I am currently reading up on the use of group theory in physics using Peter Woit's book draft (available on his homepage). I do understand the mathematical concepts but have a bit of a problem making the connection to the physical notation I am used to.
Specifically, if I have U(1) symmetry, the only irreducible representation is one dimensional meaning that the Hilbert space is also one dimensional or a direct sum of one dimensional Hilbert spaces. A state in this Hilbert space is then represented as a vector with complex numbers.
Now, how does this relate to a state represented in the occupation number formalism? When I see U(1) explained in physical texts, it is always represented as acting on the wave function or creation and annihilation operators with a global phase, but I do not see how to translate this to the one dimensional or multidimensional sum Hilbert space dictated by U(1).
Edit: By translate I mean the following: In the SU(2) case, I know that it admits a representation in terms of Pauli matrices which act on two dimensional vectors with complex entries and these complex entries represent my physical state. So it is very clear how I construct a representation of my group and how the space it acts on represents physics. In the U(1) case, I have simple exponentials labelled by the charge and they act on complex numbers. This complex number is then taken to be the phase of the wave function. To me, that is very different from the SU(2) case because the phase is not the wave function but might be a very different object such as an element of Fock space when we say that a many-body state is U(1) invariant.
I'd be grateful for any answer helping me to clear up this problem!