So, I know that spin is very important in quantum mechanics/elementary

[CyberShot]

I can explain it to a third grader, but it would take about ten years.

If I had to try to do it in less time than that, I would just say that each species of particle is identified by a short list of numbers, and that "spin" is one of those numbers. I might add that the fact that spin belongs on that list is a consequence of the rotational invariance of space, but I guess they would have stopped listening before the sentence is over.

Don't you think you're being vague here? It's not just you, however, as I see that "definition" given everywhere. You haven't defined spin, you've said what characterizes it. Am I being extremely stubborn here, or is there just no simple, physical meaning of spin?

 

[strangerep]

[...] is there just no simple, physical meaning of spin?

I've posted a reference to it a few times. See, e.g., this post:

https://www.physicsforums.com/showthread.php?p=2024618

Quick summary:

- Spin is a classical concept, also known as "intrinsic angular momentum",
which is a distinct notion from orbital angular momentum.
You'll have to look up the MTW reference I gave for the details since
I don't have time to type it in here, sorry.

- The passage from classical to quantum causes total angular momentum
to come only in discrete amounts.

- The classification of elementary particles involves these discrete
angular momentum quantum numbers.

 

[KOSS]

First, referring to the posts on pages & 2 of this thread - collectively they provide a satisfactory definition of spin. Though some kind PF mentor could probably do us all a favour and summarize.

Second. just modify your third grader version: instead of "a consequence of the rotational invariance of space" you need to say instead that "because spin is a way of accounting for the symmetry properties of particles when rotated or exchanged."

Third, why fret if the definition is only a characterization? That is all you can expect from any definition of any physical quantity! Do you suppose you know what mass or charge are in any simpler terms than the above definition of spin? If you do, please tell me! It'd be a breakthrough in meta-physics.

 

[chafelix]

If you have a problem with spin, try some more exotic notion such as charm.
It was Feynman who said that if we cannot explain something in simple terms, it means we do not understand it well. And yes, Feynman also said "nobody understands quantum mechanics" and this is normal, because our experience is with larger scales. Spin is certainely not something that spins around-that would mean speeds much larger than the speed of light. It's an internal degree of freedom. What does that mean? Take a car for example. It means that as long as we are limited to small energies, a car is a body that behaves as one particle: All of its parts move in unison. All we need to specify is position and momentum(classically speaking). But if you want to look inside, there are other degrees of freedom, such as belts moving, passengers, engines and so on. Their motion does not affect the car motion to a first approximation, but for some experiments it might. You can understand spin this way. Nonrelativistically and non-methematically, there is no deeper explanation

 

[Goldstone1]

You probably say about spin itself.
But spin-related things need Relativity even in QM.

For example, when we consider the external magnetic field (B) in nonrelativistic Schrodinger equation,

$$H = \frac{p^2}{2m} = \frac{(\sigma p)^2}{2m} \to H=\frac{(\sigma(p-eA))^2}{2m} + e\varphi$$

Here we pick up one part such as

$$(-\frac{-e}{2m})(\sigma_i p_i \sigma_j A_j + \sigma_j A_j \sigma_i p_i) = \mu_B \sigma\cdot B = g_e \mu_B S\cdot B$$

So the interaction between external magnetic field (B) and electron spin appears even in nonrelativistic Schrodinger equation.

But the spin-orbital (=internal magnetic field) interaction needs Relativity.
Actually Thomas precession factor uses "complicated" Relativity.
And if we don't use this Thomas precession, we need to use relativistic Dirac equation to explain the spin-orbital interaction (=fine structure).

And the antisymmetric property of spin 1/2 particle needs Dirac equation, doesn't it ?

The important point is that we don't know about what spin itself actually is.
(for example, electron must spin faster than light to get 1/2 angular momentum, and it returns by 2 rotation.)
This strange property can not be changed even if we use Relativity.

So we can't take anything else but Pauli matrices to explain spin itself.
(We can't move from Pauli matrices to more concrete things, because we don't know anything about spin itself.)
This is why nonrelativistic Pauli matrices is always valid even in $$\gamma$$ matrices of relativistc Dirac equation.

Nice post.

 

[Goldstone1]

In an abstract way, spin is defined by the behavior of the single-particle state vectors for a particle at rest (i.e., for massive particles).

Massless particles are a bit more special. Here a more costumary quantity instead of spin is helicity.

