# Is Dirac's equation valid for point charges?

1. Jan 24, 2005

### what_are_electrons

I've read that Dirac's equation for explaining spin does not hold if electrons are defined as true point charges. Is that correct?

2. Jan 24, 2005

### dextercioby

WHATTTTTTTTTTTTTTTTTTTT??????????????Hell,no.What said/wrote that was a big LIAR.

Daniel.

3. Jan 24, 2005

### Tom Mattson

Staff Emeritus
Dirac's equation is written for point charges, and there's nothing ill-defined about it.

What's the source of this statement?

4. Jan 24, 2005

Staff Emeritus
Maybe he's worrying about self-action at zero distance?

5. Jan 24, 2005

### marlon

Perhaps he is wondering about the fact that in QFT, the position of some particle is not exact and therefore you cannot point out exactly where the particle is. It is situated inside a little cloud, if you will, expressing the region where you have a certain propability to find such particle...Perhaps he is just referring to orbitals ???

marlon

6. Jan 24, 2005

### dextercioby

Marlon,don't speculate & perform a bad tasting mixing of QM & QFT...
Let us allow the OP to come up with the source of missinformation...

Daniel.

7. Jan 24, 2005

### marlon

Err, how can you mix QM with QFT, dexter ???

marlon

8. Jan 24, 2005

### dextercioby

U did that...Not me...If you're claiming that "orbitals" and QFT live happily ever after,then you're wrong...

Daniel.

9. Jan 24, 2005

### marlon

So are you saying that the concept of orbitals does not exist in QFT ???

I hope not...

You do know that all the principles of QM are incorporated into QFT. I did not invent this (unfortunately), many other more "ingenious" minds did...

marlon

ps : and are you saying that uncertainty does not exist in QFT ???

Last edited: Jan 24, 2005
10. Jan 24, 2005

### dextercioby

UNCERTAINTY EXISTS IN QFT,ORBITALS NOT.That's (the second part,really) what i meant by mixing things...Concepts...

Daniel.

11. Jan 24, 2005

### marlon

:surprised
Ok, then if orbitals do not exist in QFT, this implies that the spherical harmonics also don't exist in QFT. Are you really serious now ???

marlon

Let's not fall into a discussion on personal interpretations...stick to the physics...we might as well be discussing your favorite actor

Last edited: Jan 24, 2005
12. Jan 24, 2005

### Hans de Vries

The Dirac's equation hasn't anything to do with this.

Observable spin effects like the spin angular momentum and the magnetic
momentum can not come from a spinning point charge. A spinning charge
with the classical electron radius (2.817940285 10-15 meter) is also much to
small to support the measured values.

To get for instance the measured magnetic moment at this radius means
that the charge needs to "rotate" with a speed much higher than the speed
of light.

If we use classical equations to get an indication of the required size then
it turns out that one gets more realistic values ($\leq$ c) if you assume a "radius"
of circa 137 times the classical electron radius: ~ 3.8616 10-13 meter, where
1/137.03599911 is the finestructure constant.

You get this radius if you apply Heisenbergs Uncertainty Principle:
Take the electron's rest energy of 0.5109892 MeV as the uncertainty in
energy. This corresponds with an uncertainty in momentum of 0.5109892
MeV/c.

Now if you use the equation: $\Delta x \Delta p \ = \ \hbar$ then you'll get the radius above
of ~ 3.8616 10-13 meter as the uncertainty in position. The corresponding
rotation frequency becomes $\approx$ the 1.235 1020 Hz: The rest frequency of
the electron.

In a similar way one can obtain the Bohr Radius and the ground state
frequency of the hydrogen atom from the orbital angular momentum and
the 13.6 eV ionisation energy of the ground state.

So to conclude:

Spin and orbital angular momentum as well as the spin and orbital
magnetic momentum are in "the wave" and not in "the particle"

Regards, Hans

13. Jan 24, 2005

### dextercioby

It's just that.Personal interpretations... :tongue2: To me,"orbital(s)" is a term not pertaining to QM,but to chemistry and their idea of "doing" QM.It's very deluding.It derives from "orbit" and hence the old theory of Bohr.No respectable physicst teaching QM would not use this word...I think chemistry teachers have created a "passion" for it...

If u saw a QM course mentioning the term "orbital",then the author was not a physicist...Was a moron...

Daniel.

14. Jan 24, 2005

### Tom Mattson

Staff Emeritus
Since the original poster hasn't explained the remark, it's not really possible to say. Dirac's equation does indeed describe the mechanics of spin-1/2 point particles. If we take the OP at face value, it says that there is some problem with that description. What you have explained is why classical spinning cannot be the same as quantum mechanical spin.

