The biggest misconception about spin 1/2

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“If you rotate a spin 1/2 particle by 360 degrees, it doesn’t go back to its original state, rather you need 720 degrees”. This is only technically correct if you interpret the words “rotate” and “state” in the way it’s intended, and here is the video on what they really mean, and what this sentence is saying.
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Many people have talked about spin 1/2 before, but I do want to chime in because there are a few things that I myself am not satisfied with all the explanations.
(1) I personally don’t like *any* “demonstration” of spin 1/2, aimed at demonstrating in some physical situations you need two full rotations to get back to where you started, like the Dirac belt trick / spinning your hand trick. This is simply because physically, you literally can’t tell the difference before and after rotation - even in principle. You can only tell the difference when you have a superposition. And these demonstrations, to me, give the false perception that physically there is something different. This is, I think, an extremely important point that people miss out when talking about spin 1/2 and how “weird” this is.
(2) From U(2) to SU(2): the “usual” explanation for the choice of determinant is that the phase factor does not matter, so in passing from U(2) to SU(2), we sort of “remove the redundancies'' in the description of our transformation. But why did we leave the ±1 factor redundancy untouched? Either you remove *every* redundancy, or **none**. It doesn’t make sense (at least to me) that you don’t also remove that sign redundancy as well.
This is actually because the projective representations (obtained using Lie algebra methods) of SO(3) *must* have determinant 1, and this is known prior to constructing such a (projective) representation. This might be briefly explained if I decide to make a video on Lie algebras / groups / representations. I think that using the Lie algebra method means that we are imposing the constraint about analyticity and genuine representation in the neighbourhood of the identity, but I can’t be sure about this.
(3) When actually constructing the (projective) representation, the usual “trick” is to do some conjugation - but actually with Bloch sphere, we can visualise the representation extremely visually. That’s almost all the reason why I love the Bloch sphere visualisation method. The conjugation trick is not wrong, just that it seems way more complicated to me. For more information on this old trick, see this: indico.cern.ch/event/243629/a...
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There are alternative perspectives out there. What I have described is *how* SU(2) and SO(3) *acts**. But there are explanations using algebraic topology to describe **what* SU(2) and SO(3) really *are* as manifolds. For more on this, see www.damtp.cam.ac.uk/user/exam...
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In a similar vein, people often think that spin-s particles are just particles that magically goes back to its original state after 2pi / s rotations. Even Wikipedia says this! Except this is not even technically correct - it is just plain wrong. I’ll talk a bit about that, again if I decide to make a video about higher spins / Lie groups / algebras / representations.
Hopf fibration / quaternions:
en.wikipedia.org/wiki/Hopf_fi...
Video chapters:
00:00 Introduction
00:47 Chapter 1: "State"
07:42 Chapter 2: "Rotate"
17:46 Chapter 3: The construction
25:41 Chapter 4: The "spin-1/2 property"
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[@mathemaniac]

Sorry that I only had about 5 days to make the video, and there are a few glitches in it before I have to put out this video, but hopefully they are minor. Please read the description on some of the thoughts I have around the topic, because something that I mentioned / did not mention might be a deliberate editorial / pedagogical choice. If you enjoy the video, apart from liking, sharing, commenting, please consider heading to squarespace.com/mathemaniac because it helps support the channel greatly! (And the websites produced do look professional!)

[@hyperduality2838]

Equivalence, similarity = duality. Spin up is dual to spin down -- the Dirac equation. Duality:- two equivalent descriptions of the same thing -- Leonard Susskind, physicist. "Always two there are" -- Yoda.

[@annaclarafenyo8185]

This video is an example of "a little bit of knowledge is worse than none at all". Here is an experiment you can do--- take some silver atoms and split them by a Stern Gerlach experiment into two beams with opposite spin. Pass one of the beams through a magnetic field that either 1. does nothing. or 2. rotates the spin by 360 degrees. Then combine the beams. When you turn on the magnetic field, the locations of constructive interference are flipped. This is what it means to say that a 360 degree rotation is physical. It EXACTLY MEANS that rotating the state by 360 DOES NOT RETURN YOU TO THE ORIGINAL POSITION. There is no mis-speaking involved. There is no mistake. You are simply wrong in this video.

[@hyperduality2838]

@@annaclarafenyo8185 Watch the following video about spinors:- kzbin.info/www/bejne/nZfak5luo5lkebs Vectors are dual to co-vectors. Spinors are dual to dual spinors. Both spinors & vectors are dual. Electro is dual to magnetic -- Maxwell's equations. The inner product is dual to the cross product. SINE is dual to COSINE -- the word "co" means mutual and implies duality! Positive charge is dual to negative charge -- electric fields. North poles are dual to south poles -- magnetic fields. Electro-magnetic energy is dual. Null vectors or lines (light) are dual to dual to null spinors or points (matter) -- Twistor theory. Points are dual to lines -- the principle of duality in geometry. "Always two there are" -- Yoda.

[@robmorgan1214]

Good video, but it's not quite right for building correct intuition. You do, in fact, need to consider that factor of two as a tier one object/ concept of physical significance to recover the full measurable physics of the system. It shows up everywhere in experiment with measurable consequences as something more than a mathematical property of a mathematical abstraction. It shows up in the statistical mechanics as extra DoF but with the constraint of single state occupation due to the sign flip which manifests in the commutator (i.e., Pauli exclusion principle see Peskin-Schroeder p57 ch3.5 eqs 3.96-3.98 and subsequent discussion preceding "the Quantized Dirac Field"). Moreover, normalization assumes the born rule, which is COMPLETELY unnecessary and hides some of the most important ideas of quantm mechanics and the importance/interpretation of spinnors behind the "choice" of su2 representation. See Wojciech Zurek's lecture on "Decoherence and the Quantum Theory of the Classical" on YT (he's the guy responsible for the no cloning theorem). In his lecture, he shows that the normalization of the state, and hence the mathematics of probability itself, are not arrived at arbitrarily or a result of an axiomatic use of Born's rule. It is instead an artifact of Unitarity and orthogonality of basis states in a hemitian space. The 720-degree rotation is important and not an artifact of the choice of SU2 representation it happens in a very real physical sense relative to the classical rotation of extended macroscopic physical objects occupying pointer states. What's misleading is about Quantum mechanics is a result of an (decades long) inattention to detail pointed out by Zurek in his lecture regarding a second non-trivial solution to an equation for trivially entangled states that allows you to recover and derive/prove Born's rule instead of assuming it. He demonstrates it for a toy model that's easy to comprehend/ follow, but it's been proven to hold for the general case. This places an important step in your explanation in a very different light and completely changes its physical interpretation.

[@robmorgan1214]

​@annaclarafenyo8185 technically, he's on firm mathematical footing... using the standard interpretations and axioms of QM pre-2010ish. Physicists are guilty of a few decades of handwaving here that really muddied the waters, but they got away with it because the theory worked so well when you followed a specific use case/interpretation/prescription. See my above comment and see the lecture by Zurek for the missing piece of the puzzle that shows WHY his analysis is MATHEMATICALLY incorrect as well as inconsistent with experimental evidence. It's very subtle and was hiding in plain sight for decades in the equations for the zeros of trivially entangled states. It turns out there are TWO cases and we only examined one of them when building and interpreting QM theory. If it matters to you the guy who figured it out is Wojciech Zurek (he also figured out the no cloning theorem). The significanceof his talk is the difference between assuming and proving. It fills in the gaps in the mathematics where the creator of this video got led astray. It seems so simple in hindsight. If we figured this out when we figured out Bell's theorem no one would be saying "it's impossible to understand QM". Zurek's observation about the significance of the non trivial solution to the zeros of his entanglement eq is that important as it puts to rest many controversial things and places them on a firm mathematical footing, that's a provable consequence of orthogonality instead of assumptions and axioms.

[@enterprisesoftwarearchitect]

THANK YOU! when I tell people that it’s a rotation in SU(2) spin space, not SO(3) actual space, they think I am a lunatic.