All this is closely related to the representation theory of the Poincare group for relativistic particles or the Galilei group for nonrelativistic ones (in the latter case of course only for massive particles since there's no nonrelativistic limit for massless particles).

Massive particles are restrained to spin? Photons are certainly not massive but contain an angular momentum. Spin is an intrinsic property of all fields, the Klein-Gorden equation describes particles with zero spin - these psuedoscalar fields have never been observed.

 

[Cruikshank]

To OP: I have the very same question. Let me try to summarize what I've learned so far:
1.Classical spinning objects (tops, the earth, etc) tend to keep spinning. So there is this classical stuff we call "spin" and more formally "angular momentum" which has units of energy*time, or kg m^2/s, and it is conserved in the universe.
2.Now for electrons orbiting a nucleus, we have these weird "smeared out clouds" of probability representing them, such that the electron can "orbit" the nucleus without "accelerating", (as a classical object would, a=v^2/r). The math plays out such that the electron would "trip over its own wave" unless it sticks to specific orbitals. It turns out that these orbitals have 0,1,2,3...units of angular momentum, measured in multiples of hbar, Planck's constant over 2pi. So *orbital* angular momentum is quantized, and it really does map over to classical as the same "stuff" because if you do some experiment with a magnetic field and yank all the electrons around into new orbits, the whole object ends up rotating the other way. So classical spin can get tied up in electron orbits, and pulled back out, roughly. So orbital angular momentum counts when you add angular momentum up to conserve it.
3.Now for the weird part: particle "spin."
As near as I can tell, particles have a property with quantized values that are integer multiples of hbar/2, not hbar. This property follows the same math relationships as orbital angular momentum in QM. There are apparently ways to add orbital (L) and "intrinsic" (S) spins to get totals (J). I think--not sure--that you can do the same sort of magnet trick and use particle spins to make a classical object move by conservation of angular momentum.
4.So, it almost sounds as if there's no big(ger) mystery here; we could posit that these alleged point particles have some sort of radius and are like spinning balls, and that prospect is probably why it got named "spin" in the first place. BUT--the intrinsic spin comes in values half the size of what ought to be possible when you try to wrap an electron wave around a sphere. I read in a book somewhere that "it is as if the electron has to turn around twice to get back where it started."
5.What I would love is a geometrical depiction of that. Where I personally got stuck was in the details of the proof that the wave wrapping in 3D Euclidean space had to be integer and not half integer multiples of hbar.
6.Like you, I want a picture, a model, something to hang my intuition on and make it easier to *see* answers to QM problems instead of blindly calculating and hoping I got it right. That's what models are for; most of the physics community seems to be having a "sour grapes" reaction to our collective inability to make that model for QM, and some even get annoyed with those of us who stubbornly refuse to give up.
7.Luck to us both. Luck to us all.

 

[Q-reeus]

Cruickshank - you have raised some very interesting issues. Touches on what I posed in #30 but got zero response to. So will ask it anew. A bar magnet contains a collection of tightly oriented electron spins S, + orbital angular momentum L. In ferrites for instance the two contributions are roughly equal. Let's assume the orbital part L behaves macroscopically in a classical manner re a rotation of the bar magnet. What is the situation though for the spin part S. In QM there is this 1/2 integer spin -> 'rotation through 4pi' thing to restore everything. But is this reflected in the macroscopic behavour of the bar magnet as gyroscope? If not, why not?

 

[strangerep]

As near as I can tell, particles have a property with quantized values that are integer multiples of hbar/2, not hbar. This property follows the same math relationships as orbital angular momentum in QM.

Actually, it's more like the other way around. One derives half-integral quantized spin from the general su(2) Lie algebra. But when the generators of this algebra are restricted to be of the form q x p (i.e., orbital angular momentum), this causes a consequential restriction of quantized angular momentum to integer values. Details can be found in Ballentine.

"it is as if the electron has to turn around twice to get back where it started."
5.What I would love is a geometrical depiction of that.

Have you tried "Dirac scissors", a.k.a the "belt trick" ?

 

[Cruikshank]

Thank you for the suggestions. I must own 30 QM textbooks and 10 on group theory, and I'm not sure any of those things are even mentioned. Every answer I've gotten that sounds knowledgeable seems to include a lot of group theory terminology I've never heard of, even though I have a Master's in Math and took Abstract Algebra. I've gotten two textbooks on spinors and after reading a chapter of each could not find any content and gave up. It was as if the book was just spouting random trivia, not describing an entity or having a point. Frustrating.

