We investigate the origin of quantum geometric phases, gauge fields and forces beyond the adiabatic regime. In particular, we extend the notions of geometric magnetic and electric forces discovered in studies of the Born-Oppenheimer approximation to
arbitrary quantum systems described by matrix valued quantum Hamiltonians. The results are illustrated by several physical relevant examples.
A general method to derive the diagonal representation for a generic matrix valued quantum Hamiltonian is proposed. In this approach new mathematical objects like non-commuting operators evolving with the Planck constant promoted as a running variabl
e are introduced. This method leads to a formal compact expression for the diagonal Hamiltonian which can be expanded in a power series of the Planck constant. In particular, we provide an explicit expression for the diagonal representation of a generic Hamiltonian to the second order in the Planck constant. This last result is applied, as a physical illustration, to Dirac electrons and neutrinos in external fields.
We investigate the origin of quantum geometric phases, gauge fields and forces beyond the adiabatic regime. In particular, we extend the notions of geometric magnetic and electric forces discovered in studies of the Born-Oppenheimer approximation to
arbitrary quantum systems described by matrix valued quantum Hamiltonians. The results are illustrated by several physical relevant examples.
A generalized Peierls substitution which takes into account a Berry phase term must be considered for the semiclassical treatment of electrons in a magnetic field. This substitution turns out to be an essential element for the correct determination o
f the semiclassical equations of motion as well as for the semiclassical Bohr-Sommerfeld quantization condition for energy levels. A general expression for the cross-sectional area is derived and used as an illustration for the calculation of the energy levels of Bloch and Dirac electrons.
It has been recently found that the equations of motion of several semiclassical systems must take into account anomalous velocity terms arising from Berry phase contributions. Those terms are for instance responsible for the spin Hall effect in semi
conductors or the gravitational birefringence of photons propagating in a static gravitational field. Intensive ongoing research on this subject seems to indicate that actually a broad class of quantum systems might have their dynamics affected by Berry phase terms. In this article we review the implication of a new diagonalization method for generic matrix valued Hamiltonians based on a formal expansion in power of $hbar$. In this approach both the diagonal energy operator and dynamical operators which depend on Berry phase terms and thus form a noncommutative algebra, can be expanded in power series in hbar $. Focusing on the semiclassical approximation, we will see that a large class of quantum systems, ranging from relativistic Dirac particles in strong external fields to Bloch electrons in solids have their dynamics radically modified by Berry terms.