[@fangjiunnewe3634]

I'm cackling in my room near midnight because of the ChatGPT segue 😂😂

[@denishclarke4470]

😂😂

[@setheisenberger247]

Same lmao

[@petevenuti7355]

segue to S-egg vs s-eeg-wāy What's the etymology, french?

[@KieranORourke]

Geometric Algebra simplifies so many of the concepts here. Matrix algebra is marvelous but, by definition, it only works on vectors so representing other geometrical objects means sprinkling difficult-to-interpret matrices ~(like the Pauli Matrices) all over a representation.

[@annaclarafenyo8185]

Geometric Algebra is a fraud.

[@rv706]

How would "geometric algebra" (which I take to mean "the theory of Clifford algebras) simplify the treatment of spin 1/2?

[@narfwhals7843]

Often these explanations include a statement like "In statespace (1,0) and (0,1) are orthogonal, so 90° apart, but on the Bloch sphere they are on opposite poles, so 180° apart." Tadaa 90° corresponds to 180°! But i never quite understood why I'm supposed to care about this correspondence in the first place. This video helped me with that.

[@lexinwonderland5741]

YES!! PLEASE ANOTHER VIDEO ON THIS TOPIC!! great work as always!!

[@tanchienhao]

You explained the notion of “intrinsic property” much better than any other sources out there! Awesome video

[@SmogandBlack]

When I started to read about the Standard Model I didn't think that I would like to go into its Math: I felt happy just to know about the existence of Fermions and Bosons. But time passes and the brain is an unresting structure that wants satisfying answers to its questions... so thank you very much for this video.

[@annaclarafenyo8185]

This video is wrong. This person is not good at making these types of videos, because this person does not understand the material well enough.

[@niczvr]

@@annaclarafenyo8185 if you believe that so certainly, you could give us a clear example, or explain it better than him yourself.

[@annaclarafenyo8185]

@@niczvr I'm thinking about doing that, but only after I finish my current paper.

[@annaclarafenyo8185]

@@niczvr The best explanation, I think, might be to start with something called "Lattice Gauge Theory". If you are good at programming, you can write a model code to generate configurations in a day or two (I did it years ago, it was interesting). This will teach you the basic ideas of gauge theory. But the rest of the standard model is harder. I have to think about how to present it. I don't think any of the standard presentations are any good.

[@goldentrout4811]

@@annaclarafenyo8185 Looking forward to it!

[@davidhand9721]

The whole geometric description of a spinor thing prevented me from understanding them for years. You aren't rotating a geometric object in 3D space, and the moment someone showed me what they were actually rotating, I got it immediately. I caution everyone who wants to understand spinors against watching any explanation of "how to visualize spinors", the belt trick, the coffee cup trick, etc. Obviously there is no object that must be physically rotated twice before looking the same.

[@martine7456]

Finally a video that goes into the math of spin. I wish I hadn't forgotten all the math I studied 15 years ago, ha!

[@ghassanehajji7591]

Amazing explanation. Clearest discussion of the subject I've encountered so far. Thank you for your efforts, and I'm really looking forward to be watching your next videos.

[@NoNameAtAll2]

eigenchris is doing video series about spinors, explaining SU2 symmetry and everything

[@quadrannilator]

I was in my head thinking of quaternions ever since the video started and was also delighted they got their due mention at the end. Also was suprised seeing Hopf Fibration but intuitively got it why its there in relation to the projection type presentation. Keep up the amazing videos coming.

[@quadrannilator]

I actually only recently learned that Maxwell's Electromagnetism equations were originally expressed using quaternions whereas it was Heaviside and others that popularized the vector form that is taught to us. I have not read those original papers by JCM but just saw some sections of it in modernized form, and was actually amazed at how the quaternion form is condensed yet elegant in illustrating how the forces are related to each other. While we cannot visualize objects in 4D, but the extension of complex numbers and easy seperation into scaler and vector parts that include all the 3 axes, is what makes me grasp a whole lot of stuff intuitively all at once better with quaternions.

[@CptAhab-vf1ny]

You should read Eric Dollards versor algebra.

[@quadrannilator]

@@CptAhab-vf1ny Thanks! I'll lopk it up

[@japanada11]

Fantastic video, very nice derivation of the projective representation of SU(2)! Just wanted to leave a note regarding the comment in the description on spin 1/2 "demonstrations": while I agree that they can easily be misleading if presented incorrectly (i.e. as a description of what physically happens to a spin 1/2 particle during rotation), as they almost always are, I respectfully disagree that they necessarily misrepresent the nature of spin 1/2 particles. If explained properly, I would argue they describe the nature of spin 1/2 particles _extremely_ well, to the extent that you can actually use arm-twisting to perform accurate calculations involving the behavior of entangled states. At their core, all these demonstrations are exhibiting the behavior of _paths_ in SO(3): whether you use a belt (fixing one endpoint and allowing the other to rotate) or your arm (fixing your shoulder position and allowing your hand to rotate), in every case there is an extended object (belt/arm) that parametrizes a continuous trajectory from the trivial rotation to some other rotation in 3d space. When you rotate the belt end or your hand 360 degrees, the belt/arm now parametrizes a _loop_ in SO(3). What the demonstrations show is that the fundamental group of SO(3) has an element of order 2: there is a loop that cannot be contracted to the identity path, but if you repeat that exact same path twice, then there _is_ a contraction to the identity. Now paths from a fixed base point (up to homotopies that fix both endpoints) are in 1-to-1 correspondence with the universal cover. So one way to interpret these demonstrations is just that they are _performing calculations in SU(2)._ Every matrix multiplication in SU(2) can be perfectly replicated, with no information loss, as a sequence of rotations of the belt/arm! This already gives a nice explanation for the purpose of the demonstrations: even if you don't want to attach any physical significance to them, at the very least they perfectly explain the _mathematical structure_ that is required for performing calculations on spin states. But I would go further and argue that the demonstrations actually _do_ have a very nice physical analogy. Most explanations make the mistake of comparing the particle to the entire extended object (belt/arm), saying that the 360 degree rotation has "twisted" the particle somehow. In a more accurate interpretation, the arm is a _timeline_ of the spin state of a particle; that is, it describes what quantum operators were applied to the spin state to put it into its current state. Only the very end of the belt, or the hand at the end of the arm, represents the current state of the particle. When it comes time to make a measurement of the hand, we have absolutely no way of telling whether a 0 or 360 degree rotation was applied, because all we have access to is the current position of the hand, which has no memory; the history of the particle (the shape of the arm) is lost to the depths of time. However, this all changes once entanglement enters the picture, because this allows us to keep track of histories _between_ particles! In particular, the twisted/untwisted arm exactly conveys the difference between the "twisted" (a1-a2) and "untwisted" (a1+a2) entangled states. Instead of treating shoulder=past and hand=present, we now interpret shoulder=entangled particle #1 and hand=entangled particle #2 (initialized in the exact same state a1=a2, giving entangled state a1+a2). By applying a 360 degree rotation operator to the hand (i.e. to the spin state of one of the entangled particles), we've encoded into the wave function (arm) the fact that one must follow a nontrivial loop in SO(3) (ie apply the quantum "360 degree rotation on spin state" operator) to get from the state of one particle to the state of the other particle. If you just look at my shoulder, or just my hand, you would have no way of knowing that I did anything to either of them - but the twisting of my arm in the middle is evidence that I performed some odd number of 360 degree rotations to the hand, putting me in state a1-a2. This "twist in the arm" can be measured experimentally _only_ because we've set up an entangled system: even though the particles have both ended up in the exact same state (up to projective equivalence), the twist allows us to detect the fact that at some point different operators were applied to each particle. And again, while this is just an analogy, it's a very precise one, in the sense that SU(2) acts in the exact same way on the arm as it does on the entangled pair! That is, if you want to know which sequence of operators will recover the a1+a2 state, all you need to do is figure out which rotations of the hand bring the arm to the untwisted state. tl;dr: the arm (belt) demonstrations captures the behavior of an _entangled pair_ of spin 1/2 particles; the hand (end of the belt) alone captures the much less interesting behavior of individual spin 1/2 particles.