 

[syberraith]

As Einstein once said, you don't truly understand something in physics unless you can explain it to a third grader.

Now, assume I know nothing about physics. Please explain it to me in layman and concrete terms, preferably even a single sentence. :-)

I'm just learning about this myself and I lack knowledge to explain it in mathematical form. Although I think I can provide the general concept.

It's called spin because it has some of characteristics of a rotating mass like a gyroscope. Spin tend to point in one direction like a gyroscope standing up by itself until some thing causes it to change its alignment. With particles they keep spinning because there's a lack of any friction to slow them down like a gyroscope.

I know with neutrons and electrons and presumably protons too, spin is related to magnetism. That is a magnetic field can change the orientation of the spin.

Also the spin of particles seem to have particular magnitude when they interact with other particles.

So there you have it, at least with an electron spin has some of the character of a magnet gyroscope. that is it has an axis and magnitude of rotation and is subject to magnetic fields.

 

[strangerep]

Thank you for the suggestions. I must own 30 QM textbooks and 10 on group theory, and I'm not sure any of those things are even mentioned. Every answer I've gotten that sounds knowledgeable seems to include a lot of group theory terminology I've never heard of, even though I have a Master's in Math and took Abstract Algebra. I've gotten two textbooks on spinors and after reading a chapter of each could not find any content and gave up. It was as if the book was just spouting random trivia, not describing an entity or having a point. Frustrating.

In Misner, Thorne & Wheeler, "Gravitation", there a diagram showing a point connected by flexible strings to a sphere at infinity. They use this diagram to try and illustrate how spinors are related to a kind of orientation entanglement.

But more helpful (perhaps) is Penrose & Rindler's "Spinors & Spacetime" which delves into such geometric pictures in vast detail. That's where I first read about the Dirac scissors thing and was so astonished that I immediately took off my belt and tried it then and there (fortunately in the privacy of my own home :-)

 

[Q-reeus]

Seems I have an invisible presence on this thread. What was posed in #30 and #43 was a genuine query. Has no-one here an answer, or is it 'too dumb' a question? Bad manners to ignore - if someone here can explain, please provide the explanation!

 

[Cruikshank]

Q-reeus: As far as I know, you can make macroscopic rotation out of particle spins as well as their orbital angular momentum. But someone more knowledgeable about experimentation could be more definitive.

 

[Cruikshank]

Strangerep: thank you very much for the text references. I was thinking it was time for another assault on MTW anyway...and I'll look up the other as well.

 

[Q-reeus]

Q-reeus: As far as I know, you can make macroscopic rotation out of particle spins as well as their orbital angular momentum. But someone more knowledgeable about experimentation could be more definitive.

Well thanks for a response Cruikshank, but that doesn't help me as a QM dumbo to get a handle on the physical consequences of 1/2 integer spin as it relates - or doesn't - to operations on a macroscopic collection as per precessing bar magnet. I'm assuming the orbital L component can be treated classically re macroscopic properties - so let's ignore that. No question that the gross macroscopic *magnetic* properties of a bar magnet can be modeled by that of a collection of classically spinning charged spheres/loops/whatever - the S contributions. There is no 'double rotation funny business' going on magnetically - rotate the magnet through 360 degrees about a perpendicular axis and everything returns. Otherwise dynamos for one would act rather oddly. Seems to me this would also apply to the S contributed magnet angular momentum, if I understand right the meaning of the gyromagnetic ratio g ~ 2, the magnet's S contributed gyroscopic properties should similarly act classically. Or not? In short, why does or doesn't double rotation manifest macroscopically - I can't see any gradual transition via correspondence principle applying here. Any links to a 'visual' tutorial on how 1/2 integer spin manifests at the particle interaction level would help here.

 

[Nano-Passion]

I'd say from the point of view of a physicalist, the words "a fundamental characteristic property of elementary particles, composite particles (hadrons), and atomic nuclei" are pretty vague. What exactly is this fundamental property?

Yes, but then the theory doesn't paint a very accurate description of reality, now does it? Sure, it makes sense mathematically, as an idealization that only makes our computational tasks simpler, but does nature really care about our computational struggles or what the maths says? I'm sorry to be so very stubborn, but I can't simply accept things without the deeper meaning behind them being revealed. That's probably why Einstein so despised QM. Now where's the fun in that! =/

Our understanding is far from complete. Maybe you might help figure nature at its deepest level in the future. But there is no use in complaining of quantum mechanics. Not knowing the deeper meaning behind certainly does not mean quantum mechanics is false either. I'm sure you don't have the slightest clue about the meaning of entanglement or the non-deterministic nature of quantum states. I don't think any of us really do.

