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Although the standard theory of superconductivity based on the BCS theory isa successful one, several experimental results indicate the necessity for afundamental revision. We argue that the revision is on the origin of the phasevariable for superconductivity; this phase appears as a consequence of theelectron-pairing in the standard theory, but its origin is a Berry connectionarising from many-body wave functions.  When this Berry connection is non-trivial, it gives rise to a collective modethat generates supercurrent; this collective mode creates number-changingoperators for particles participating in this mode, and these number-changingoperators stabilize the superconducting state by exploiting the Cooperinstability. In the new theory, the role of the electron-pairing is tostabilize the nontrivial Berry connection; it is not the cause ofsuperconductivity.  In BCS superconductors, however, the simultaneous appearance of thenontrivial Berry connection and the electron-pairing occurs. Therefore, theelectron-pairing amplitude can be used as an order parameter for thesuperconducting state. We revisit the Andreev$-$Saint-James reflection and theJosephson effect. They are explained as consequence of the presence of theBerry connection. Bogoliubov quasiparticles are replaced by the particle-numberconserving Bogoliubov excitations that describe the transfer of electronsbetween the collective mod and single particle mode.

Superconductivity by Berry connection from many-body wave functions: revisit to Andreev$-$Saint-James reflection and Josephson effect