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Chern-Simons Th

  1. Dec 6, 2008 #1
    Hi everybody!!!

    Is there someone that can help me to prove that

    [tex]
    \omega^2E-k^2E=-ip_0k\times E+i\omega p\times E
    [/tex]

    imply that the dispersion relation is

    [tex]
    (k^\mu k_\mu)^2+(k^\mu k_\mu)(p^\nu p_\nu)=(k^\mu p_\mu)^2
    [/tex]

    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
     
  2. jcsd
  3. Dec 6, 2008 #2
    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

    [tex]
    \begin{pmatrix}
    0 & v3 & -v2 \\
    -v3 & 0 & v1 \\
    v2 & -v1 & 0
    \end{pmatrix}
    [/tex]​

    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
     
  4. Dec 6, 2008 #3
    Muiti obrigado Dave!!

    Best from Brazil!

    Zwicky
     
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