Proceeding from a coherent-state functional-integral approach we give a first-principles theory of generalized pairing phases in a dense neutral system of electrons and protons. Apart from a standard stationary phase approximation the approach is general, it requires no adiabatic separation of time scales, and it can be applied for arbitrary temperatures. For the resulting mean-field theory, we show that pairing of both electrons and protons is possible at low temperatures, and especially so when an appropriately defined electron-proton order parameter becomes sufficiently large. As a preliminary to the experimentally important case where the protons order in a crystalline phase, the case of continuous symmetry is first presented. Among generic results is the prediction, through a stability analysis, of a charge-density wave ͑and repairing͒, and the location of a critical point, both discussed in light of recent experiments on the high-pressure states of hydrogen.

We show that the standard Dirac phase factor is not the only solution of the gauge transformation equations. The full form of a general gauge function (that connects systems that move in different sets of scalar and vector potentials), apart from Dirac phases also contains terms of classical fields that act nonlocally (in spacetime) on the local solutions of the time-dependent Schrödinger equation: the phases of wavefunctions in the Schrödinger picture are affected nonlocally by spatially and temporally remote magnetic and electric fields, in ways that are fully explored. These contributions go beyond the usual Aharonov-Bohm effects (magnetic or electric).

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