Scattering and intrinsic irreversibility

The search of a physical explanation for the evolution towards equilibrium of quantum systems has a great interest for quantum statistical mechanics. For many years a great number of papers were devoted to this problem. The microscopic explanation of the approach to equilibrium was related to the so called ’intrinsic irreversibility’ of quantum systems. B.Misra, I.Prigogine, M.Courbage [1] [2] pointed out the existence of a time operator for the statistical description of classical and quantum systems. The mean value of this operator is the ’age’ of the system, which is a growing function of time. A. Bohm et al. [3] [4] related the intrinsic irreversibility to the existence of generalized eigenvectors of the Hamiltonian with complex eigenvalues, corresponding to poles of the analytic extension of the scattering matrix. Complex eigenvalues have been obtained by E. Sudarshan et al. [5] by analytic continuation in a generalized quantum mechanics. The Friedrichs model, a prototype of decaying system describing the interaction between a quantum oscillator and a scalar field, was extensively analyzed in the literature for the one exited mode sector. It is an exactly solvable model, in which the quantum oscillator decays to the ground state for all initial conditions. E. Sudarshan et al. [5] computed the complex spectral decomposition. The spectral decomposition was also obtained by T.Petrosky, I.Prigogine, S.Tasaki [6] using subdynamic theory. The spectral decomposition with complex eigenvalues was interpreted in terms of Rigged Hilbert spaces by I.Antoniou, I.Prigogine in reference [7] and by I.Antoniou, S.Tasaki in reference [8]. When it is necessary to deal with systems with a huge number of particles, the standard procedure is to start with N particles in a box of volume V , making the limit N → ∞, V → ∞ with NV = c < ∞ in the last step of the calculations. This method was used in subdynamic theory (I.Antoniou, S.Tasaki [8], T.Petrosky, I.Prigogine [9]), where the collision operator, with complex eigenvalues, is responsible for the evolution to statistical equilibrium. It is not surprising that the time evolution of Friedrichs model can be successfully described using the methods of non equilibrium statistical mechanics that can be used, for example, to describe the approach to statistical equilibrium of a quantum gas. In both cases the interaction eliminates constants of motion. In Friedrichs model the discrete eigenvalue disappear and in the gas the momentum of each particle is no more a constant of motion when the interaction is present. In this paper we want to discuss ”intrinsic irreversibility” in connection with pure scattering processes, where the total and the free Hamiltonian have the same continuous spectrum. For this purpose, it is important to use a formalism where ”final” states (t →∞) are well defined. For finite systems with continuous spectrum, the usual formalism of quantum mechanics fails to give a description of the ”final” states in terms of wave functions or density operators. To overcame this difficulty we will use in this paper the formalism developed by I. Antoniou et al. for quantum systems with diagonal singularity [10] [11] [12]. The quantum states of this theory are functionals over the space of observables O. Mathematically this means that the space S of states is contained in O×. Physically it means that the only thing we can really observe and measure are the mean values of the observables O ∈ O in states ρ ∈ S ⊂ O×: namely 〈O〉ρ = ρ[O] ≡ (ρ|O). This is the natural generalization of the usual trace Tr(ρ̂Ô) which is ill defined in systems with continuous spectrum. For finite quantum systems with continuous spectrum, some observables (for example the Hamiltonian) are represented by operators with diagonal singularities, and as they should have well defined mean values, diagonal singularities also appears in the states. In section 2, the resolvent formalism including creation, destruction and collision superoperators is obtained in general for quantum systems with diagonal singularities.