Canonical quantization of dissipative systems

J�rgen Burzlaff

Massimo V. Fischetti, William G. Vandenberghe

Vasily Evgenevich Tarasov

Antonio Maria Scarfone

Nivaldo Agostinho Lemos

An old result of Brittin to the effect that dissipative systems belonging to a certain class lend themselves to canonical quantization in the Heisenberg picture but not in the Schr\"odinger picture is shown to be incorrect. It is shown that canonical quantization is impossible in the presence of nonconservative forces, no matter which of the two pictures one chooses. This restores the equivalence of the Heisenberg and Schr\"odinger pictures.

Susumu Ôkubo

Some one-dimensional dissipative systems can be found to admit canonical-quantization procedure, contrary to opinions expressed occassionally in the literature. Also, a natural way of quantizing an equation $\stackrel{\ifmmode\ddot\else\textasciidieresis\fi{}}{q}+\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{V}(q)=0$ for a function $V(q)$ is found to use a nonstandard commutation relation $[q,\ensuremath{\pi}]=i\ensuremath{\hbar}\ensuremath{\pi}$ together with Hamiltonian $H=\ensuremath{\pi}+V(q)$. Here, $\ensuremath{\pi}=\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{q}$ stands for the velocity operator rather than the momentum operator. Finally, a quantization problem of a Newtonian geodesic equation of motion in nonflat space is discussed.