Can quantum field theory explain the g-factor in the Dirac equation?

[kof9595995]

To quote Weinberg Vol1, Pg 14 :

(iii) One of the great successes of the Dirac theory was its correct
prediction of the magnetic moment of the electron. This was particularly
striking, as the magnetic moment (1.1.8) is twice as large as would be
expected for the orbital motion of a charged point particle with angular
momentum $$\frac{\hbar }{2}$$; this factor of 2 had remained mysterious until Dirac's theory. However, there is really nothing in Dirac's line of argument that
leads unequivocally to this particular value for the magnetic moment. At
the point where we brought electric and magnetic fields into the wave
equation (1.1.23), we could just as well have added a 'Pauli term'
$$\kappa {\alpha _4}[{\gamma ^\mu },{\gamma ^\nu }]\psi {F_{\mu \nu }}$$
with arbitrary coefficient к. (Here F_uv is the usual electromagnetic field
strength tensor) This term could be obtained by first adding a term to the free-field equations proportional to $$[{\gamma ^\mu },{\gamma ^\nu }]{\partial _\mu }{\partial _\nu }\psi$$which of course equals zero, and then making
the substitutions (1.1.22) as before. A more modern approach would be
simply to remark that the term (1.1.32) is consistent with all accepted
invariance principles, including Lorentz invariance and gauge invariance,
and so there is no reason why such a term should not be included in the
field equations. (See Section 12.3.) This term would give an additional
contribution proportional to к to the magnetic moment of the electron, so
apart from the possible demand for a purely formal simplicity, there was
no reason to expect any particular value for the magnetic moment of the
electron in Dirac's theory.

And immediately he said:

As we shall see in this book, these problems were all eventually to be
solved (or at least clarified) through the development of quantum field
theory.

So to speak, Dirac equation alone cannot determine g-factor uniquely, but quantum field theory can? How?

 

[dextercioby]

Yes, the term Weinberg is mentioning which would change the magnetic moment of the electron (or g-factor), when added to the interacting QED lagrangian, prevents the new theory from being renormalizable.

Si it is QFT which clears all the possible aspects of the Dirac equation and its implications.

 

[torquil]

So to speak, Dirac equation alone cannot determine g-factor uniquely, but quantum field theory can? How?

Well, from your quote it would seem that the ultimate answer is probably somewhere later in very same book from which you are quoting

When I look up "Magnetic moment" in the index it leads me to pages 454-457, where the result seems to be derived.

 

[kof9595995]

I see, thank you.

 

[dr_uri]

kof. Neither the Dirac equation, nor QFT are required to invoke either g=2, or the two component spin of the electron. Read the book by Greiner, "Quantum Mechanics, an introduction" chapter 13 titled "A nonrelativistic wave equation with spin". Writing a non-relativistic wave equation linear in d/dx, d/dy, d/dz will give g=2 and also the 2 component spin of the electron.

 

[kof9595995]

Yes, I've read that before,but Weinberg's argument still applies for Greiner's derivation. All Weinberg has done is to insert a 0 term in the free equation, which becomes non-zero after minimal coupling.

 

[QuantumClue]

To quote Weinberg Vol1, Pg 14 :

And immediately he said:


So to speak, Dirac equation alone cannot determine g-factor uniquely, but quantum field theory can? How?

The g-factor has often muddled my understanding of the Dirac Equation. I don't understand why it is mysterious... I mean this in the sense that I personally have a solution which is mathematical in respects to the energy of a particle. I will write this up, but I have some questions first.

1) Does the g-factor have any relationships to the instrinsic energy of the particle?

The reason why I see no mystery is that if you take the net magnetization $$\vec{M}$$ as the sum of individual magnetic moments,

$$\vec{M}= \sum^{N}_{i=1} \vec{\mu}_i$$

The sum of the magnetic moments have a magnitude which is proportional to an energy $$\frac{e \hbar}{2Mc}$$. Any contribution of energy supplied to the particle through possibly a Langrangian on the particle taken to higher orders, or possibly a perturbation of energy by some external field, then the magnitude of energy $$\frac{e \hbar}{2Mc}$$ could be seen as giving rise to a magnetic moment.

You could work out further how to make the energy as a shift in the magnetization rates by a factor of $$sin (\theta + \pi) = -\sin \theta$$.The g-factor then would arise as a perturbation by shifts of $$\pi$$ as a plane of energy on the system.

This is obviously not what mainstream believes - this post is proposed as a question rather than an empiracle fact (just before I get penalised for it).

 

[dextercioby]

kof. Neither the Dirac equation, nor QFT are required to invoke either g=2, or the two component spin of the electron. Read the book by Greiner, "Quantum Mechanics, an introduction" chapter 13 titled "A nonrelativistic wave equation with spin". Writing a non-relativistic wave equation linear in d/dx, d/dy, d/dz will give g=2 and also the 2 component spin of the electron.

