# Symmetry groups of EM Field

by Mentz114
Tags: field, groups, symmetry
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 PF Gold P: 4,087 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 ?
 Sci Advisor HW Helper Thanks P: 25,235 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.
 PF Gold P: 4,087 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...
PF Gold
P: 4,087
Symmetry groups of EM Field

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
Attached Files
 SU12 group .pdf (189.7 KB, 4 views)

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