The Berry connection and Born–Oppenheimer method

Abstract

By performing the most general Born–Oppenheimer procedure, the (non-Abelian) Berry connection for quantum systems in a quantum environment is derived. This method is then applied to the rapid rotation of a particle about a slowly changing axis, as exemplified by the electronic motion of a diatomic molecule. The angular part of the resulting dynamics for the quantum environment is equivalent to that of a monopole.

Cheshire Cat hadrons

“…We now turn to the problem of constructing conserved angular momentum in a diatomic molecule in which a Berry potential couples to the dynamics of slow degrees of freedom, corresponding to the nuclear coordinate R, a system that has been studied extensively [34]. The procedure is in complete parallel to that reviewed above.…”

“…In order to understand this situation, we first discuss here a case in which a nonabelian gauge structure arises in a relatively simple quantum mechanical system. To do so, we will study a simple toy-model example of the induced nonabelian gauge fields and Berry phases in the Born-Oppenheimer approximation [34]. When suitably implemented, the treatment can be applied to a realistic description of the spectrum of a diatomic molecule, wherein this approximation is usually described as a separation of slow (nuclear) and fast (electronic) degrees of freedom.…”

Quantum phases of Yang-Mills matrix model coupled to fundamental fermions

“…A modern treatment of the B-O scheme has been discussed in [7], and we will adapt it to our matrix model below.…”

“…This situation is reminiscent of the BornOppenheimer (B-O) approximation in quantum molecular dynamics, where the atomic nuclei are slow degrees of freedom (precisely because the contribution of their kinetic energy to the total Hamiltonian is small), and the surrounding cloud of electrons the fast degrees of freedom. The dynamics of the nuclei is treated adiabatically, and a careful treatment of the B-O approximation leads to an adiabatic scalar potential induced in the space of slow variables [6,7], in addition to the well-known adiabatic Berry connection.…”

Pseudoforces in quantum mechanics

“…As each observer can apply an overall unitary operator on his Hilbert space and still obtain an equivalent description, we see that the structure group should be the group of all unitary operators on the Hilbert space [8]. Thus there is an underlying "gauge freedom" which can be used to transform the natural parallel sections into constant sections and do away with the need to use all Hilbert spaces at once.…”