Title: Forces exerted by quantum systems on Classical systems
Bahar Mehmani - Dec 07 2009
I shall mostly talk about our recent work on a slow classical system [particle] coupled to a fast quantum
system with discrete energy spectrum. We adiabatically exclude the quantum system and construct an
autonomous dynamics for the classical particle in successive orders of the small ratio ε of the
It is known that in the order ε0 the particle gets an additional [Born-Oppenheimer]
potential, while in the order ε1 it feels an effective magnetic field related to the Berry
In the order ε2 the motion of the classical particle can be reduced to a free [geodesic]
motion on a curved Riemannian manifold, with the metric generated by the excluded quantum system. This
motion has a number of unusual features, e.g., it combines subspaces of different (Riemannian and
pseudo-Riemannian) signature for the metric tensor.
In the order ε3 the motion of the classical particle is still described by a Lagrangian,
but the latter linearly depends on the particle's acceleration. This implies the existence of a spin tensor
[non-orbital angular momentum] for the particle. This spin tensor is related to the momentum via an analogue
of the zitterbewegung effect.
The Hamiltonian structure of the system is non-trivial and is defined via non-linear Poisson brackets. The
linear dependence of the effective classical Lagrangian on higher-order derivatives is seen as well in the
higher orders εn.