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Understanding definitions

  1. Jul 7, 2012 #1
    Hi guys!
    There is something I would like to get your help with...

    I am looking at the equation:

    [itex]W^{\mu}=-\frac{1}{2} \varepsilon^{\mu\nu\lambda\sigma}M_{\nu\lambda}p_{\sigma}[/itex]

    Which is, if I understand correctly,a Casimir Operator.
    Now, I wish to look at a particle in its rest reference, meaning,
    [itex]p_\mu=(m,0,0,0)[/itex]

    Why would these conditions yield :
    [itex]W^\mu =\frac {1} {2} m\varepsilon^{\mu\nu\lambda0}M_{\nu\lambda}[/itex]
    ?
    I can seem to understand how the indices change...

    The next thing I want to do, is understand what happens if I take [itex]m^2<0[/itex]

    Why does this condition mean that the momentum vector would be
    [itex]p_\mu=(0,0,0,m)[/itex]
    ?
    Thank you
     
  2. jcsd
  3. Jul 7, 2012 #2

    Nugatory

    User Avatar

    Staff: Mentor

    The expression you've quoted uses the Einstein summation convention, in which repeated indices are summed over: AiBi is a convenient short way of writing [itex]\sum[/itex]AiBi

    And because the only non-zero element of p is p0, when you do the summation over σ, all the terms are zero except the one in which σ is zero.
     
  4. Jul 7, 2012 #3
    Thank you!

    But why does
    [itex]p_{\mu}=(0,0,0,m)[/itex] relate to [itex]m^2<0[/itex]?

    And likewise,

    [itex]p_{\mu}=(p,0,0,p)[/itex] relate to [itex]m=0[/itex]?

    I understand why
    [itex]p_{\mu}=(m,0,0,0)[/itex] relate to massive particle,
    My logic here is [itex]p_0=E[/itex] and [itex]E\approx m[/itex]
    and [itex]\vec{p}=0[/itex] (because we are looking at the reference frame)
    But same logic does not work for me regarding the two eq. above...
    Thank you!
     
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