[@japanada11]

I actually think this observation deals with a lot of the complaints you had (e.g. with Wikipedia's phrasing): if you replace every instance of "spin 1/2 particle" with "tethered spin 1/2 particle" (i.e. instead of considering the state alone, you consider the state with respect to a fixed reference state, for instance by setting up an entangled pair) then everything works nicely as stated, even if you allow "state" to mean projective equivalence classes of states. *Even though a and -a are projectively equivalent, the pairs (a,a) and (a,-a) are not.* So you can meaningfully talk about the smallest value of θ such that {a, D(R(n,θ)) a} is physically indistinguishable from {a, a}. For spin 1/2 particles, you really do have to apply a 720 degree rotation operator to recover the original state; the 360 degree rotation is _physically_ different from 0 (e.g. because the Pauli exclusion principle applies to one but not the other), while 0 and 720 are identical. The "tether is twisted" in the 360 degree case, in the sense that there is a negation relation to the fixed reference state (despite the two states being physically indistinguishable). This "tether" is of course referring to the path in SO(3) / arm / belt that associates D(R(n,θ))a to the fixed reference state (e.g. entangled particle) a.

[@Thisisnotmyrealname8]

Excuse me, does this mean I could, literally, dance into another dimension?

[@japanada11]

@@Thisisnotmyrealname8 _All_ dance is intrinsically 4-dimensional, because it involves movement through both space and time! The hand/arm trick isn't doing anything new other than choosing to recognize this fact.

[@Thisisnotmyrealname8]

@@japanada11 I mean if your Chakra is fully activated, though.

[@cleon_teunissen]

This sentence drew my particular attention: "What the demonstrations show is that the fundamental group of SO(3) has an element of order 2: there is a loop that cannot be contracted to the identity path, but if you repeat that exact same path twice, then there is a contraction to the identity." To my understanding: to account for the observed patterns in splitting of atomic spectrum lines it was necessary to attribute a spin to electrons. Over time mathematical structures were developed for the purpose of accounting for spin states. We have that there is a unified mathematical structure for accounting for spin for all quantum entities, which obviously will leave very little wriggle room, if any. Still: it seems to me: it should not be assumed that there is a 1-on-1 correspondence between the mathematical model and physical reality. Could it be that the mathematical models inevitably incorporate superfluous mathematical structure? The equation describes how the quantum entity developes over time. Is the mathematical operation of "contraction" part of that development process? Or is that "contraction" purely an operation that is performed in the course of *evaluation* of the mathematics?

[@macronencer]

Intuitively, I get that you are trying to lay some myths to rest here. I only wish I could follow the maths (I have a maths degree, but it was in 1986 and I haven't kept studying since...). I felt very "adrift" while watching this, as I began to lose track of which parts were tools brought in for convenience and which parts were linked to reality via observation and experiment. Perhaps one day I'll have time to come back and watch this in pieces, pausing every time I get lost to go and research the subtopics. Until then, I'm afraid I give up. Obviously not your fault as it looks as if you've done a really thorough explanation here!

[@meofamily4]

I got lost a lot as well. You are not alone. All too often a statement was made with zero explanation of its basis. In my case, I think it is the fault of the presenter.

[@user-qh9vh2me9v]

Hopefully this notification won't be a nuisance, but I would encourage to come back and follow it again

[@jessfucket]

this is a tremendous way to explain it! you allow me to see electromagnetism (the propagation of photons - Complex rotation) in terms of hypergeometry rather than just numbers. the momentum has two components. if all you can see is momentum and angle, you'll be all confused by the component in the other dimension. I also understand I As an orthogonal direction more clearly now. thank you thank you thank you!

[@samuelefraizzoli1070]

Great video! This one the best Physics channel on KZbin, in my opinion (I graduated in Physics some decades ago...). Congrats and keep making videos. Thanks Samuele

[@karelknightmare6712]

A way to visualize cos^2 of theta with normalized vectors is to draw a circle of diameter 2b, and then project a on it. The length on the origin side is cos^2 the remainder is sin^2. 😊

[@MarceloRobertoJimenez]

Congratulatios for the video. Definitively, we need a video on Lie groups and Lie algebras.

[@friendly_hologram9597]

Thank you! Explanations of spin are often disappointing, but this one was great, I feel like I have a much better understanding now. I study pure maths and I was a little disillusioned with physics (I used to study it but swapped to maths recently), but mathematical explanations like this are making me regain some of my appreciation for the subject :) (especially the mention of the hopf fibration!)

[@annaclarafenyo8185]

This one was wrong. Incorrect. Factually false.

[@friendly_hologram9597]

@@annaclarafenyo8185 What's wrong with it?

[@annaclarafenyo8185]

@@friendly_hologram9597 It thinks phases in the wavefunction aren't measurable. This is completely and totally false, it's a common misconception among first-year university students. You measure the phases in a wavefunction by comparing to other parts with different phases.

[@Achrononmaster]

Here is a better explanation: A spinor is not a particle, it's a description of what we need to do to translate/transform a measurement system (unitary operator or "observable" in the formalism) onto the frame of a particle. The spinor comes from the geometric algebra, and is the factor in the state function for determining the rotation part of the translation. In the proper spacetime algebra it is much clearer because instead of using matrix algebra one uses the proper multivector graded algebra, which reveals the spinor is a rotor in the even subalgebra of spacetime, and so acts double-sided on physical quantities to effect the transform (the rotation or boost). Because the rotor acts double-sided, to transform a rotor under a general rotation one needs to use an angle of 4\pi. That's not "rotating around 720deg" it is just transforming a rotor under another rotor. Rotors are not physical objects, so there's nothing weird about this. Rotors are part of normal Euclidean or Riemannian geometry, and there is thus nothing "quantum" about them. The classical rotation groups have the same half-integer representations. The reason we do not use rotors so much in classical mechanics is because we use an entirely different formalism for keeping track of particles, namely phase space time evolution. One can get a phase space representation for QM too, but it's awkward, and it is just simpler to move to a measurement representation. If you formulated classical mechanics as a measurement theory (only care about the boundaries of a cobordism) you'd also have fermions. But you don't bother, since in classical mechanics you can imagine you can precisely track particles in phase space, so you don't ever care about the nuances of measurements, since you have ħ→0. The reason you cannot use a spinor for a boson is because all bosons are massless without the Higgs gauged symmetry, and massless particles cannot be rotated out of their plane of motion, so can transform single-sided in the geometric algebra (like "vectors" although they are elementary particles or stringy stuff, not actual vectors). The boson wave function is again only a mathematical instruction, it's not "the particle." Don't confound mathematical objects with physical objects, they're not the same. Mathematics is description/epistemology, not ontology.

[@annaclarafenyo8185]

@@Achrononmaster This is mostly nonsense, partly correct. It's a word-salad that the author does not understand.

[@bboysil]

Love how this ties into Lie theory, which I happened to learn last year, needed for work :D. Excelent video!

[@funnyman4744]

I finally understand the difference between spin up and spin down! Thank you so much for these fantastic videos! I am super excited for your video on Lie algebra as well, please make it the best it can possibly be!

[@tac0519]

It looks like clock. 12:00(Noon) + 2Pi = 24:00(Midnight), 12:00(Noon) + 4Pi = 12:00(Noon) .

[@profdc9501]

This also ties into the geometric phase (Berry phase or Pancharatnam phase) which generally for a spin x particle is going to be 2 pi x, for example, for the spin 1/2 particle, rotating it through 360 degrees added a pi phase shift. Of course, this leads into all kinds of directions, such as adiabatically varying systems (Foucault's pendulum, etc.), adiabatic and diabatic transitions in molecules, etc.

