What is the property of a particle known as spin?

[JDude13]

What is the property of a particle known as spin?
I ask this because I read somewhere that particles don't actually spin around and that no one really knows what spin is.
If no one knows what spin is, how can we measure it?

 

[tom.stoer]

Spin is a purely quantum mechanically property which must not be confused with orbital angular momentum where something is going round in circles - except for the fact that the algebraic properties (rotation group SO(3) and SU(2)) are used for both orbital angular momentum and spin.

 

[ytuab]

What is the property of a particle known as spin?
I ask this because I read somewhere that particles don't actually spin around and that no one really knows what spin is.
If no one knows what spin is, how can we measure it?

Yes, no one really knows what spin is.
(Somewhere I have read about Penrose's (or somebody's) spin.
He tried to explain why the electron spin doesn't return by one-revolution.
But it is still difficut to explain this property.)

Spin is "empirically" defined. (=spin angular momentum 1/2 (not g factor) is empirically defined.)
I think the anomalous zeeman effect is historically more important than Stern-Gerach experiment.
When the spin angular momentum in the direction of magnetic field is +1/2 or -1/2, only the "Bohr magneton" can be measured.
This case is the same as the Paschen-Back (or normal Zeeman effect), which is not specific to the spin itself.

In the anomalous Zeeman effect, this value 1/2 in the direction of external magnetic field is changing due to the precession.

This is almost all I know about electron spin.

 

[tom.stoer]

Spin can be related to the symmetry structure of spacetime. Our spacetime has (according to GR) 3+1 dimensions with the Lorentz group as its (local) symmetry group. This is the group of "rotations" in 4-dim. Minkowski space. SO(3,1) is locally isomorphic to SU(2)*SU(2).

Each SU(2) factor generates so-called representations which are labelled by integers and half-integer, i.e. J = 0, 1/2, 1, 3/2, 2, ... The fact that half-inter spin exists in nature can be traced back to a specific property of space-time

 

[element4]

I think it is somewhat misleading to say that we don't know what spin is. We know very well what it is, in the sense that we know how to measure it and how to manipulate it. We know very well what it is theoretically, and its deep connection to relativistic symmetries (as explained by tom).

Spin is an internal degree of freedom which doesn't have an classical counterpart, and this is probably what people mean by "we don't know what spin is".

 

[humanino]

I think it is confusing to say "spin is quantum mechanical" or "spin is relativistic". Just classical SO(3), ordinary 3D isotropic flat space, already has SU(2) as universal cover. It is precisely the fact that SO(3) is not simply connected which gives rise to spin 1/2. If you attach a random object with pieces of string and rotate the object by 2pi you can not disentangle the strings with translations. However, you can always do so with 4pi rotation.

Just take a book, slide one edge of a long band into the book and attach the other edge of the band somewhere else, say in another book. Each side of the band represents two such strings attach to your book. Experience with it, you will handle a spin 1/2.

 

[element4]

SO(3,1) is locally isomorphic to SU(2)*SU(2).

Is this correct? If I'm not wrong, [tex]SU(2)\times SU(2)[/tex] is the universal covering group of [tex]SO(4)[/tex], and they are therefore locally isomorphic (isomorphic Lie algebras). How can it be true that [tex]SO(3,1)[/tex] is locally isomorphic to [tex]SU(2)\times SU(2)[/tex]?

I ask this because I found it very confusing when we covered this in a QFT course few months ago (the lecturer claimed that [tex]SO(3,1) = SU(2)\times SU(2)[/tex] not only locally, which is cleary wrong! One is compact and simply connected while the other isn't!). After a lot of thinking, I remember I concluded that the complexifications of the Lie algebras are isomprphic [tex]\mathfrak{so}(3,1)_{\mathbb C} \approx(\mathfrak{su}(2)\oplus\mathfrak{su}(2))_{\mathbb C}[/tex], and this is probably what he meant. But I guess that the relation [tex]\mathfrak{so}(3,1)_{\mathbb C} \approx\mathfrak{sl}(2,\mathbb C)\oplus\mathfrak{sl}(2,\mathbb C)[/tex] is the truly important one for the (projective) representations of [tex]SO(3,1)[/tex]. Am I totally off?

 

[element4]

I think it is confusing to say "spin is quantum mechanical" or "spin is relativistic". Just classical SO(3), ordinary 3D isotropic flat space, already has SU(2) as universal cover. It is precisely the fact that SO(3) is not simply connected which gives rise to spin 1/2. If you attach a random object with pieces of string and rotate the object by 2pi you can not disentangle the strings with translations. However, you can always do so with 4pi rotation.

Just take a book, slide one edge of a long band into the book and attach the other edge of the band somewhere else, say in another book. Each side of the band represents two such strings attach to your book. Experience with it, you will handle a spin 1/2.

Isn't the point, that classically a >>point particle<< cannot have spin? The situation is of course different if you have many degrees of freedom (like your strings).

Another way of playing with "spin 1/2" classically: put a book in your right hand and and try to rotate the book/hand. After a [tex]2\pi[/tex] rotation your arm is twisted, while your back in the initial position after a [tex]4\pi[/tex] rotation.

 

[tom.stoer]

@element4

The algebras so(3,1) and su(2)+su(2) are identical.
The groups SO(3,1) and SU(2)*SU(2) are locally isomorphic, but not globally.

