Relativity and Irreversibility

The dynamical laws of physics in their conventional formulation are time reversible. The description of irreversible phenomena was always thought to be incompatible with the dynamical laws. We have recently discussed [1, 2, 3, 9, 10, 11, 12] the origin of irreversible behaviour due to resonances in various Poincaré nonintegrable quantum systems. The analytic extension of the Hamiltonian or Liouvillian in such systems leads to new solutions describing unstable states, which evolve irreversibly. Such states defined in the wave function space are usually called Gamow vectors [4, 6]. Thus it was shown that the presence of resonance leads to intrinsic irreversibility of dynamical systems. Therefore, irreversibility is compatible with dynamics. The aim of the present paper is to show that the irreversible description is compatible with relativistic invariance as well. To consider this question clear is very important, in view of recent publications on this topic, for example, [8]. In order to demonstrate this compatibility, we study the relativistic transformation of decaying unstable Gamow modes using a relativistic invariant model of two interacting fields introduced in [2]. We suggest that the transformation of the Gamow states is induced by the Lorentz transformation of field modes which are eigenstates of the fourdimensional energy momentum of the system. The formal application of the Lorentz boost to the Gamow state leads to a complex value of the momentum k′ after the transformation [2]. However, we can keep real the transformed momentum if we consider this transformation at the level of wave packets. Our arguments are based on the fact that the Gamow state |ΨG(k)〉 with momentum k is obtained as pole contribution from