15. Jan 24, 2005

### marlon

I am sorry but what you say is wrong. Orbitals are just the squared spherical harmonics. They are nothing else then a probability...But why orbitals, well, because these spherical harmonics are the eigenfunctions of the equations for L² and L_z...They only depend on the two angular degrees of freedom. This combined with the name of L² and L_z, you have the explanation for the word ORBITAL... Nothing is orbiting here

marlon

16. Jan 24, 2005

### Hans de Vries

Classical spinning isn't quantum mechanical spin OK. But the gap is not
that big. Dirac included Pauli spinors in his equation to account for the
spin 1/2 behavior. It was the (relativistic) classically derived Thomas
Factor of 2 which convinced Pauli of the existence of the electron spin.
Tomonaga showed that this approach predicts other angular interactions
and energy levels in the atom as well.

The conclusion remains the same: spin angular and magnetic moment
can not stem from an arbitrary small sized particle. It must come from
the quantum field.

Regards, Hans

17. Jan 24, 2005

### Tom Mattson

Staff Emeritus
Yes it is that big. It's huge, in fact. Take a spin-1/2 particle and rotate it through 2π radians. You'll find that you do not arrive at the initial state, but rather the negative of the initial state. Something is very much at odds with classical angular momentum here.

Dirac included Dirac spinors (4 components) in his equation. They aren't the same as Pauli spinors, which have 2 components. And I don't think it's accurate to say that he included them to account for spin-1/2 particles. The mapping of Dirac's theory to spin-1/2 particles emerges naturally from the linearization process on the relativistic Hamiltonian, if we take the gamma matrices to be of the lowest possible dimension.

How does this address the fitness of the Dirac equation with point charges?

In QFT the particles are excitations of the quantum field, and they carry angular momentum (including spin). So it's not surprising that angular momentum must come from the quantum field. But we weren't talking about where angular momentum comes from, we were talking about an alleged problem with the Dirac equation.

18. Jan 24, 2005

### what_are_electrons

I've read that Dirac's equation for explaining spin does not hold if electrons are defined as true point charges. Is that correct?

Physics Prof. I.A.Sellin (Univ of Tenn.) wrote a section (p 17-32) titled "Atomic Structure and Spectra" in a 1988 book called " Spectroscopy Source Book" published by McGraw Hill. On p23 he wrote: "A spinning electron can crudely be pictured as a spinning ball of charge, imitating a circulating electric current (though Dirac electron theory assumes no finite electron radius - classical pictures fail.) This circulating current gives rise to a magnetic field distribution very similar to that of a small bar magnet, with north and south magnetic poles symmetrically distributed along the spin axis above and below the spin equator."

On page 31 he goes on to write: "Since in Maxwell's theory, only accelerating charges radiate, atomic electrons in stationary states do not immediately collpase into the nucleus. Their freedom from this catastrophe can be interpreted as a proof that electrons may NOT be viewed as pointlike planetary objects orbiting atomic nuclei."

"There is still no generally accepted explanation for why electrons do NOT explode under the tremendous Coulomb repulsion forces in an object of small size."

Robert Gourian worte a book in 1967 titled "Particles and Accelerators" (World Univ Libr). On p.27 of this book he wrote: "The electron cannot be considered as a geometric point wich is a perfect conception of zero, for its electric field would then be infinite. But if it is slightly extended the electric charge tends to repel itself, and would lead to dissolution unless a contrary force opposed it in order to maintain cohesion; but this 'self-force' could only accumulate an infinte self-energy, giving to the electron infinite mass!"

In the same book on p. 112 in a section titled "Parity and its 'non-conservation', Gourian wrote: "The free electron only exists in a non-symmetrical form."

In a book written by Malcolm MacGregor (recently retired from LLNL) called "The Enigmatic Electron" published by Kluwer in 1992, he quoted Gottfried and Weisskopf book called "Concepts of Particle Physics Vol 1": "At one time it (the Dirac Equation) was thought to 'explain' the spin s=1/2 of the electron, but we now know that this is not so. Equations of the Dirac type can be constructed for any S. At this time we have no understanding of the remarkable fact that the fundamental fermions of particle physics (electrons, neutrinos, quarks, etc.) all have spin 1/2."