There is a local bijection between the groups, but not a global one.

I think you understood everything correctly, you only overlooked the "locally" in my statement

 

[humanino]

Isn't the point, that classically a >>point particle<< cannot have spin?

How does that viewpoint change with relativity or quantum mechanics ?

 

[element4]

@element4

The algebras so(3,1) and su(2)+su(2) are identical.
The groups SO(3,1) and SU(2)*SU(2) are locally isomorphic, but not globally.

There is a local bijection between the groups, but not a global one.

I think you understood everything correctly, you only overlooked the "locally" in my statement

Thanks for your reply. I actually got the "local" part, but I cannot see how it can be correct. We know that [tex]\mathfrak{so}(4)\approx \mathfrak{su}(2)\oplus\mathfrak{su}(2)[/tex], because [tex]\text{Spin}(4)\approx SU(2)\times SU(2)[/tex] is the universal cover of [tex]SO(4)[/tex]. But since [tex]\mathfrak{so}(4)[/tex] and [tex]\mathfrak{so}(3,1)[/tex] are different, I cannot see how [tex]\mathfrak{so}(3,1)[/tex] is identical to [tex]\mathfrak{su}(2)\oplus\mathfrak{su}(2)[/tex]? (But I think that their complexifications are isomorphic).

Maybe I am misunderstanding something, I will think about it.

 

[element4]

How does that viewpoint change with relativity or quantum mechanics ?

I am on very shaking ground here, so don't take what I say too seriously.

I think that the difference between classical and quantum physics is that in quantum physics physical observables must transform according to projective representations (in contrast to usual representations). This makes spinor representations possible.

In relativistic quantum theory the relevant group is the Poincaré group, and for massive particles the relevant little group is [tex]SU(2)[/tex]. But I don't know what happens in non-relativistic quantum theory, since I don't know about the representation theory of the Euclidean group.

I get the feeling that you know all this (much better than me), and that there are many subtleties I haven't thought about.

 

[NanakiXIII]

I think it is confusing to say "spin is quantum mechanical" or "spin is relativistic". Just classical SO(3), ordinary 3D isotropic flat space, already has SU(2) as universal cover. It is precisely the fact that SO(3) is not simply connected which gives rise to spin 1/2. If you attach a random object with pieces of string and rotate the object by 2pi you can not disentangle the strings with translations. However, you can always do so with 4pi rotation.

Just take a book, slide one edge of a long band into the book and attach the other edge of the band somewhere else, say in another book. Each side of the band represents two such strings attach to your book. Experience with it, you will handle a spin 1/2.

Could you elaborate on what you said here? It sounds like an interesting way to "understand" this, but I don't quite follow what you mean.

 

[ytuab]

Some people show interesting classical models of electron spin.
But I don't understood three points yet.

1. Spinor can be expressed as
[tex]\begin{pmatrix} \cos \frac{\theta}{2} e^{-i\phi/2} \\
\sin \frac{\theta}{2} e^{+i\phi/2} \end{pmatrix}[/tex]
So the rotations in all directions apply to this "two-valued" property.
In the cases of string, when we rotate a thing in several directions at the same time, the string becomes more twisted?

For example, in the hand/book case, when we rotate the right hand + book by [tex]2\pi[/tex] on the x-y plane, and then rotate it in clockwise or counterclockwise direction by [tex]2\pi[/tex] on the x-z plane, the right hand becomes more twisted in one case of them?

2. According to the interference experiment, the spinor need to change as [tex]e^{i\phi/2}[/tex].
Not only after two revolutions, the twisted string or hand can change like this in the process of the rotation?

3. We can define the above [tex]\theta[/tex] and [tex]\phi[/tex] freely.
(We can define these variables from our observers's viewpoint.)
In this case, the object is still (not moving), and when we rotate around it by [tex]2\pi[/tex], we will see a different thing. By [tex]4\pi[/tex] rotation, we will see the same thing as the first object.

 

[strangerep]

On the assertions that "no one really knows what spin is", and
that "spin is quantum mechanical", I offer my $0.02 ...

Spin := "intrinsic angular momentum". This concept is well-defined
in a classical context, distinct from orbital angular momentum...

The most direct way of explaining this that I've seen is in
Misner, Thorne & Wheeler "Gravitation", Box 5.6, pp157-159.

The distinction between "intrinsic" and "orbital" angular momentum
of a system is that the latter depends on where (relative to the
system CM world line) you're measuring it. In contrast, the
intrinsic angular momentum only depends on the CM 4-velocity
and an integral over a spacelike hyperplane orthogonal to the
4-velocity. Like I said, see MTW for details on this. IMHO, their
treatment makes it pretty clear what "spin" is. :-)

But classically, this spin is not constrained to come in half-integral
amounts. However, if we now demand that the angular momentum
Lie algebra is represented on a Hilbert space (with the usual
Hermitian inner product), this extra requirement alone is sufficient to
derive the "half-integer" property. Just turn the mathematics handle
(as shown in, eg, Ballentine, and plenty of other texts). Is this sense,
half-integral spin is indeed a QM feature, but intrinsic angular
momentum in general is a well-defined concept both classically
and quantum-mechanically.