# QM Spin

1. Jan 23, 2005

### da_willem

"What is spin?"

This is the abstact from an article published in the American Journal of Physics (54 (6) June 1986) by Hans C. Ohanian titled "What is spin?":

I haven't studied quantum field theory, and didn't understand the whole article. But if this is the case why is the general concensus that spin is something we can't understand classically and has nothing to do wth the rotation of mass (or energy)?

Last edited: Jan 23, 2005
2. Jan 23, 2005

### dextercioby

I'm sorry,i don't believe it.You have to come up with the calculations.I'll try to find Belinfante's article at the library,i can promiss u that.
But isn't it a bit weird that NO QUANTUM MECHANICS TREATISE speaks about "circulation of energy"???

Daniel.

EDIT TO ADD:This is (should be) QM,not QFT.In QFT spin has a clear explanation and it's nothing "mysterious" about it...

Last edited: Jan 23, 2005
3. Jan 23, 2005

### da_willem

That's why I asked! I saw the calculations and they looked pretty solid, but as I said I'm not an expert...

4. Jan 23, 2005

### dextercioby

I cannot formulate an opinion based upon things i cannot see/feel.If i don't see those calculations,i cannot say whether it's bull or not.
Anyway,for the record,Belinfante was a good friend with Pauli and Fierz and did some nice things...

Daniel.

5. Jan 23, 2005

### da_willem

Well he starts from the symetrized energy-momentum tensor wich sais the momentum density in the Dirac field is:

$$\mathbf{G}=\frac{\hbar}{4i}[\psi^\dagger \nabla \psi - \frac{1}{c} \psi^\dagger \mathbf{\alpha} \frac{\partial \psi}{\partial t}]+hc$$

Where hc denotes the hermitian conjugate of the preceding term. The time derivative can be eliminated by means of the Dirac equation to yield:

$$\mathbf{G}=\frac{\hbar}{4i}[\psi^\dagger \nabla \psi + \psi^\dagger \mathbf{\alpha} (\mathbf{\alpha} \cdot \nabla)\psi]+hc$$

Wich (he says) can be written using the commutation relations of $\alpha_k$:

$$\mathbf{G}=\frac{\hbar}{2i}[\psi^\dagger \nabla \psi -(\nabla \psi ^\dagger)\psi]+\frac{\hbar}{4}\nabla \times (\psi^\dagger \mathbf{\sigma} \psi)$$

Now it comes: "The first term in this momentum density is associated with the translational motion of the electron, whereas the second term is asociated with circulating flow of energy in the rest frame of the electron."

Then he calculates this term for a Gaussian packet (wich represents in the nonrelativistic limit an elektron of spin up with zero expectation value of the momentum):

$$\psi=(\pi d^2)^{-3/4}e^{-(1/2)r^2/d^2}w^1(0)$$

wich yield for the second term circular flow of energy! :

$$\mathbf{G}=\frac{\hbar}{4} (\frac{1}{\pi d^2})^{-3/2} \frac{e^{-r^2/d^2}}d^2 (-2y \hat{x} +2x \hat{y})$$

Such a circulating flow will give rise to angular momentum, spin! Then he proceeds showing that the second term indeed comes down to the expectation value of the QM spin operator.

6. Jan 23, 2005

### dextercioby

That's weird...I don't like this version/treatment of Dirac field.I don't claim it to be incorrect,but the fact that he pulls out of the hat the "circulating flow of energy in the rest frame of the electron" gives me the creeps.
Besides,where does that time derivative come from ??

The way i know it,the momentum density for the Dirac field is:
$$\vec{P}=\frac{1}{2}[\bar{\Psi}_{\alpha}i(\gamma_{0})^{\alpha} \ _{\beta} \nabla \Psi^{\beta} -\nabla\bar{\Psi}_{\alpha}i(\gamma_{0})^{\alpha} \ _{\beta}\Psi^{\beta}]$$

which doesn't have any time derivative.I find kind of awkward his notation involving "alpha" matrices.Alpha matrices don't come into QFT since 1930... :yuck: :yuck:

And u said the guy wrote it in 1986... :yuck: Who taught him QFT,Dirac??

Daniel.

Last edited: Jan 23, 2005
7. Jan 23, 2005

### da_willem

About the first equation with the time derivative, a footnote says:

And for the alpha matrices: $$\sigma_1=-i \alpha_2 \alpha_3 , \sigma_2=-i\alpha_3 \alpha_1 , \sigma_3=-i\alpha_1 \alpha_2$$

Hew does a similar thing thing for the magnetic moment. He separates the standard electric current density of the Dirac field into two parts by means of the Klein-Gordon decomposition formula. This yields again two terms. One is associated with the translational motion of the electron and the other is nonzero even in the rest frame of the electron (these are separately conserved he says) wich constitutes the magnetic moment.

Is it this seperation that gives you the creeps? If it yields an intuitive appealing picture, in terms of a energy or current flow wich explains the spin and magnetic moment of the electron and is legitimate, it's fine by me!

