A sum rule for the so-called quantum phases is established, without assuming adiabatic evolution. By considering two interacting quantum systems, conventionally designated as a boson and a fermionic one, we evaluate the effective bosonic action after integrating out the fermionic variables. This effective action contains a nontrivial topological part, which has the same origin as the quantum adiabatic phase. However, the derivation does not rely on any adiabatic assumption.

• Received 18 December 1987

DOI:https://doi.org/10.1103/PhysRevA.37.4084