[@Timespacefractal]

Thanks for your Manim-style videos explaining these compelling physics topics.

[@TheSandkastenverbot]

That was an extraordinarily good explanation of spin. Thank you!

[@Chr15T]

Great video. Just one concern: you call two spin states differing just by a phase "physically equivalent". That is true unless the phase plays a role in the experiment. Consider the Bohm-Aharonov-effect. Here, the wavefunction of a spin-1/2-particle is first split into two parts, then one of the two parts is sent into a magnetic field that "rotates" it (by precession) 360°, and then the two parts of the wavefunction are again combined, allowing and forcing them to interfere. And we see destructive interference!, showing that by the 360^-rotation, the spinor phase has been multiplied by -1. So the phase is not really irrelevant, and two spin states differing just by a phase can result in a different physical measurement.

[@KatarzynaMatylla]

I've seen a ton of squarespace ads on YT, but I really love how you transition form the content to the ad. That was best. I mean' I really like listening about quantum mechanics, but this was still the best part of the video

[@pseudolullus]

Great video! Thanks for discussing the topic at a greater depth than many other users

[@kylebowles9820]

Love your videos, very nicely presented!

[@tanchienhao]

Awesome video explaining the projective representation!

[@LucaFanciullini]

Simply fantastic, I look forward for the next video, hopefully

[@TheRenaSystem]

This was by far the best explanation of spin in layperson's terms that I've ever heard, bravo!

[@vasumehra960]

I haven't watched the whole video yet But from what I've seen, the video is as good as always Great video man

[@alphadek]

A really nice video that teaches one new points of view and tickle one's curiosity, giving this motivation to yet understand more. I would be glad if you had some sources on where all that reasoning comes from, especially the beginning on why one needs this additional intrinsic quantity to compute time evolutions of quantum states ?

[@byronwatkins2565]

If you allow the polar angle, theta, to have range [0, 2pi) -- like the azimuthal angle -- then you also cover the Bloch sphere twice. The interval [0, pi) is what we have considered historically and the interval [pi, 2pi) is also what we have considered historically except that the 'north' and 'south' poles (and the chirality) are interchanged. This might be philosophically more attractive since the trigonometry need not be broken in half.

[@Nickita11101]

I'm so glad I've found this channel! I've faced mathematical representation of spin in my quantum mechanics classes, but I've never got intorduced to the way it can be derived. Only few questions left, and they are not the mathematical ones: why 2s+1 complex components in vector corresponding to spin? And why would people assume this quantity that we call spin is connected with particles actually spinning around some axis back in the day?

[@rv706]

When we carry an electron through a 360 degrees rotation* the (projective) state of its spin doesn't change, but the state _vector_ gets a phase factor of -1. * (we assume the electron to be in the initial state "spin up" along, say, the axis z, and the rotation to be around an axis orthogonal to z)

[@4pharaoh]

@32:45 the real problem of understanding this 720 degree spin is identified. “In practice, this rotation is done using a magnetic field, and neutron interferometry.” The results of this experiments are never discussed, only their interpretations. Thanks for not using the stupid belt in your explanation.

[@ProCoderIO]

Nicely dovetails with the spinor series coming from @eigenchris!

[@BartdeBoisblanc]

It is satisfying to see another example of the use of imaginary and complex number. These also show up in electronics all the time.

[@luudest]

Intresting video. When does the second part come out?

[@LeoDaSR]

Great videos, keep it up!

[@SpotterVideo]

Conservation of Spatial Curvature: Both Matter and Energy described as "Quanta" of Spatial Curvature. (A string is revealed to be a twisted cord when viewed up close.) Is there an alternative interpretation of "Asymptotic Freedom"? What if Quarks are actually made up of twisted tubes which become physically entangled with two other twisted tubes to produce a proton? Instead of the Strong Force being mediated by the constant exchange of gluons, it would be mediated by the physical entanglement of these twisted tubes. When only two twisted tubules are entangled, a meson is produced which is unstable and rapidly unwinds (decays) into something else. A proton would be analogous to three twisted rubber bands becoming entangled and the "Quarks" would be the places where the tubes are tangled together. The behavior would be the same as rubber balls (representing the Quarks) connected with twisted rubber bands being separated from each other or placed closer together producing the exact same phenomenon as "Asymptotic Freedom" in protons and neutrons. The force would become greater as the balls are separated, but the force would become less if the balls were placed closer together. Therefore, the gluon is a synthetic particle (zero mass, zero charge) invented to explain the Strong Force. An artificial Christmas tree can hold the ornaments in place, but it is not a real tree. String Theory was not a waste of time, because Geometry is the key to Math and Physics. However, can we describe Standard Model interactions using only one extra spatial dimension? What did some of the old clockmakers use to store the energy to power the clock? Was it a string or was it a spring? What if we describe subatomic particles as spatial curvature, instead of trying to describe General Relativity as being mediated by particles? Fixing the Standard Model with more particles is like trying to mend a torn fishing net with small rubber balls, instead of a piece of twisted twine. Quantum Entangled Twisted Tubules: “We are all agreed that your theory is crazy. The question which divides us is whether it is crazy enough to have a chance of being correct.” Neils Bohr (lecture on a theory of elementary particles given by Wolfgang Pauli in New York, c. 1957-8, in Scientific American vol. 199, no. 3, 1958) The following is meant to be a generalized framework for an extension of Kaluza-Klein Theory. Does it agree with some aspects of the “Twistor Theory” of Roger Penrose, and the work of Eric Weinstein on “Geometric Unity”, and the work of Dr. Lisa Randall on the possibility of one extra spatial dimension? During the early history of mankind, the twisting of fibers was used to produce thread, and this thread was used to produce fabrics. The twist of the thread is locked up within these fabrics. Is matter made up of twisted 3D-4D structures which store spatial curvature that we describe as “particles"? Are the twist cycles the "quanta" of Quantum Mechanics? When we draw a sine wave on a blackboard, we are representing spatial curvature. Does a photon transfer spatial curvature from one location to another? Wrap a piece of wire around a pencil and it can produce a 3D coil of wire, much like a spring. When viewed from the side it can look like a two-dimensional sine wave. You could coil the wire with either a right-hand twist, or with a left-hand twist. Could Planck's Constant be proportional to the twist cycles. A photon with a higher frequency has more energy. ( E=hf, More spatial curvature as the frequency increases = more Energy ). What if Quark/Gluons are actually made up of these twisted tubes which become entangled with other tubes to produce quarks where the tubes are entangled? (In the same way twisted electrical extension cords can become entangled.) Therefore, the gluons are a part of the quarks. Quarks cannot exist without gluons, and vice-versa. Mesons are made up of two entangled tubes (Quarks/Gluons), while protons and neutrons would be made up of three entangled tubes. (Quarks/Gluons) The "Color Charge" would be related to the XYZ coordinates (orientation) of entanglement. "Asymptotic Freedom", and "flux tubes" are logically based on this concept. The Dirac “belt trick” also reveals the concept of twist in the ½ spin of subatomic particles. If each twist cycle is proportional to h, we have identified the source of Quantum Mechanics as a consequence twist cycle geometry. Modern physicists say the Strong Force is mediated by a constant exchange of Gluons. The diagrams produced by some modern physicists actually represent the Strong Force like a spring connecting the two quarks. Asymptotic Freedom acts like real springs. Their drawing is actually more correct than their theory and matches perfectly to what I am saying in this model. You cannot separate the Gluons from the Quarks because they are a part of the same thing. The Quarks are the places where the Gluons are entangled with each other. Neutrinos would be made up of a twisted torus (like a twisted donut) within this model. The twist in the torus can either be Right-Hand or Left-Hand. Some twisted donuts can be larger than others, which can produce three different types of neutrinos. If a twisted tube winds up on one end and unwinds on the other end as it moves through space, this would help explain the “spin” of normal particles, and perhaps also the “Higgs Field”. However, if the end of the twisted tube joins to the other end of the twisted tube forming a twisted torus (neutrino), would this help explain “Parity Symmetry” violation in Beta Decay? Could the conversion of twist cycles to writhe cycles through the process of supercoiling help explain “neutrino oscillations”? Spatial curvature (mass) would be conserved, but the structure could change. ===================== Gravity is a result of a very small curvature imbalance within atoms. (This is why the force of gravity is so small.) Instead of attempting to explain matter as "particles", this concept attempts to explain matter more in the manner of our current understanding of the space-time curvature of gravity. If an electron has qualities of both a particle and a wave, it cannot be either one. It must be something else. Therefore, a "particle" is actually a structure which stores spatial curvature. Can an electron-positron pair (which are made up of opposite directions of twist) annihilate each other by unwinding into each other producing Gamma Ray photons? Does an electron travel through space like a threaded nut traveling down a threaded rod, with each twist cycle proportional to Planck’s Constant? Does it wind up on one end, while unwinding on the other end? Is this related to the Higgs field? Does this help explain the strange ½ spin of many subatomic particles? Does the 720 degree rotation of a 1/2 spin particle require at least one extra dimension? Alpha decay occurs when the two protons and two neutrons (which are bound together by entangled tubes), become un-entangled from the rest of the nucleons . Beta decay occurs when the tube of a down quark/gluon in a neutron becomes overtwisted and breaks producing a twisted torus (neutrino) and an up quark, and the ejected electron. The production of the torus may help explain the “Symmetry Violation” in Beta Decay, because one end of the broken tube section is connected to the other end of the tube produced, like a snake eating its tail. The phenomenon of Supercoiling involving twist and writhe cycles may reveal how overtwisted quarks can produce these new particles. The conversion of twists into writhes, and vice-versa, is an interesting process, which is also found in DNA molecules. Could the production of multiple writhe cycles help explain the three generations of quarks and neutrinos? If the twist cycles increase, the writhe cycles would also have a tendency to increase. Gamma photons are produced when a tube unwinds producing electromagnetic waves. ( Mass=1/Length ) The “Electric Charge” of electrons or positrons would be the result of one twist cycle being displayed at the 3D-4D surface interface of the particle. The physical entanglement of twisted tubes in quarks within protons and neutrons and mesons displays an overall external surface charge of an integer number. Because the neutrinos do not have open tube ends, (They are a twisted torus.) they have no overall electric charge. Within this model a black hole could represent a quantum of gravity, because it is one cycle of spatial gravitational curvature. Therefore, instead of a graviton being a subatomic particle it could be considered to be a black hole. The overall gravitational attraction would be caused by a very tiny curvature imbalance within atoms. In this model Alpha equals the compactification ratio within the twistor cone, which is approximately 1/137. 1= Hypertubule diameter at 4D interface 137= Cone’s larger end diameter at 3D interface where the photons are absorbed or emitted. The 4D twisted Hypertubule gets longer or shorter as twisting or untwisting occurs. (720 degrees per twist cycle.) How many neutrinos are left over from the Big Bang? They have a small mass, but they could be very large in number. Could this help explain Dark Matter? Why did Paul Dirac use the twist in a belt to help explain particle spin? Is Dirac’s belt trick related to this model? Is the “Quantum” unit based on twist cycles? I started out imagining a subatomic Einstein-Rosen Bridge whose internal surface is twisted with either a Right-Hand twist, or a Left-Hand twist producing a twisted 3D/4D membrane. This topological Soliton model grew out of that simple idea. I was also trying to imagine a way to stuff the curvature of a 3 D sine wave into subatomic particles. -----------

