higgs - When do we consider states under a $U(1)$ transformations to be physically different?

Consider the Goldstone model of a complex scalar field $\Phi$. It has $U(1)$ global symmetry, so if we apply the transformation $\Phi \to e^{i\alpha} \Phi$ the Lagrangian is left invariant.

Furthermore, we have an infinite set of possible vacua all with the same non-zero vacuum expectation value. But the vacuum changes under a $U(1)$ transformation, so $U(1)$ symmetry is spontaneously broken.

• In this case we assume that for each value $\alpha$, $e^{i\alpha} |0 \rangle$ corresponds to a different state, right?


• But since we can't measure a phase, wouldn't it be more natural to consider them the same states? On the other hand, if I think of the real and imaginary parts as two independent fields, I would say that they shouldn't be the same states.

Let's now couple $\Phi$ to a gauge field, such that the Lagrangian is invariant under local $U(1)$ transformations. Then we do consider $e^{i\alpha} | \Psi \rangle$ to be all the same states, right? But then, doesn't this mean that the Higgs vacuum is unique?