objects which have more orthogonal states at a different spacing? I can't really see how the geometry of the phase space would hold up though..
More to the point, a vector can be an object with a number of different of amplitudes along different orthogonal directions. That is, we have Hilbert spaces where there is a ket encoding the probabilities of for example a ton of energy eigenstates. Why can we not describe spinors like this too?
The 2D circle I show for QM is really a partical view of the 3D Bloch Sphere. And the 2D circle for light polarizations is a partial view of the 3D Poincaré Sphere (the two are mathematically equivalent). I talk about them both in videos 2-5.
The word "vector" means different things in math. The abstract meaning is "an element of a vector space". The kets in a Hilbert space are this kind of abstract vector. There's also the more "physical" definition where a vector is a 3D pointed stick. A Pauli spinor is what you get when you decompose this 3D pointed stick into 2 pieces using a special mathematical trick. Both Pauli spinors and 3D poimted-stick vectors can be "kets" in an abstract vector sapce.
I realize it has been stated before but, AWESOME! Thanks for this!