[@IronLotus15]

Can't wait for the lie group / algebra video :) I just puzzled out how the exponential map on matrices made sense myself, so I look forward to your presentation.

[@annaclarafenyo8185]

Puzzled? Just exponentiate the matrix. It's a convergent power-series. There's no puzzle.

[@IronLotus15]

@@annaclarafenyo8185 puzzled out = made sense of. So also no puzzle here :) If my wording wasn't the confusion, sure, it's a convergent power-series, computationally it makes sense. But I wasn't sure why such a mapping would be useful until I realized that the key property d/dt e^(Mt) = M * e^(Mt) still holds, so it can still describe rotations and such handily.

[@annaclarafenyo8185]

@@IronLotus15 You should think of infinitesimal steps, not differential equations. That's just a dumb way of 'making things rigorous' with no gain in rigor but only loss of comprehensibility. The 'exponential map' is then obvious, it's taking lots of multiplicative steps one after the other after the other. You can also take steps in different noncommutative directions, and then it's a "Wilson loop" or a "Path integral", depending on context (it's always drearily the same ideas, people have no creativity).

[@IronLotus15]

@@annaclarafenyo8185 Could you help explain the infinitesimal step view? The first however-many terms in the series seems rather "finitesimal" to me.

[@annaclarafenyo8185]

@@IronLotus15 Infinitesimal means an infinite product (1 + epsilon A)^N where N is t/epsilon. You're multiplying things again and again and again, and they're only infinitesimally different from the identity. This is how you're supposed to think about the so-called 'exponential map', it's an infinite product of infinitesimal steps, but you won't find this important intuition in any math books, only in physics books. In multiplication, as opposed to addition, you want to be only slightly different from the multiplicative identity to be infinitesimal, so it's 1+ epsilon rather than just epsilon. The infinite product becomes an exponential power series by the same mechanism that produces e from (1+1/n)^n (binomial expand). Since the power-series is finite and well behaved, and the infinitesimal thing requires some thinking to define, mathematicians take the road with the shortest proofs and least intuition, as always.

[@alexanderbeliaev5244]

Simply amazing, waiting for the next video on Lie algebra and Lie groups, thank you :)

[@KipIngram]

2:09 - I *include* spin in the state; I think when you say you "know the state," then you know the spin. "State" includes all that you can know about the particle.

[@eliacampigotto2632]

Great video!!

[@lukebielawski2777]

This is much more intuitive than what I learned in school !

[@chudleyflusher7132]

This used to happen to me all the time before I quit drinking. The world would spin twice for every step forward.😹

[@ZeroPlayerGame]

Great video! I wish you'd go into more detail about contexts in which this becomes physically relevant, because it seems contradictory to you saying that it's just about frame of reference change. Do you have reading recommendations on that?

[@General12th]

This was great!

[@GIRGHGH]

I've not got much knowledge of this area but you explained it very well. The only thing confusing was when rather than using a new symbol. you reused one and primed it instead. With all the primed characters and redundant characters the math got a little hard to read.

[@deltalima6703]

You need to know where grammas house is before you start this journey thru the forest or you will be hopelessly lost. QM is confuzzling.

[@keithdow8327]

Thanks!

[@pyropulseIXXI]

Good video. I always said the physical explanation was lame and had nothing to do with the actual ‘rotation’ back to the original spin state It isn’t a physical rotation

[@coolconner1029]

what a hero

[@subhaschandrakangjam3215]

waiting for this a long time ago ,thank you and first comment

[@tim57243]

The greatest misconception I had about spin was assuming that the Stern Gerlach experiment sorted particles by spin. It sorts them by angular momentum. Spin is quantized and angular momentum is quantized but they aren't the same thing. The misconception is that I thought they were. For a spin 1/2 particle like an electron or neutron, you can use SG to sort the angular momentum so the upper beam has angular momentum up and the lower beam has angular momentum down. Discard the lower beam so now we have a beam of particles with angular momentum up. Now apply an electromagnet to rotate the angular momentum on that beam and apply another SG to the result. As you turn up the electromagnet, the angular momentum rotates, and the second SG will have its upper beam fade and a lower beam appear. Now keep turning up the magnet , the lower beam will fade and the upper beam will reappear. You have rotated the particle 360 degrees. The new beam has the same angular momentum as the beam before the rotation, but opposite spin. Getting the original spin back requires doubling the magnetic field for a total rotation of 720 degrees. The claim that a 360 degree rotation gives opposite spin despite having the same angular momentum could be tested with an interference experiment. Split the beam of angular momentum up particles before the rotation, use an electromagnet to rotate one side of the split 360 degrees, and make the two sides interfere. The two sides should have destructive interference. With a rotation of 0 or 720 degrees, the two sides should have constructive interference. Unfortunately I haven't yet found someone actually doing this experiment. So far, the SG experiments I have found are separate from the experiments that rotate and do interference. Granted, I haven't looked very hard. I would like to see a KZbin video with a physical demonstration of this.

