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

[zwicky]

Hi everybody!

Is there someone that can help me to prove that

<br /> \omega^2E-k^2E=-ip_0k\times E+i\omega p\times E<br />

imply that the dispersion relation is

<br /> (k^\mu k_\mu)^2+(k^\mu k_\mu)(p^\nu p_\nu)=(k^\mu p_\mu)^2<br />

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

<br /> \begin{pmatrix}<br /> 0 &amp; v3 &amp; -v2 \\<br /> -v3 &amp; 0 &amp; v1 \\<br /> v2 &amp; -v1 &amp; 0<br /> \end{pmatrix}<br />​

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

 

[zwicky]

Muiti obrigado Dave!

Best from Brazil!

Zwicky