I might get bashed for this, seeing the optimism in this thread of people feeling that we have completely deduced the nature of nature (of spin at least). But I beg to differ. As for accusing the question of "what is charge or mass" as metaphysics.. preposterous. People hundreds of years ago might have said the same thing of why we are attracted to the earth.. heck it was taken as "reality" and the way things are for most people. But now we know this is not true. Be careful of assuming we have an all consuming form of knowledge.

There are reputable physicists trying to figure out what causes mass. Regarding the big unknown questions as metaphysics completely pisses me off. Reality shouldn't be taken point blank, we are increasingly seeing this as we move from classical mechanics to quantum mechanics and other forms of physics deducing reality (high energy etc.)

----

As for spin, cybershot, the reason you are having trouble following the interpretation of spin is that what we have is a mathematical truth of its property. I contend that it is not based on your everyday macroscopic observation of reality.

 

[strangerep]

Seems I have an invisible presence on this thread. What was posed in #30 and #43 was a genuine query. Has no-one here an answer, or is it 'too dumb' a question? Bad manners to ignore - if someone here can explain, please provide the explanation!

Well, I know how it feels to be ignored when one asks a genuine question. Usually I don't respond to exclamation marks, but I'll make an exception for you here because I think your question is not "dumb" at all and deserves an answer. So I'll try to say something vaguely helpful even though I probably can't answer it to your full satisfaction since the gory details of ferromagnetism are not my strong point.

A bar magnet contains a collection of tightly oriented electron spins S, + orbital angular momentum L. In ferrites for instance the two contributions are roughly equal. Let's assume the orbital part L behaves macroscopically in a classical manner re a rotation of the bar magnet. What is the situation though for the spin part S. In QM there is this 1/2 integer spin -> 'rotation through 4pi' thing to restore everything. But is this reflected in the macroscopic behavour of the bar magnet as gyroscope? If not, why not?

Can you put your hands on a copy of Ballentine? Specifically, section 7.6 "Rotation through 2pi", pp182 -183.

In case you can't, here's some brief extracts:

We are accustomed to thinking of a rotation through 2pi as a trivial operation that leaves everything unchanged. Corresponding to this belief, we shall assume that all dynamical
variables are invariant under 2pi rotation. That is, we postulate
$$R(2\pi) A R^{-1}(2\pi) ~=~ A ~,~~~~~ or ~~~~[R(2\pi), A] = 0 ~,$$
where A may represent any physical observable whatsoever. But $R(2\pi)$ is not a trivial operator (that is, it is not equal to the identity), and so invariance under transformation by $R(2\pi)$ may have some nontrivial consequences.

It is important to distinguish consequences of invariance of the observable from those that follow from invariance of the state. [...]

He then elaborates further about how expectation values of an observable for two different states can be the same even if the states are different, and derives a few more interesting properties such as the superselection rule between integer and half-integer spin. So what I'm suggesting is to try and tease apart which aspects of your scenario correspond to an observable, and which correspond to state.

It also helps to understand one of the "direct" observation experiments of spin-half properties of fermions under 2pi rotation in an interferometer experiment. (I think this is described in a brief footnote in that Penrose & Rindler textbook I referenced earlier.) Basically, one side of the interferometer is arranged so that it can rotate such fermions (using their magnetic moment, iirc). Then when the beams are recombined, one sees cancellations because the states from both sides are being added coherently.
(Well, it's not really a "direct" observation, but it's about as close as one can get, afaik.)

Probably, the above doesn't answer the question properly but maybe it will help you articulate a followup question and/or prompt others to give a better answer than I can.

 

[Q-reeus]

Well, I know how it feels to be ignored when one asks a genuine question. Usually I don't respond to exclamation marks, but I'll make an exception for you here because I think your question is not "dumb" at all and deserves an answer.

That's the attitude I respect strangerep - thanks. Looks like I should be far more parsimonious with '' and "" and **. OK I'll try to reform, but without making it an excuse, one can be driven to use of such out of sheer frustration at times. :grumpy:

Can you put your hands on a copy of Ballentine? Specifically, section 7.6 "Rotation through 2pi", pp182 -183.

