# A Pauli and a Dirac electron, momentum density and spin.

1. Jul 8, 2012

### Spinnor

In the article, "What is spin", by Hans C. Ohanian we are shown how to take the wave-function for a Dirac electron with spin up and localized in space and then determine the momentum density in the Dirac field. The momentum density divides into two parts, a part that depends on the motion of the electron and another part that "is associated with circulating flow of energy in the rest frame of the electron". For a localized packet we learn that a circular "flow of energy" can be interpreted as electron spin.

Is the momentum density of a localized up Pauli electron similar or the same as the momentum density of a localized up Dirac electron? Do we still get a circular "flow of energy" with the Pauli electron?

A snippet of the paper follows, thanks for any help!

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2. Jul 8, 2012

3. Jul 8, 2012

### Bill_K

A Pauli electron and a Dirac electron walk into a bar...

It's interesting to see this paper by O'Hanian, whose main claim to fame I believe is a book on general relativity. As I guess you know, Am J Phys's purpose is to explain non-cutting-edge topics to undergrads, and here O'Hanian is recounting a 1939 result by Belinfante which tied the spin density for the Dirac equation to the momentum density.

I'd say the first thing to do is figure out if the interpretation he gives is valid. Our understanding of the variables in the Dirac equation has changed since 1939. Especially, we now know that x is not really the position operator. The replacement for it, the Newton-Wigner position operator X, was not introduced until 10 years after this paper, in 1949. Likewise there is a replacement for the spin density, ΣM, which is called the "mean spin".

4. Jul 8, 2012

### Hans de Vries

The same is true for the Pauli equation. The circular convection
current is the curl of the magnetization. The difference is that the
Dirac equation is the relativistically correct one: The magnetization
is part of the magnetization-polarization tensor $\bar{\psi}\sigma^{\mu\nu}\psi$.

Note that you can view this from either a spin-density or a magneti-
zation point of view. The first leads to the particles spin and the
second to the particles magnetic moment. Hans Ohanian addresses
the latter in section IV.

You can find the same result also in other treatments of the Gordon
Decomposition. For instance Sakurai's Advanced Quantum Mechanics,
paragraph 3-5 page 108/109.

I mention this in the opening chapter of my book in section 1.9 here:
http://physics-quest.org/Book_Chapter_EM_basic.pdf

Hans.

5. Jul 8, 2012

### Spinnor

Thanks to both Bill and Hans for the fast replies! A follow up if I might, if I were able to graph the wave-function of a localized spin-up Pauli electron in space-time should this "circular energy flow" be evident by just looking at the graph?

Thanks for all help!