# Help: Gordon decompositon of the current

1. May 14, 2012

### elmerx25

Hello:

A spinless electron can interact with $A^\mu$ only via its charge; the coupling is proportional to $(p_{f} + p_{i})^{\mu}$. An electron with spin, on the other hand, can also interact with the magnetic field via its magnetic moment. This coupling involves the factor $i\sigma^{\mu\nu}(p_{f} - p_{i})$The relation between the Dirac current and the Klein-Gordon current can be studied as
follows: Define the antisymmetric sigma tensor as:

$$i\sigma^{\mu\nu} = \frac{i}{2} (\gamma^{\mu}\gamma^{\nu} - \gamma^{\nu}\gamma^{\mu})$$

And the Gordon decomposition of the Dirac current can be made:

$$\bar u_{f}\gamma^{\mu}u_{i} = \frac{1}{2m} \bar u_{f} [(p_{f} + p_{i})^{\mu} + i\sigma^{\mu\nu} (p_{f} - p_{i})_{\nu} ] u_{i}$$

To identify the magnetic moment interaction $(-{\mu} . B)$ it suffices to show that:

$$\int[-\frac{e}{2m} \bar \psi_{f} i\sigma_{\mu\nu} (p_{f} - p_{i})^{\nu}\psi_{i}] A^{\mu} d^{3}x = \int\psi^{f*}_{A}(\frac{e}{2m}{\sigma} . B) \psi^{i}_{A} d^{3}x$$

Can someone please tell me how I can desmostrate this ecuation?
Thanks.

P.S.: Exercise 6.2 of "Quarks and Leptons. An Introductory Course in Modern Particle Physics - F.Halzem,A.Martin"

2. May 16, 2012

### Bill_K

elmerx25, All three of your posts have been asking for solutions to homework problems. You need to take a look at the Forum Rules. In particular where it says, "If you are seeking help with a homework/coursework/textbook related issue please post your problem in the appropriate forum in our Homework & Coursework Questions area."

3. May 12, 2014

### elmerx25

How can I send this post to the homework/coursework/textbook area?
Thanks.

4. May 12, 2014

### Staff: Mentor

What did you find out so far?

5. May 12, 2014

### elmerx25

Hello:

I have had a sabbatical year without physics. Now I try to continue with my study.