Electric curvature and the time component of the adiabatic connection

Generalizing the concept of the Berry phase, I show that an additional gauge structure associated with a time-component vector potential $A_0$ is present in the adiabatic evolution of a quantum system. The gauge structure, which generates the usual dynamical phase, is revealed by a formalism in which the time variable is treated on an equal footing with other parameters on which the Hamiltonian depends. The invariant curvature associated with $A_0$, an electriclike field, determines the phase difference between low- and high-energy paths in a generalized phase-space and time picture. The covariance of the Born-Oppenheimer approximation illustrates the results.