On the Canonical Treatment of Dissipative Systems with Broken Time-Reversal Symmetry

It will be shown how dissipative systems can be consistently described by effective Hamiltonians which also allow canonical quantization. It is of importance to distinguish between a physical, dissipative level, where no proper Hamiltonian for the dissipative systems alone is known and where an effective quantum mechanical description leads to non-linear Schrödinger equations (NLSE), and a formal canonical level where quantization leads to linear theories. The non-unitary connnection between the explicitly time-dependent Hamiltonian of Caldirola and Kanai and a logarithmic NLSE will be discussed and it will be shown how to avoid the violation of the uncertainty principle. The key point is the fact that physical and canonical levels are connected via non-canonical (classical) or non-unitary (quantum mechanical) transformations. Finally, on the canonical level, a corresponding transformation leads to a Hamiltonian for a dissipative system which is a constant of motion and uniquely connected with the energy of the the system.