Abstract The Kato perturbative scheme is recovered and generalized. The procedure followed is inspired by the technique used in statistical mechanics to study the asymptotic evolution of the density matrix.
In the frame of the theory of subdynamics a general picture of the time evolution of a many-body system has been developed by the Brussels school of statistical mechanics. In this paper, we avoid the perturbation scheme so that we get compact notations and demonstrations. We derive here a general expression for the commutator of an arbitrary superoperator W with the " asymptotic " evolution superoperator Σ(t) = ΠU(t) where [formula] is the complete evolution superoperator and where Π is the usual superprojector of the theory of subdynamics. In the frame of relativistic non-equilibrium statistical mechanics, the application of this commutator to all the generators of the Poincaré group is then given and we get back in particular the fundamental property of commutation with Π of the Lorentz-Liouville superoperator.
Abstract We study the resonant scattering for discrete time quantum walks on graphs with some tails. In our arguments, we reduce the study of resonances to the perturbation of eigenvalues of a finite rank matrix associated with the internal graph. Then we can apply Kato’s perturbation theory of matrices, and the reduction process of generalized eigenspaces allows us to derive an explicit asymptotic expansion of the scattering matrix. As a consequence, we obtain the resonant scattering at resonant energies.
Abstract In this paper are introduced several novel techniques having as their objects, first, the gain of a new understanding of the divergence difficulties of orthodox quantum theory, and secondly, the systematic development of the quantum mechanics of fields in terms of the density matrix. There is included a new presentation of a form of perturbation theory which has the outstanding advantages over the usual one that it is much more quickly convergent, and leaves the density matrix normalized at every stage so that the diagonal elements representing occupational probabilities cannot diverge. Exact general formulae are derived for self-energies and cross-sections for the purpose of examining their convergence properties. The general theory of fields is developed abinitio, and it is shown that the use of the density matrix in place of the wave vector illuminates and simplifies the customary theory. The second quantization of the density matrix which follows throws unexpected light on the existence of particles with negative expectation values. Finally, the whole theory is applied to the crucial example of the electron or positron in an electromagnetic field. It is confirmed that the self-energy of the electron in the orthodox theory cannot be made finite without the introduction of negative energy photons.