Unfortunately not, but maybe someone knows of an equivalent online resource?

"...But R(2π) is not a trivial operator (that is, it is not equal to the identity), and so invariance under transformation by R(2π) may have some nontrivial consequences..."

He then elaborates further about how expectation values of an observable for two different states can be the same even if the states are different, and derives a few more interesting properties such as the superselection rule between integer and half-integer spin. So what I'm suggesting is to try and tease apart which aspects of your scenario correspond to an observable, and which correspond to state.

As a general statement I can appreciate that different states could leave a given observable (say magnetic moment) unchanged - provided some other observable has changed. Otherwise what meaning is there to saying the states differ?
Further though, what exactly does Ballentine mean by "...invariance under transformation by R(2π) may have some nontrivial consequences..."? That is where it all hits the fan for me. My naive expectation here is that invariance is synonymous with 'nothing changes', otherwise we have an oxymoron? Or is he simply saying here one must have a multiple of 4pi for fermions?

It also helps to understand one of the "direct" observation experiments of spin-half properties of fermions under 2pi rotation in an interferometer experiment. (I think this is described in a brief footnote in that Penrose & Rindler textbook I referenced earlier.) Basically, one side of the interferometer is arranged so that it can rotate such fermions (using their magnetic moment, iirc). Then when the beams are recombined, one sees cancellations because the states from both sides are being added coherently.
(Well, it's not really a "direct" observation, but it's about as close as one can get, afaik.)

This is getting closer to something useful as to physically observable consequences, but I would like more detail. Is this cancellation undone by a further 2pi rotation of the same fermion? We are talking about an interference pattern, or something else?

[EDIT: Just did some reading at http://en.wikipedia.org/wiki/Spin-%C2%BD" and under 'Complex Phase' reads:
"...Say you send a particle into a system with a detector that can be rotated where the probabilities of it detecting some state are affected by the rotation. When the system is rotated through 360 degrees the observed output and physics are the same as at the start but the amplitudes are changed for a spin-½ particle by a factor of -1 or a phase shift of half of 360 degrees. When the probabilities are calculated the -1 is squared and equals a factor of one so the predicted physics is same as in the starting position. Also in a spin-½ particle there are only two spin states and the amplitudes for both change by the same -1 factor so the interference effects are identical unlike the case for higher spins. The complex probability amplitudes are something of a theoretical construct and cannot be directly observed.

If the probability amplitudes changed by the same amount as the rotation of the equipment then they would have changed by a factor of -1 when the equipment was rotated by 180 degrees which when squared would predict the same output as at the start but this is wrong experimentally. If you rotate the detector 180 degrees the output with spin-½ particles can be different to what it would be if you did not hence the factor of a half is necessary to make the predictions of the theory match reality."

First paragraph above seems to be saying there are no single-particle observable consequences there, but presumably that does not apply to a 2-particle interaction as per interference fringes? In the second paragraph it is not explained what the relation is between say electron spin axis and equipment rotation axis. I take it for instance there is no equivalent in electron diffraction to vertical and horizontal polarization of light.]

Probably, the above doesn't answer the question properly but maybe it will help you articulate a followup question and/or prompt others to give a better answer than I can.

Apart from the above, I suppose just how it finally relates to whether our bar magnet has any macroscopic properties that peculiarly relate to a 4pi rotation!

 

[edguy99]

Well thanks for a response Cruikshank, but that doesn't help me as a QM dumbo to get a handle on the physical consequences of 1/2 integer spin as it relates - or doesn't - to operations on a macroscopic collection as per precessing bar magnet. I'm assuming the orbital L component can be treated classically re macroscopic properties - so let's ignore that. No question that the gross macroscopic *magnetic* properties of a bar magnet can be modeled by that of a collection of classically spinning charged spheres/loops/whatever - the S contributions. There is no 'double rotation funny business' going on magnetically - rotate the magnet through 360 degrees about a perpendicular axis and everything returns. Otherwise dynamos for one would act rather oddly. Seems to me this would also apply to the S contributed magnet angular momentum, if I understand right the meaning of the gyromagnetic ratio g ~ 2, the magnet's S contributed gyroscopic properties should similarly act classically. Or not? In short, why does or doesn't double rotation manifest macroscopically - I can't see any gradual transition via correspondence principle applying here. Any links to a 'visual' tutorial on how 1/2 integer spin manifests at the particle interaction level would help here.