Ah, yes, the work by Levy-Leblond. Great reference, indeed.

 

[kof9595995]

The g-factor has often muddled my understanding of the Dirac Equation. I don't understand why it is mysterious... I mean this in the sense that I personally have a solution which is mathematical in respects to the energy of a particle. I will write this up, but I have some questions first.

1) Does the g-factor have any relationships to the instrinsic energy of the particle?

The reason why I see no mystery is that if you take the net magnetization $$\vec{M}$$ as the sum of individual magnetic moments,

$$\vec{M}= \sum^{N}_{i=1} \vec{\mu}_i$$

The sum of the magnetic moments have a magnitude which is proportional to an energy $$\frac{e \hbar}{2Mc}$$. Any contribution of energy supplied to the particle through possibly a Langrangian on the particle taken to higher orders, or possibly a perturbation of energy by some external field, then the magnitude of energy $$\frac{e \hbar}{2Mc}$$ could be seen as giving rise to a magnetic moment.

You could work out further how to make the energy as a shift in the magnetization rates by a factor of $$sin (\theta + \pi) = -\sin \theta$$.The g-factor then would arise as a perturbation by shifts of $$\pi$$ as a plane of energy on the system.

Hmm, although I don't really understand what you're trying to say, but I think I can safely say g-factor has little bearing with energy. If you read the reference that dr_uri gives, you'll realize the Clifford algebra, which finally gives Pauli equation, is not induced by dispersion relation of energy, but by the attempt of writing both t and x derivatives in 1st order.

 

[QuantumClue]

Hmm, although I don't really understand what you're trying to say, but I think I can safely say g-factor has little bearing with energy. If you read the reference that dr_uri gives, you'll realize the Clifford algebra, which finally gives Pauli equation, is not induced by dispersion relation of energy, but by the attempt of writing both t and x derivatives in 1st order.

Right ok. However, I don't know if it can be fully justified that the g-factor has little bearing of energy on the particle. Keep in mind that the g-factor is related to the magnetic moment by our universal equation $$\mu= g \mu_B S/ \hbar$$ and the magnetic moment experiences torque which is related to energy as $$\Delta E = -\mu \cdot B$$. In fact there are equations which describe the magnetic interaction energy $$\Delta E = g \mu_B M B$$ so is there really little influence?

edit/ sorry, latext \dot $$\Delta E = -\mu \cdot B$$.

 

[QuantumClue]

Also, as an undergrad, I do realize the non-importance of non-relativistic equations. They are not useful in that sense. So why would the Pauli equation be absolute in the sense that dr-dru gave it? If I see anything, I see a flaw, because the QFT needs relativistic unification.

 

[fzero]

Yes, the term Weinberg is mentioning which would change the magnetic moment of the electron (or g-factor), when added to the interacting QED lagrangian, prevents the new theory from being renormalizable.

Si it is QFT which clears all the possible aspects of the Dirac equation and its implications.

The Pauli term does not prevent the theory from being renormalizable. Weinberg's point is that we don't add the Pauli term with an arbitrary coefficient. Rather we start with the simplest terms in the QED Lagrangian (dimension 4) and the Pauli term is generated in the quantum theory with a coefficient that is suppressed by a factor of the fine-structure constant. So the Pauli term contributes a fairly small correction to g=2.

 

[kof9595995]

Right ok. However, I don't know if it can be fully justified that the g-factor has little bearing of energy on the particle. Keep in mind that the g-factor is related to the magnetic moment by our universal equation $$\mu= g \mu_B S/ \hbar$$ and the magnetic moment experiences torque which is related to energy as $$\Delta E = -\mu \cdot B$$. In fact there are equations which describe the magnetic interaction energy $$\Delta E = g \mu_B M B$$ so is there really little influence?

edit/ sorry, latext \dot $$\Delta E = -\mu \cdot B$$.

Because you said "intrinsic energy", so what I was trying to point put is that g=2 does not comes from nonrelativistic or relativistic energy dispersion relation themselves, but from the attempt to write time and space derivatives in 1st order.
However, after I saw Weiberg's argument, it becomes fuzzy to me how to really make sense of derivation of g=2.

 

[kof9595995]

The Pauli term does not prevent the theory from being renormalizable. Weinberg's point is that we don't add the Pauli term with an arbitrary coefficient. Rather we start with the simplest terms in the QED Lagrangian (dimension 4) and the Pauli term is generated in the quantum theory with a coefficient that is suppressed by a factor of the fine-structure constant. So the Pauli term contributes a fairly small correction to g=2.

Emm..it seems you are making more sense, I'll return to this when I get there of my QFT study.

 

[QuantumClue]

Because you said "intrinsic energy", so what I was trying to point put is that g=2 does not comes from nonrelativistic or relativistic energy dispersion relation themselves, but from the attempt to write time and space derivatives in 1st order.
However, after I saw Weiberg's argument, it becomes fuzzy to me how to really make sense of derivation of g=2.

ok thank you.