8. Jan 23, 2005

### marlon

How is this statement made (obout the circulating flow of energy)...What energy are we talking about ??? How is this energy represented and where does it come from...I am sorry but i am lost here... i don't follow...

How do you justify that this last term is even energy???

marlon

9. Jan 23, 2005

### marlon

and what the hell is the wave field of an electron...i suppose it is just the fermionic matterfield that describes the electron right ???

marlon

10. Jan 23, 2005

### dextercioby

I resent the notation altogether.I said i do not contest the result (the guys form AJP would have been some idiots to accept into publishing),yet the fact that in 1986 the notation used in the '30 is still "available" and "usable",that is unacceptable...

Bjorken & Drell wrote 2 good books...But they're OLD AND OUTDATED...

I been taught QFT following Bailin & Love's book "Introduction to gauge field theory" fropm 1993...

And i'm a great supporter of path integral approach into QFT...

Daniel.

P.S.And yes,how the f*** is that term energy???

11. Jan 23, 2005

### dextercioby

And is my eyesight misleading me,or those fermionic fields HAVE 2 components???????????????????????????Why the hell does he use Weyl spinors??????????Hasn't he heard in 1986 of Dirac spinors??????????

Daniel.

12. Jan 23, 2005

### marlon

I am a bit sceptical here...I am not saying it is wrong, i am just saying we need more info. Here are my questions :

1) why is this term energy
2) how is it linked to spin or how do you introduce spin
3) how about the connection between spin, L and the magnetization ?
4) Is the Einstein de Haas experiment still respected. this somewhat relates to the previous question.
5) What are the dynamics of this rotational flow of energy???

I find the content a bit strange because it tries to explain spin in terms of QFT. But remember that the principles of QM (of which spin is one that has been proved both by theory and experiment) are incorporated into the very base of QFT, since the latter is the unification of QM and special relativity...How about that ???

marlon
just wondering...
is this a peer reviewed article ???

13. Jan 23, 2005

### da_willem

Some quotes from the article:

"He (Belinfante) established that this picture of the spin (circulating flow of energy in the wave field) is valid not only for electrons , but also for photons, vector mesons, and gravitons."

He complains a while about the idea not getting the attention it deserves . Mainly because neither Belinfante nor Gordon loudly proclaimed a new physical explanation of spin. Although it is completely consistent with the standard interpretation of QM.

"Belinfante showed that by a sitible choice of the term $\partial_{\alpha} U^{\mu \nu \alpha}$, it is always possible to construct a symmetrized energy-momentum tensor. ($T^{\mu \nu} = T^{\nu \mu}$)"

This is essential in the calculations. He takes for granted this is the correct one as this is demanded by GR.

I Quote (directly under the equation):

he first does some classical calculations on Em radiation as an introduction. He also has a drawing of the circular lines of the 'energy flow in the spinor wavepacket'.

I have no idea. He talks about the wave field, and the Dirac field...

Hre concludes by saying that 'spin is intrinsic but not internal'!

14. Jan 23, 2005

### dextercioby

Wiat a minute.QFT is one thing,coupling classical fields to gravitational field is another.In the context of QFT is irrelevant whether the energy-momentum 4-tensor is symmetric or not.Think of the Dirac Lagrangian density.It has 2 possible forms:the unsymmetrized (nonreal wrt the involution of the Grassmann algebra of fields) leads to an unsymmetrized energy momentum 4-tensor.Calculations and quantizing Dirac field are not affected by this choise of Lagrangian.However,this Lagr.Density (just for the reason presented before) is not suitable for coupling to the gravitational field (by the famous vierbein-spin connection procedure).Using the symmetrized Lagr.density in QFT couldn't possibly change anything...I cannot conceive that the fact that $T^{\mu} \ _{\nu}$ is not symmetric could affect physical relevence of phenomena not involving the gravity field...

For the photon field,it's the same discussion.To describe (electrically) charged BH,u need to solve the Einstein-Maxwell equations which involve a symmetrized energy-momentum 4-tensor...

Daniel.

15. Jan 23, 2005

### da_willem

As I didn't understand half of the article I can't help you with all of these questions, and can only advise you to read the article. And when you have inform me about it's validity .

About the dynamics of the energy flow. He discusses the analogous case for an EM wave. Using a circularly polarized plane wave vector potential he shows some of the properties: 'circular flow lines represent the time-avaraged energy flow, or the momentum density, in a circularly polarized electromagnetic wavepacket'.

He mentiones the Einstein-de Haas effect in his conclusions:
And I wouldn't know if the article was peer-reviewed. Can you see this somewhere? It was received 5 februari 1984; accepted for publication 1 May 1985.

16. Apr 18, 2006

### arivero

Last edited by a moderator: May 2, 2017
17. Jun 28, 2010

### centry57

can anyone send the paper “F.J. Belinfante，On the spin angular momentum of mesons” to my Email：centry57@gmail.com
and HC Ohanian‘s “What is spin?”

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