# What is the effect of the global $U(1)$ spontaneous symmetry breaking on the physical observables in superconductors?

I have been taught that in Ginzburg-Landau model for superconductors the transition to the superconductive states lead to the spontaneous symmetry breaking of global $$U(1)$$ symmetry.

While I have understood the significance of this from a mathematical point of view I have not yet a clear idea of what is the physical significance of this.

If I consider the ground state, clearly I have a collection of configuration with the same energy but different global phase. However, I don't see how this is affecting the physical observable of the system.

The condensate density $$|\psi|$$ is clearly unaffected. The current $$\vec j_s \propto \nabla \theta + \vec A$$ depends only on the gradient of the phase and the magnetic field $$\vec B = \nabla \times \vec A$$ is independent.

Are these states different only in the mathematical representation but, in fact, physically indistinguishable? In this case why do we even care about this symmetry breaking?