[@mathemaniac]

At the end of the video, I described neutron interferometry. Is that what you are looking for?

[@tim57243]

@@mathemaniac The neutron interferometry experiments I have seen measured the magnetism along the beam path and assumed the published magnetic moment is right and showed you need a rotation of 720 degrees to get the constructive interference right. There was no measurement of angular momentum. Having the magnetic moment off by a factor of 2 or having the magnetic field strength off by a factor of 2 would also explain the observation just as well. Being able to demonstrate that you have one beam with angular momentum up and constructive interference, and later an identical looking beam with angular momentum up and destructive interference, would make it more clear that spin and angular momentum are different things. And it would be more fun to have a video of real equipment operating than a paper with diagrams.

[@angeldude101]

If the difference between the ϕs is the only thing that matters and not the ϕs themselves, then why are we even bothering to store the ϕs in the first place? Really, a general vector should look more like [cos(θ/2), sin(θ/2)exp(iϕ)], which has only 3 components. But wait... doesn't that just mean it's a normal 3D vector? _Yes!_ (mostly). _Actually,_ we already have a form of 3D vector that not only predates quantum mechanics, but _already represents rotations with half the angle! Quaternions._ It's worth noting that quaternions have a matrix representation... that isn't unique because the matrix has more degrees of freedom than the quaternions themselves. Sound familiar? I'm honestly _shocked_ that I haven't seen more descriptions of spin 1/2 that even mentions quaternions. Here they were at least mentioned at the very end despite 1) representing the same thing, and 2) having _much_ simpler notation (even if slightly obfuscated).

[@NormanVN]

As he said at the end, the phis do matter, it's an encoding of 'phase' which requires a linear combination of states to detect.

[@Miguel_Noether]

The most important thing is to let from 0:00 to 2:38 to really sink in 👌

[@MohdIrfanZ7]

Looking forward to your video on lie groups and lie algebra.Thanks for making these videos, anyways

[@Jaantoenen]

The point is Robert, if you do not give enough and clear information for someone else to do recreate your devices, then you are just talking to the wind, or loving the sound of your voice and don't actually stand to the test of what your devices claim. It is very frustrating to see something apparently good and useful and not have clear details to replicate it myself.

[@samuelthecamel]

I lost you at putting complex numbers in vectors. Thanks for clearing the spin 1/2 confusion up though.

[@zlm001]

Awesome.

[@anywallsocket]

30 minutes and you disprove yourself, amazing

[@tatytatytaty95]

Why do we only consider rotations around the z-axis? What is so special about them and what is their physical meaning?

[@gcewing]

A car engine might be a better physical analogy. The crankshaft and the camshaft are geared together with a 2:1 ratio. The camshaft is hidden inside, so if you turn the crankshaft by 360 degrees, everything looks the same from the outside, but internally the engine is in a different state. Also, to get the state change you have to rotate the crankshaft relative to the engine block. If you rotate the whole engine by 360 degrees, nothing changes, not even internally.

[@praveenb9048]

So .. if I understand correctly... when Dirac came up with the old belt trick, he created the first quantum mechanics clickbait.

[@Andres-is3lj]

But why should rotations on the 2-component vector correspond to an intuitive rotation on the Bloch sphere?

[@Electronics4Guitar]

If you haven’t done them already, similar analyses for spin 0, 1 and 2 would be interesting too.

[@5ty717]

Very very good

[@TheoWerewolf]

The moment I got to the 'rotate' part, I just wanted to yell 'point particles cannot rotate because they have no dimensional size'!!!!! Bizarrely every discussion about 'spin' as intrinsic angular momentum goes to great lengths to point out it can't be a real spin because point particles can't rotate - then slide RIGHT into the 1/2 spin 'rotation' problem. Also, "statistical determinism" - it's a thing.

[@johnm.v709]

Fundamental particle not point like kzbin.info/www/bejne/qneQYpd8Zcp1qtU Basic state of the cosmos

[@tatytatytaty95]

Why is continuity important if it doesn't affect the physical result?

[@tatytatytaty95]

Why is the identity matrix the only possible representation of D for 3D rotations?

[@jakublizon6375]

The non real values are the source of local phase invariance. Aka charge.

[@showmikbiswas4272]

How does this argument work for bosons? Why do they have integer spins?

[@ccd30132]

I still don't get why you used theta/2 and why the trivial representation is produced by the first two (stronger) conditions.

[@mathemaniac]

theta/2 is so that theta (now in the range [0, pi]) can be the azimuthal angle on the sphere. If we don't do that, then theta is in the range [0, pi/2], and this does not correspond nicely to the range of the azimuthal angle on the sphere. That's why I used "it turns out", because to explain that would require Lie groups / algebra. This is to motivate why we have to consider projective representations rather than just representations.

[@JustMeAsIamHere]

Spinor is quaternion

[@willemesterhuyse2547]

I can use my intuition of angular momentum since I do not believe in point particles. Point particles smacks of infinity.

[@johnm.v709]

Point particle - No kzbin.info/www/bejne/qneQYpd8Zcp1qtU Basic state of cosmos

[@JaagUthaHaivaan]

But what about particles with integral spin? Like Bosons?

[@mathemaniac]

Same framework with the projective representations with (2s + 1) x (2s + 1) matrices. But as I have said, constructing those projective representations is not easy, and requires the knowledge of Lie groups / Lie algebras. It's just that in the case of s = 1/2, we can use the Bloch sphere to construct this directly.

[@JaagUthaHaivaan]

@@mathemaniac Thanks for your explanation! I always suspected the layperson explanations of spin numbers seem way too simplistic.

[@ludovicrebouillat3128]

it's like the path on a mobius ribbon

[@darrennew8211]

All I know is I have to rotate my USB connectors 720 degrees before they'll plug in, minimum.

[@bitcoininfinity9640]

If time is not real, how come they make it correct? 😂

[@Achrononmaster]

Are think you are highly confused. A spinor is not a particle, it's a description of what we need to do to translate/transform a measurement system (unitary operator or "observable" in the formalism) onto the frame of a particle. The spinor comes from the geometric algebra, and is the factor in the state function for determining the rotation part of the translation. In the proper spacetime algebra it is much clearer because instead of using matrix algebra one uses the proper multivector graded algebra, which reveals the spinor is a rotor in the even subalgebra of spacetime, and so acts double-sided on physical quantities to effect the transform (the rotation or boost). Because the rotor acts double-sided, to transform a rotor under a general rotation one needs to use an angle of 4\pi. That's not "rotating around 720deg" it is just transforming a rotor under another rotor. Rotors are not physical objects, so there's nothing weird about this. Rotors are part of normal Euclidean or Riemannian geometry, and there is thus nothing "quantum" about them. The classical rotation groups have the same half-integer representations. The reason we do not use rotors so much in classical mechanics is because we use an entirely different formalism for keeping track of particles, namely phase space time evolution. One can get a phase space representation for QM too, but it's awkward, and it is just simpler to move to a measurement representation. If you formulated classical mechanics as a measurement theory (only care about the boundaries of a cobordism) you'd also have fermions. But you don't bother, since in classical mechanics you can imagine you can precisely track particles in phase space, so you don't ever care about the nuances of measurements, since you have ħ→0. The reason you cannot use a spinor for a boson is because all bosons are massless without the Higgs gauged symmetry, and massless particles cannot be rotated out of their plane of motion, so can transform single-sided in the geometric algebra (like "vectors" although they are elementary particles or stringy stuff, not actual vectors). The boson wave function is again only a mathematical instruction, it's not "the particle." Don't confound mathematical objects with physical objects, they're not the same. Mathematics is description/epistemology, not ontology.

