Adiabatic Approximation for Path Integrals and Geometrical Potentials

Abstract

Geometrical concepts have played an important role in quantum mechanics. In the Schrodinger representation, the Berry–Simon connection and the quantum geometric tensor appear as gauge potentials in the effective Hamiltonian for a heavy system under the Born–Oppenheimer approximation. On the other hand, the Berry phase should arise as the amplitude of each path in the pathintegral formalism. In their pioneering work, Kuratsuji and Iida have developed the treatment of the Born– Oppenheimer approximation in the path integral quantization, where the Berry phase of the light system naturally appears in the effective action for the heavy system. However, they did not derive the gauge potentials themselves in the effective Hamiltonian. In this short note, we show that the gauge potentials which relate to the geometry of the Hilbert space of the light system are also derived in the adiabatic scheme of path integrals. Our result presents the path-integral formalism to consider the effective dynamics of the heavy system under the adiabatic approximation. Let us consider the following Hamiltonian: