DIRECTED QUANTUM CHAOS

Quantum disordered problems with a direction (imaginary vector potenti al) are discussed and mapped onto a supermatrix sigma model. It is arg ued that the OD version of the sigma model may describe a broad class of phenomena that can be called directed quantum chaos. It is demonstr ated by explicit calculations that these problems are equivalent to th ose of random asymmetric or non-Hermitian matrices. A joint probabilit y of complex eigenvalues is obtained. The fraction of states with real eigenvalues proves to be always finite for time reversal invariant sy stems.