[@FunkyDexter]

So what is then an electron? It's not the wave function, it's not the spinor, it's not something rotating... I would think electrons didn't actually exist.

[@Aziqfajar]

Haven't watch fully, will come back later. Might need to introduce imaginary number to formula involving 1/2-spin.

[@konradswart4069]

I thought I had understood the spin of an electron at least somwehat. But after _this_ video, I am _completely confused!_ Nevertheless, I find this a great video. I intend it to watch several times. _Maybe_ what it says becomes more clear after some time. My own understanding of the spin of an electron is different, and based on the work of Dennis Morris on higher-dimensional complex numbers. kzbin.info/www/bejne/hZ7EnYBugN2MZpY He asserts, that the spin of a particle in 4D spacetime can best be described by a 4D quaternion or a 4D anti-quaternion. It so happens, that rotations in spaces of an even number do not have a rotational axis _in that space!_ For examjple, a rotation in 2D space (plane) needs an axis that 'sticks out'. Which means, that the axis of rotation is not in the 2D plane. In the same manner, a higher dimensional rotation in a 4D space(time) can not be around an axis _in_ that 4D space(time). At best, when there is a rotational axis in a 4D space(time) it must exist in a fifth dimension, just like the axis of a rotation in a 2D plane is orthogonal to the plane in which the rotation takes place. I must also remark, that a 4D rotation is not an ordinary rotation according to Dennis Morris. It is a rotation whereby the 'angle' of rotation is a 3D vector, and not just a number. Dennis Morris has a terrible notation for his work. So do all other mathematicians have. I have a better notation. I reject the sine function, and rename it, in the multi-dimensional framework of Dennis Morris as sin (theta) = Cos2Y(theta). The 'ordinary cos function I describe as cos (thet) = Cos2X(theta) In three dimensional space you have three trigonometric functions having two-dimensional vectors as angles. In that space we have Cos3X(theta, phi) as the projection of a point in on a unit 3d 'sphere' in 3D space on the x-axis, Cos3Y(theta, phi) as the projection of that same point on the y axis, and Cos3Z(theta, phi) for the z-axis. The 'sphere'is then _not_ given by the equation x^2 + y^2 + z^2 = 1, but it is given by the equation: (x^3 - xyz) + (y^3 - xyz) + (z^3 - xyz) = 1, or its 'dual': (x^3 - xyz) + (y^3 - xyz) + (z^3 - xyz) = -1 Both look like 'funnels' in opposite directions, if you draw them both. You can see pictures of all of those constructions on my facebook page.

[@jinjunliu2401]

Quaternions can be represented by 2x2 complex matrices, the i, j, k numbers behave exactly like the R matrices used in this video :)

[@konradswart4069]

@@jinjunliu2401 Indeed. That is what Dennis Morris also said! According to Dennis Morris, complex numbers are just 2-dimensional numbers. There are two kinds of two-dimensional numbers, the 'ordinary' Euclidean complex numbers, whereby i^2 = -1, and the other Hyperbolic kind, whereby j^2 = 1 There are 4 kinds of 3 dimensional 'complex' numbers, which define 4 3D geometries, which are isomorphic to one another. And there are 8 kinds of 4 dimensional 'complex' numbers, and therefore 8 kinds of 4 dimensional spaces. One of these spaces is defined by the Quaternion, and the other is defined by the anti-Quaternion. The spaces defined by the Quaternion and anti-Quaternion are isomorphic to one another.

[@angeldude101]

If we're talking 3+1D spacetime, I'd argue the best representation is a 4D rotor from Clifford Algebra: a + bγ₂₃ + cγ₃₁ + dγ₁₂ + eγ₀₁ + fγ₀₂ + gγ₀₃ + hγ₀₁₂₃, which can also be represented as exp(bγ₂₃ + cγ₃₁ + dγ₁₂ + eγ₀₁ + fγ₀₂ + gγ₀₃). Those four components at the start happen to align with the quaternions and spatial rotations, while eγ₀₁ + fγ₀₂ + gγ₀₃ correspond to spacetime boosts. The final term is used for double-rotations, which only occur in 4D and higher. (They also correspond to screw motion when using 3D homogeneous coordinates.) When we have only a single rotation, 4D rotations still have an axis of rotation, but the axis is a _plane_ rather than a line. In N-dimensions, rotation axes are always N-2 dimensional subspaces. They just happen to be lines in 3D, which most mathematicians and physicists are already comfortable with. This also means that 2D rotation axes are points, not lines that somehow _escape_ the space you're working in. Clifford Algebra really makes changing the dimension you're working in very easy, to the point that it's not too hard to formulate physics in a hypothetical 4+1D spacetime. You'd have 5D rotations around 3D volumetric axes that can be represented with 10 components (which not coincidentally happens to be sum of the number of linear and planar degrees of freedom in 4D), with a total of 16 when exponentiated into a full rotor. Of those 10 degrees of freedom, 4 would be hyperbolic boosts and 6 would be "ordinary" 4D spatial rotations. There's still a limit of 2 axes of rotation for a given transformation though; you don't get a third until 6D. Going backward, the same pattern appears when considering that 0D and 1D have no axes of rotation, while 2 and 3D have a limit of 1 per transformation.

[@konradswart4069]

