Symmetry groups of EM Field

  1. Mentz114

    Mentz114 4,191
    Gold Member

    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. Dick

    Dick 25,824
    Science Advisor
    Homework Helper

    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. Mentz114

    Mentz114 4,191
    Gold Member

    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. Mentz114

    Mentz114 4,191
    Gold Member

    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).

    M
     

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