Is U(1) the group of unit circle rotations e^(iθ)?

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Matterwave
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I keep hearing statements like "XXX theory is based on U(1)" or some such, but I haven't heard what this group actually is. If U(N) are the NxN unitary matrices, then U(1) are the 1x1 matrices such that x*=x^-1. So, I just want to confirm then, that U(1) is simply the 1 parameter Abelian Lie group given by [itex]e^{i\theta}[/itex]? This is simply the unit circle right, and should be isomorphic to rotations about a unit circle?
 

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Matterwave said:
U(1) … is simply the unit circle right, and should be isomorphic to rotations about a unit circle?

Yup! :smile:
 
Matterwave said:
I keep hearing statements like "XXX theory is based on U(1)" or some such, but I haven't heard what this group actually is. If U(N) are the NxN unitary matrices, then U(1) are the 1x1 matrices such that x*=x^-1. So, I just want to confirm then, that U(1) is simply the 1 parameter Abelian Lie group given by [itex]e^{i\theta}[/itex]? This is simply the unit circle right, and should be isomorphic to rotations about a unit circle?

Yes, it's not very impressive. Though when we say a U(1) gauge theory we mean the physics is invariant to a LOCAL U(1) gauge transformation (i.e exp(i theta(x)), not just a global one (i.e. exp(i* constant)). However, the reason we label such a simple thing is because we also then establish theories which are invariant under SU(2) and SU(3) local gauge transformations which gives us the weak and strong force physics.
 
Ok, last question. Is U(1) identical (homomorphic or diffeomorphic?) to SO(2) then?