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    $\begingroup$ Of course a phase is meaning less. What makes sense is a phase difference, hence the citation in your question. As in any situation with quantum mechanics, a phase difference is related to interference phenomena. The specific point with superconductivity is that the related particles are charged, and so interference is connected with a charged current (the one flowing in your computer, or your house). An important example of the role of phase difference in the phenomenology of superconductors is the Josephson effect. Also, the superconducting vortices can be explained using the phase. $\endgroup$
    – FraSchelle
    Commented Apr 5, 2017 at 17:59
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    $\begingroup$ In quantum field theory, the phase (or the U(1) global rotation as it is named in your question) is related to the number of particles though the Noether theorem. There are enormous amonts of materials already on this website so I do not write an other answer. Check for Noether, Josephson, Superconductivity + Higgs, Ginzburg-Landau model, ... Note the phenomenology of the superconductivity can be understood as a Higgs mechanism for the U(1) gauge redundancy (when one makes the global U(1) symmetry/rotation a dynamical gauge field. i.e. give it the gauge structure of Maxwell's theory). $\endgroup$
    – FraSchelle
    Commented Apr 5, 2017 at 18:05
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    $\begingroup$ This is a rather open-ended question... and physicsl meaning is a subjective notion - it typically means relating things to the already familiar cocnepts. $\endgroup$
    – Roger V.
    Commented Mar 29, 2021 at 2:18