I am big on visual representations, I like to draw them and talk about them (hopefully no one takes offence to this one). Consider a representation of an electron starting with a cross-section:

Now consider the full particle:

and in full motion http://www.animatedphysics.com/videos/electrons.htm" .

The principle of this type of drawing is that the red lines (magnetic) lines are twice as long as the blue (electrical) lines (the blue vertical line is the axis of the spinning blue ring). Hence, 2 turns of the the inner blue ring are required to bring the outer lines back to their original orientation. (Ie, after a turn of 360 degrees, the particle is "upside down"). It also represents up/down spin, precession and antiparticles. Finally, it is basically an extension of the spinning particle http://en.wikipedia.org/wiki/Angular_momentum" where in addition, the spinning electrical current has captured an equivalent amount of spinning (orthogonal) magnetic energy.

Different particles have very different moments of inertia as illustrated http://en.wikipedia.org/wiki/List_of_moments_of_inertia" and I would be very interested if anyone has any good idea on how to calculate the moment of inertia of this type of particle.

 

[Q-reeus]

Some interesting animations there edguy99 (and decently viewable once right click options are discovered), but I somehow think not really standard QM fair! Having trawled through various sites like
http://en.wikipedia.org/wiki/Spin_(physics)
http://en.wikipedia.org/wiki/Spin_quantum_number
http://math.ucr.edu/home/baez/spin/node15.html
http://math.ucr.edu/home/baez/spin/node18.html

it seems the only explanations are in abstract mathematics. Wonder though if rotating our bar magnet end-over-end might be considered a QM 'measurement' operation in some sense that somehow collapses the collective spin wavefunctions to a classical system in a continual manner. Just my rambling - ignore as usual folks! :shy: :zzz:

 

[strangerep]

[...Ballentine...]
Unfortunately not, but maybe someone knows of an equivalent online resource?

If you have an Amazon account, you can probably use the "look inside" feature to read a little -- at least the pages I mentioned. Also try Google books.

But if you have a serious interest in Quantum Physics, then acquiring your own copy of Ballentine will not be a wasted investment.

Regarding your other questions, see whether you can read enough of Ballentine via these sources to at least follow the R(2π) stuff -- since it answers some of questions directly. Then decide which of your questions remain.

(Unfortunately, the Wiki page you mentioned doesn't cover the kind of split-rotate-recombine arrangement I was talking about.)

 

[jambaugh]

Decided to throw my two cents worth into the ring...

I see two aspects to the OP's question the first of which has been pretty well answered, i.e. spin is intrinsic angular momentum. For a classical analogue one need simply consider the Earth and Sun. The Earth has orbital angular momentum due to its orbit around the Sun but in addition has spin due to its rotation about its own axis.

The second aspect is the implied question from the quantum case "what is a spinor". I would begin with the canonical duality between observables and generators of kinematic transformations.

Spatial translations correspond to and are canonically generated by linear momenta.

Rotations correspond to and are canonically generated by angular momenta.

The intrinsic spin of a quantum particle then manifests as its representation under rotations. A spin 1 particle transforms as a vector, a spin 2 particle transforms as a symmetric 2-tensor and so on... then there are the spinor representations...

For all my study I didn't fully understand spinors until I studied Clifford algebras in detail.

Begin with the antisymmetric tensor representations of rotations (or Lorentz transformations in relativistic theory... or SO(p,n) in abstract generalizations).

For rotations in n dimensions you have scalar (dim 1=n choose 0) vector (dim n = n choose 1) bi-vector=rank 2 antisymmetric tensor (dim n choose 2) and so on up to rank n antisymmetric tensors (dim n choose n = 1).

In three dimensions these correspond to scalar, vector, bi-vector and pseudo-scalar of dimension 1, 3, 3, and 1 respectively. Now we can represent rotations of each of these by writing the corresponding rotation matrix acting on the corresponding column vectors but with Clifford algebras we have a nice adjoint representation.

