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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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    $\begingroup$ About the uncertainty relation, you may find this post of interest : physics.stackexchange.com/q/284314/16689 I commented about the symmetry breaking mechanism of superconductivity there : physics.stackexchange.com/q/133780/16689 The relation between Higgs-Anderson and superconductivity is done in the book by Weinberg, S. (1995). The Quantum Theory of Fields (Volume 2). Cambridge University Press. $\endgroup$
    – FraSchelle
    Commented Apr 3, 2017 at 8:26
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    $\begingroup$ A pedagogical overview of the statement is given in Greiter, M. (2005). Is electromagnetic gauge invariance spontaneously violated in superconductors? Annals of Physics, 319(1), 18. Superconductivity. doi.org/10.1016/j.aop.2005.03.008 / arXiv:arxiv.org/abs/cond-mat/0503400 $\endgroup$
    – FraSchelle
    Commented Apr 3, 2017 at 8:26

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