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Symmetry groups of EM Field

by Mentz114
Tags: field, groups, symmetry
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Feb15-07, 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^2-B^2 but is it a member of the preceeding ?
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Feb16-07, 08:33 AM
Sci Advisor
HW Helper
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.
Feb16-07, 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...

Feb17-07, 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).

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

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