Explicit construction of a time superoperator for quantum unstable systems

A time superoperator T conjugate to the Liouville superoperator L-H = [H,] is constructed for a quantum system with one excited state or unstable part icle. While there is no time operator conjugate to the Hamiltonian in the w ave function space due to the positivity of energy, T may exist in the dens ity matrix space as the spectrum of L-H covers all the real axis. This is t he first example of an observable that can only be formulated in the Liouvi lle-von Neumann space of density matrices. In our example the expectation v alue of T gives the lifetime of the unstable particle. Once the time supero perator is obtained it is easy to define an entropy superoperator. (C) 2001 Elsevier Science Ltd. All rights reserved.