​@@angeldude101 Interesting! Dennis Morris has many critiques on the Clifford Algebra or Geometric Algebra approach as such. One is, that it uses all kinds of different multiplications, which prevent geometric algebras to really be called algebras. They subscribe to Hurwitz theorem, which states that the only division algebras that are possible are those of spaces of 1, 2, 4, 8 and 16 dimensions. In the approach of Dennis Morris, you can construct division algebras of any number of dimensions, just by using a different distance function for every dimension. In the approach of Dennis Morris, you have in all those dimensions just _a single_ multiplication, although you must sacrifice the idea of negative distances in even numbered dimensional spaces. As he says it: when you cross a dog with a dog, you must get a dog, and not a cat or frog. The approach of Geometric Algebra is bottom-up. You _construct_ mathematical structures which are both geometrical and algebraic. The approach of Dennis top-down, or axiomatic. Just a remark. You have basically two kinds of mathematics: bottom up, or calculative mathematics, also called intuitionistic mathematics (Brouwer) or top down or axiomatic mathematics (Hilbert.) Axioms are basically rules that tell you to what kinds of mathematical structures _you restrict yourself!_ Whenever you begin with an axiomatic system consisting of axioms, you are _implicitly stating_ what you are _excluding!_ And then, through logical means, you derive other theorems which tell you some surprising things you _also_ exclude, without being aware of it from the axioms. With intuitionistic mathematics you cannot make sense of concepts like 'infinity'. You cannot really _imagine_ something that is infinite. That is the limitation of intuitionistic mathematics. However, you can _understand_ infinity by seeing it as _an exclusion!_ Infinite just means 'not finite'. The set of positive integers, for example, is infinite = not finite, because there is _no largest finite number!_ This implies that _all_ mathemnatical proofs that 'prove' that some structure is infinite is proving that there is something _not_ existing! (E.g. there is _no_ largest integer!) And, very important, if you define something, and it turns out that some entity _cannot exist,_ then you have defined something _that is incomplete_ seen from the perspective of the constructivist/Brouwer perspective. That is why he denies the validity of the principle of the excluded middle = If you have proved that the non-existence of something leads to a contradiction, you _cannot_ conclude that it 'exists'. It requires _an explicit construction_ to exist on top of this proof. This is why _all_ mathematical proofs that show that some construction is infinite _always_ work with the modus tollens construction of logic = you assume the existence of somethihg, and then derive a contradiction. And then you know that it _misses_ something to be called finite, and _call_ it infinite! Axiomatic mathematics is also synthetic mathematics, because typically there are many _constructs_ that satisfy some set of axioms. These are then all _synthesized_ by the set of axioms that 'defines' them all. Axioms are therefore not definitions, but exclusions. Once you understand that, you understand that there are _two ways_ to understand anything mathematically. You can understand because you can _imagine_ so that you know what _it is,_ or you can understand because you can _exclude_ so that you know what _it is not!_ This distinction between constructive = analytic mathematics and eliminative, axiomatic = synthetic mathematics is very important for physics, and is invariably overlooked by them. When Einstein, for example, started with the Galilean principle of relativity, and with the invariance of the speed of light wrt any observer, and then derived the Lorentz transformations as _the only_ transformations satisfying those two conditions, he was unaware that such a simple question as: 'what do you _mean_ if you say that light does not require a medium. Isn't _space itself_ a medium?' was something he 'glossed over'. He didn't realize that his special theory of relativity _is incomplete_ because he didn't consider this question. That is why he wrestled his entire life with the problem, that the 'now' had no place in his theory, and he knew it! Whenever I raise this point to other physicists, I am only met with blank stares. They just _don't get_ that _not being able to answer this question_ means that you are _unaware_ of a huge omission in the very idea of 3 + 1D spacetime. You know what 3 + 1D spacetime _is not!_ Anything that _does not_ satisfy the Lorentz transformationns, or the later Tensor transformations which are a generalization of this point is what you _exclude!_ But you are _not_ telling _what 3 + 1D spacetime _is!_ In any case, Dennis Morris approach is top-down. He just takes N x N matrices, and he _imposes_ the condition on them that sets of such matrices must be division algebras = satisfying a set of axioms, and then you see what you _can_ get. And what you can get is a general theory of numbers of any number of dimensions, with their corresponding trigonometric functions and trigonometric identities. I have been in communication with him. This is because I was particularly interested in 3D numbers. This is not his main interest. He is mainly interested in 4D numbers, because he is interested in physics, and has a very keen interest in 3 + 1 D spacetime. He considers the 3D trigonometry as too simple and not interesting, because it doesn't tell anything about reality, so he thinks. I am interested in 3D spaces, because I am very interested in computer animations, and I am wondering whether I can use those 3 three-dimensional equivalents of the cosine function in my animations. Thanks for your response.

[@angeldude101]

@@konradswart4069 I actually view Clifford algebra from an axiomatic construction, namely that the algebra is defined by the geometric product, which itself is defined as a bilinear, associative product, where a (grade-1) vector squared is a scalar. "bilinear, associative" just means that it's actually useful, especially since geometric transformations are inherently both associative and non-commutative. The latter part is the Contraction Axiom, and is the fundamental property that sets the geometric product apart from other products. (If you note that the ℝeal numbers, ℂomplex numbers, and quaternions satisfy the prior properties, congratulations! You have found Cl(0), Cl(0,1), and Cl(0,2)!) You mentioned "too many products," but every single one is a special case of/defined in terms of the geometric product (and the dual, since sometimes zero-divisors make it hard to define it in terms of the geometric product). For instance, for vectors v and u, v · u = ½(vu + uv) and v ∧ u = ½(vu - uv). If these definitions look vaguely familiar, they should, since all they do is isolate the commutative and anti-commutative half of the product. They're also very related to the definitions of (hyperbolic) cosine and sine in terms of the even and odd parts of the exponential respectively. In general, the inner and outer product (inner product is used loosely in geometric algebra, since you can have objects whose inner product with themselves are negative or zero without the object itself being zero) are defined in terms of specific grade parts of the geometric product. Everything comes back around to that one associative (usually invertable) product. The part about division algebras only being dimension 1, 2, 4, 8, and 16 still applies in Clifford algebra. While you can construct Clifford algebras of higher dimensions, every Clifford algebra built from a 5D vector space or higher cannot be invertible since the full algebra would be 32 dimensions or higher. It's also worth noting that Clifford algebras always have powers-of-2 number of components for a general object, and they're often identified with a bitset in computer libraries. The basis elements of a Clifford algebra are essentially the power set of the vector space it was built from. Even in 4D and lower, it's not hard to find Clifford algebras that aren't division algebras. The dual numbers Cl(0,0,1) are a trivial example since ε² = 0 has no inverse. A common demonstration is Maxwell's Equation (no 's'): ∇F = J, or "the change in the electromagnetic field is equal to the source density." The distinction between the two forms is a confusing one, at least to me. It may have something to do with naturally preferring bottom-up thinking, but I interpret axioms as asserting what is true in your system, ignoring that which doesn't satisfy the axioms, and then building what you can prove exists _if_ you have something satisfying those axioms. Clifford algebras can be represented as matrices, and often are without being called "Clifford algebras," (this video is an example) but I don't really care about that since as long as I have _something_ that satisfies those properties, I can use them to derive other things. I guess it involves the Axiom of Choice, which from my understanding is a tad controversial. My favourite description of mathematics has been "making stuff up and seeing what happens." You can only do things consistent with the rules you've already declared, but you also are free to make those rules whatever you want so long as they don't permit something that contradicts itself. And since math is so complex, even the simplest choice of rules (axioms) can lead to surprising results.

[@UnCavi]

14:53 What about the case r = -1?

[@fullfungo4476]

It gives the same result.

[@UnCavi]

@@fullfungo4476 Don't you get double the freedom in choosing the representation D? Now another valid projective rep becomes D(R_3) = - e^{i \theta} D(R_2) D(R_1)

[@mathemaniac]

When I write the complex number in the form r e^(i theta), implicitly I assume r is the modulus and theta is the argument, because that's just the standard form for complex numbers. The modulus of a complex number has to be nonnegative.

[@mathemaniac]

Theta refers to *any* theta, if you have the negative sign, it would just be equivalent to making the substitution theta --> theta + pi.

[@UnCavi]

@@mathemaniac oof, that was embarassingly simple to answer. You're right, sorry my bad. Thank you!

[@meofamily4]

There were too many jumps in the explanation for me to follow it. Why is it that the only matrices that show rotations that produce the identity matrix are the identity matrix? Why do you say that sin and cos must be positive, and then allow the angle to go to pi, which has values for sin and cos that are negative? And more. Basically, the frequent dropping of unjustified assertions into the presentation has soured me on bothering with any other Mathematics videos.

[@mrgadget1485]

D(O(3)) = SU(2) ?

[@user-fr2jr6hd4i]

I major in electrical engineering. I haven't learned much about quantum mechanics. This video is too hard for me. I have no idea what this video is talking about. Yes, it's my fault. But I think there might be many people have the same problem. Maybe you should consider adding some more explanation to people who do not have foundations in quantum mechanics, but I know it might spend too much time to add those too basic contents to interrupt those people who already have good foundations, so just a small suggestion.

[@meofamily4]

I took quantum mechanics and I had a great deal of trouble following it.

[@johnm.v709]

kzbin.info/www/bejne/qneQYpd8Zcp1qtU Chance of learning spin & electron

[@XeiDaMoKaFE]

aether: 👁️👄👁️

[@jean-francoisguilbo7833]

Sorry I can’t see 3D rotations on 2D vectors

[@BracaPhoto]

It's an easy enough Analogy - take 2 humans the exact same mass - traveling in the same direction at the same velocity - Now tell me which one is a Socialist and which is a Capitalist - tadaaaa we dont know 😂😂😂 Tada- the math don't math because it's too constricting of a language - it's obvious you CAN divide by ZERO ❤

[@anywallsocket]

Good now do time reversal

[@RevisionWithDev]

india

[@akf2000]

When can we get a face reveal

[@atiqrahman7289]

Do not understand this what actually she is talking about.

[@alexanderpushkin9160]

wat

[@zeroonetime]

Your Thought of God spinn your Feelings