In general, the BCS ground state wave function can be written as

$$|\psi_G\rangle=\prod_k (|u_k|+e^{i\theta}|v_k|c_{k\uparrow}^\dagger c_{-k\downarrow}^\dagger)|\phi_0\rangle $$

According to Tinkham, the system satisfies the uncertainty relaion $\Delta \theta\Delta N>1$. Hence, if $\theta$ is fixed, then we have large uncertainty in the number of particles in $N$, and vise versa.

Because these are charged particles, I would think that fixing the phase $\theta$ would break a local $U(1)$ gauge symmetry. Hence, the system would experience massive modes (Higgs-Anderson mechanism). Is the indefinite number of particles in the system somehow connected to the generation of massive Higgs modes in the superconductor? I can't really find any resources that connect the two phenomenon, but it would seem as if they are connected due to the breaking of the local gauge symmetry. Any explanation or resources at the level of Tinkham would be greatly appreciated.



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