Prove Chern-Simons Dispersion Relation | Carrol, Field, Jackiw

[zwicky]

Hi everybody!

Is there someone that can help me to prove that

$$\omega^2E-k^2E=-ip_0k\times E+i\omega p\times E$$

imply that the dispersion relation is

$$(k^\mu k_\mu)^2+(k^\mu k_\mu)(p^\nu p_\nu)=(k^\mu p_\mu)^2$$

Thanks in advance ;)

p.d. The reference for this formula is the paper of Carrol, Field, Jackiw, Limits on a Lorentz and parity violating modification of electrodynamics

 

[schieghoven]

The right hand side is a linear operator on the 3-component vector E, so it can be represented by
a 3-by-3 matrix, and what you really need to do is find the eigenvalues of this matrix. It's a matrix of the form

$$\begin{pmatrix} 0 & v3 & -v2 \\ -v3 & 0 & v1 \\ v2 & -v1 & 0 \end{pmatrix}$$​

In general, the three eigenvalues of this matrix are i|v|, 0 and -i|v|. In this case [itex] v = -ip_0k + i\omega p [/tex]. That will get you to the result quoted.

Best

Dave

 

[zwicky]

Muiti obrigado Dave!

Best from Brazil!

Zwicky