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Superconductivity is reformulated as a phenomenon in which a stable velocityfield is created by a $U(1)$ phase neglected by Dirac in the Schroedingerrepresentation of quantum mechanics. The neglected phase gives rise to a $U(1)$gauge field expressed as the Berry connection from many-body wave functions.The inclusion of this gauge field transforms the standard particle-numbernon-conserving formalism of superconductivity to a particle-number conservingone with many results of the former unaltered. In other words, the newformalism indicates that the current standard one is an approximation thateffectively takes into account this neglected $U(1)$ gauge field by employingthe particle-number non-conserving formalism. Since the standard and newformalisms are physically different, conflicting results are predicted in somecases. We reexamine the Josephson relation and show that a capacitancecontribution of the Josephson junction to the $U(1)$ phase is missing in thestandard formalism, and inclusion of it indicates that the standard theoryactually does one agree with the experiment while the new one does. It is alsoshown that the dissipative quantum phase transition in Josephson junctionspredicted in the standard theory does not exit in the new one in accordancewith the recent experimental result.

Neglected $U(1)$ phase in the Schroedinger representation of quantum mechanics and particle number conserving formalisms for superconductivity