Matrix Operator Approach to the Quantum Evolution Operator and the Geometric Phase

The Moody-Shapere-Wilczek's adiabatic effective Hamiltonian and Lagrangian method is developed further into the matrix effective Hamiltonian (MEH) and Lagrangian (MEL) approach to a parameter-dependent quantum system. The matrix-operator approach formulated in the product integral (PI) provides not only a method to find the wave function efficiently in the MEH approach but also higher order corrections to the effective action systematically in the MEL approach, a la the Magnus expansion and the Kubo cumulant expansion. A coupled quantum system of a light particle of a harmonic oscillator is worked out, and as a by-product, a new kind of gauge potential (Berry's connection) is found even for nondegenerate cases (real eigenfunctions). Moreover, in the PI formulation the holonomy of the induced gauge potential is related to Schlesinger's exact formula for the gauge field tensor. A superadiabatic expansion is also constructed, and a generalized Dykhne formula, depending on the contour integrals of the homotopy class of complex degenerate points, is rephrased in the PI formulation.