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Symmetry groups of EM Field 
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#1
Feb1507, 05:48 PM

PF Gold
P: 4,087

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^2B^2 but is it a member of the preceeding ? 


#2
Feb1607, 08:33 AM

Sci Advisor
HW Helper
Thanks
P: 25,228

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^2B^2.



#3
Feb1607, 09:44 AM

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... 


#4
Feb1707, 10:38 PM

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 


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