What is QM Spin? Exploring Classical Origins

[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?":

According to the prevailing belief, the spin of the electron or of some other particle is a mysterious internal angular momentum for which no concrete physical picture is available, and for which there is no classical analog. However, on the basis of an old calculation by Belinfante [Physica 6, 887 (1939)], it can be shown that the spin may be regarded as an angular momentum generated by a circulating flow of energy in the wave field of the electron. Likewise, the magnetic moment may be regarded as generated by a circulating flow of charge in the wave field. This provides an intuitively appealing picture and establishes that neither the spin nor the magnetic moment are ``internal''—they are not associated with the internal structure of the electron, but rather with the structure of its wave field. Furthermore, a comparison between calculations of angular momentum in the Dirac and electromagnetic fields shows that the spin of the electron is entirely analogous to the angular momentum carried by a classical circularly polarized wave.

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

 

[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...

 

[da_willem]

But isn't it a bit weird that NO QUANTUM MECHANICS TREATISE speaks about "circulation of energy"?

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

 

[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.

 

[da_willem]

I'm sorry,i don't believe it.You have to come up with the calculations.

Well he starts from the symetrized energy-momentum tensor which 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.

 

[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...

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

Daniel.

 

[da_willem]

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 ??

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...

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

Daniel.

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

The notation for spinors employed here is that of J.D. Bjrken and S.D. Drell. Relativistic Quantum Mechanics (McGraw-Hill, NY 1964). The notation of Wentzel is slightly different.

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) which constitutes the magnetic moment.

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

 

[marlon]

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

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."

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

 

[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

 

[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?

 

[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.

 

[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 ?

 

[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.

How is this statement made (obout the circulating flow of energy)...

I Quote (directly under the equation):

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

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'.

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

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'!

 

[dextercioby]

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.

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.

 

[da_willem]

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 ?

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:

A circularly polarized light wave is an example of a system in which the classical macroscopic spin angular momentum arises from the addition of large number of quantum spins. Such a classical limit is also possible for electrons, but we must make the precaution of placing the electrons in different orbital states whenever we place them in the same spin state. The Einstein-de Haas effect and the magnetisation found in permanent magnets involve classical limits brought about by large number of electron pins and magnetic moments

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.

 

[arivero]

The Belinfante articles are online but access restricted, he published both in Prola, er PhysRev, and in Physica... I doubt most people quoting him has read him, just check the abstract:

F.J. Belinfante
On the spin angular momentum of mesons

http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6X42-4CB74W8-14&_user=987788&_handle=V-WA-A-W-CY-MsSAYVW-UUA-U-AAVDEYWZCE-AAVCCZBVCE-YUYUCVDBE-CY-U&_fmt=summary&_coverDate=12%2F31%2F1939&_rdoc=32&_orig=browse&_srch=%23toc%237314%231939%23999939992%23500330!&_cdi=7314&view=c&_acct=C000049881&_version=1&_urlVersion=0&_userid=987788&md5=072b74c101f499e981bdcef6f4d9d7ae
http://dx.doi.org/10.1016/S0031-8914(39)90090-X

 

[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?”