For n dimensional rotations we construct the Clifford algebra from (the identity 1 and) the grade 1 elements corresponding to a basis of the vector representation. $\gamma_1, \gamma_2, \cdots \gamma_n$
These have the property that their anticommutator is a multiple of the identity, and that multiplier is the metric:
$$\gamma_i \gamma_j + \gamma_j \gamma_i = g_{ij} \mathbf{1}$$
Then resolving all products of gammas into commutator and anti-commutator components we find that we have higher grade products corresponding exactly to higher rank anti-symmetric tensors. In particular the grade two elements $\gamma_{ij} = \frac{1}{2}[\gamma_i, \gamma_j]$ are the generators of rotations in the i,j planes (or pseudo-rotations if the metric is indefinite).

One observes then that rotations acting on the clifford algebra are represented by the adjoint action, that is to say a rotation of any element $A$ through angle $\theta$ in the i,j plane is given by:
$$R_\theta[A] = e^{\theta\gamma_{ij}/2}A e^{-\theta\gamma_{ij}/2}$$
--The grade 1 elements $\gamma_i$ will transform as if rotated as vectors.
--The grade 2 elements $\gamma_{jk}$ will transform as bi-vectors.
...
--The grade k elements will transform as rank k anti-symmetric tensors.
and so on.

So the Clifford algebra represents rotations of vectors and tensors (including the rotation generators themselves) internally via this adjoint action. This is why Clifford algebras are also called geometric algebras. Now this by itself is just a simple mathematical trick, like using quaternions or complex numbers to represent rotations (which is actually doing the same thing as these are clifford algebras when graded appropriately).

The real insight is when we express the elements of the Clifford algebra as matrices and then ask what is the vector space these matrices act on via left multiplication. The answer is that they are spinors.
$$R_\theta[\psi] = e^{\theta \gamma_{ij}}\psi$$
In this sense the space of spinors is the "square root" of the space of anti-symmetric tensors (including vectors) in the way that abstract vectors are "square roots" of matrices.
$Cliff = \Psi\otimes\Psi^*$
When all the mathematical construction is said and done, we have another set of irreducible representations of groups of rotations in the spinor space $\Psi$

For rotations in 3 dimension the Clifford algebra is the algebra generated by the Pauli spin matrices. The $\sigma_i$ are the grade 1, vector elements. Their commutators $[\sigma_i,\sigma_j]= 2i\sigma_k = 2\sigma_{ij}$ are the grade 2 generators or rotations. They are expressed as 2x2 complex matrices and the (right ideal) complex 2-vectors they act upon are the spinors of rotation in 3 dimensions.

For relativistic rotations and boosts we have the Lorentz group, the Clifford algebra of Dirac gamma matrices: $\{1,\gamma_\mu, \gamma_{\mu\nu}, \gamma_{\mu\nu\lambda},\gamma^5=\gamma_{1234}\}$
representable as 4x4 matrices acting on 4 dimensional relativistic spinors.

Getting back to what they mean physically, a point particle would have no intrinsic spin and is unaffected by rotation about its center of mass. Since this description relies on treating it as a classical particle we rather say in quantum mechanics, rotating a spin-0 particle about any origin is represented only by acting on the coordinates (x,y,z) of its wave-function.
With this then the observable of angular momentum is expressed wholly by the orbital component: $L_z = -i(x\partial_y - y\partial_x)$ etc.

For a particle with intrinsic spin, one has a vector, tensor, or spinor valued wave-function and the total angular momentum observable will be the orbital component plus the representation of intrinsic spin expressed by a matrix. In the spinor case that matrix will be $S_{3}=-i\gamma_{12}=\sigma_3$ et al.

The physics is then in the spectrum of the total angular momentum and its orbital and intrinsic components: $J = S+L$.

[Edit: I would recommend https://www.amazon.com/dp/0521551773/?tag=pfamazon01-20 by Ian Porteous as a thorough advanced reference. ]

 

[Q-reeus]

If you have an Amazon account, you can probably use the "look inside" feature to read a little -- at least the pages I mentioned. Also try Google books.

But if you have a serious interest in Quantum Physics, then acquiring your own copy of Ballentine will not be a wasted investment.

Regarding your other questions, see whether you can read enough of Ballentine via these sources to at least follow the R(2π) stuff -- since it answers some of questions directly. Then decide which of your questions remain.

(Unfortunately, the Wiki page you mentioned doesn't cover the kind of split-rotate-recombine arrangement I was talking about.)

Thanks for leads - the split between state and observable and consequences finally starts to make some sense by p184. Still can't connect to therefore there being observable consequences re the interference experiment you mentioned in #53, but clearly I would need a lot of background math study to really grasp it. Will trust that it all formally reconciles and give it a break for now at least.