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Symmetry groups of EM Field

  1. Feb 15, 2007 #1


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    I understand that [tex] E^2 - B^2 [/tex] is invariant under various transformations.

    If we consider the vector ( E, B ) as a column, then [tex] E^2 - B^2 [/tex] is preserved after mutiplication by a matrix -

    | cosh( v) i.sinh(v) |
    | i.sinh(v) cosh(v) |

    I think this transformation belongs to a group, but I can't put a name to it.
    Does anyone recognise it ?

    This matrix

    1 i
    i 1

    also seems to preserve E^2-B^2 but is it a member of the preceeding ?
  2. jcsd
  3. Feb 16, 2007 #2


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    If you look at what you are doing, this is the same as preserving the spacetime interval in 1+1 dimensions (t,x). So it's 'like' the lorentz group, though you've got complex entries and the one parameter family is not a group. Call it a subset of SU(1,1). The second matrix doesn't even preserve E^2-B^2.
  4. Feb 16, 2007 #3


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    Dick, thanks a lot.
    I thought it might be a subset of 1+1 boosts.
    I must have fumbled the calculation with the second matrix. Too much coffee...
  5. Feb 17, 2007 #4


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    Thanks again for naming the group. It is SU(1,1) in all its glory.
    I had a lucky find which I've attached. It is a great intro to the group, see
    especially section 6.1. I just noticed that the file is called SU12, that is an error,
    it really is about SU(1,1).


    Attached Files:

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