MacGregor then wrote: " Although the electric charge e of the electron, as viewed in scattering experiments, seems to be point-like, its manifestation in atomic bound states is NOT point-like. The Lamb shift reveals that the electric charge is smeared out over a region of space which is comparable to the electron Compton radius R(c) as we discuss in Chapter 7. This smearing out is attributed to two characteristic phenomena of QED: vacuum polarization, which broadens the electric field of the charge; and zitterbewegung, which broadens the spatial location of the charge. Thus we have an effective electron bound-state QED charge radius

R(QED) approx = R(C). "

These quotes are the reason that I wrote the question: I've read that Dirac's equation for explaining spin does not hold if electrons are defined as true point charges. Is that correct?

Last edited by a moderator: Jan 24, 2005
19. Jan 24, 2005

### Kane O'Donnell

What the passages you've quoted seem to be alluding to is the fact that you can't consider an electron as a *classical* point particle.

Kane

20. Jan 24, 2005

### what_are_electrons

Yes, quite correct. If my reading and comprehesion are correct, then Dirac first developed his equation(s) for an electron that was understood to be a point charge, be it a classical point charge or a quantum point charge.

21. Jan 25, 2005

### Kane O'Donnell

No, Dirac didn't do that. What he did was try and come up with a relativistic wave equation for a quantum mechanical point charge. The fact that QM changes one's ideas of position and so forth were getting to be fairly well known back then, so you would expect that Dirac knew that classical point charges were not going to come out of his equations (especially since the NR limit needed to be something like the Schrodinger equation).

Kane

22. Jan 25, 2005

### Hans de Vries

Quantum mechanical spin and orbital angular momentum (both from
spin ½ and spin 1 particles) can be transferred to macroscopic
particles where it behaves like ordinary angular momentum (Einstein,
de Haas, Dublin). You are talking about a specific property of spin
½ particles.

Dirac’s original 4x4 matrices are build from 2x2 Pauli spinors on the
diagonal Dirac already mentions Pauli’s spin theory in the introduction
of his paper. Pauli's spin theory was a 2x2 matrix version of the
Shroedinger equation using his spinors.

Pauli spinors obey Heisenberg’s matrix mechanics angular momentum
commutation rules:

sxsy-sysx = isz, sysz-szsy = isx, szsx-sxsz = isy

and in particular for spin ½ particles:

|s|2 = ½ ( ½ + 1 ) = ¾

with: $s_x \ = \ \frac{1}{2}\sigma_x, \ \ \ \ \ s_y \ = \ \frac{1}{2}\sigma_y, \ \ \ \ \ s_z \ = \ \frac{1}{2}\sigma_z$

equation using his spinors. Historically it's hard to say in which order
Dirac arrived at his equation. Did he linearized Klein Gorden while playing
with Pauli's work or not? while trying to do with Klein Gorden's equation
what Pauli did with Schroedinger's equation to include spin....

Two of the requirements for linearization:

$$\gamma_m^2 = 1, \ \ \ \gamma_m \gamma_n + \gamma_n \gamma_m = 0$$

are also fulfilled by Pauli's Spinors:

$$\sigma_m^2 = 1, \ \ \ \sigma_m \sigma_n + \sigma_n \sigma_m = 0$$

Dirac must have seen this immediately, probably to find out that he
needed to combine them in a 4x4 representation to get things right.
(There are only 3 spinors and you need 4 gamma's. That's why a
4x4 representation is needed)

Dirac’s equation doesn’t have any reference to any presumed size of the
electron. It’s therefor incorrect to say that it doesn’t work for a certain size.

The background of the presumed problem was not given so I could only
guess where the misunderstanding came from. It may as well come from
the first of the two quotes from the paper:

"The question remains as to why Nature should have chosen this
particular for the electron instead of being satisfied with the point charge"

However, in a second quote he uses the word "point charge" again but
now in combination with his equation:

"It appears that the simplest Hamiltonian for a point-charge electron
satisfying the requirements of relativity and the general transformation
theory leads to an explanation of all duplexity phenomena without further
assumption."

Regards, Hans

Last edited: Jan 25, 2005
23. Jan 25, 2005

### what_are_electrons

I need to ask 2 more questions. (1) Do bound electrons inside a normal atom (excluding Z=1) have a finite charge based size on the order of 10(-13) cm? (2) What is the size of charge of an electron after it is free from an atom?

24. Jan 25, 2005

### dextercioby

We don't really know the SPATIAL extent of the electrons and we could never find out,at least as long as QM and its principle of uncertainty is among us...

The same kinda question gets the same answer...

Daniel.

25. Jan 25, 2005

### Hans de Vries

I should first say that the only "official size" here is probably the size of the
electron's wave-function cloud in the atom. (in the order of 10-10 meter)
The other sizes are "interesting to know" but should be viewed in the light of
the presumptions to derive them.

Regards, Hans

Last edited: Jan 25, 2005