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 ?
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.
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...
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