Symmetry groups of EM Field

[Mentz114]

I understand that $$E^2 - B^2$$ is invariant under various transformations.

If we consider the vector ( E, B ) as a column, then $$E^2 - B^2$$ 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 ?

 

[Dick]

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.

 

[Mentz114]

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...

 

[Mentz114]

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