Invariant theory and exact solutions for the quantum Dirac field in a time-dependent spatially homogeneous electric field

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Abstract
On the basis of the generalized invariant formulation, the invariant-related unitary transformation method is used to study the evolution of the quantum Dirac field in a time-dependent spatially homogeneous electric field. We solve the functional Schrödinger equation for the Dirac field and obtain the exact solutions and corresponding total phase. The total phase includes both the dynamical phase and geometric phase (Aharonov-Anandan phase).