Non-equilibrium Quantum Field Theory and Entangled Commutation Relations

Non-equilibrium quantum field theory studies the time dependence of processes to which the $S$-matrix approach is inapplicable. One of the new methods of investigation in non-equilibrium quantum theory is the stochastic limit method. This method is an extension of the works by Bogolyubov, Van Hove, and Prigogine and permits the study not only of the system but also of the reservoir degrees of freedom. We consider the stochastic limit of translation invariant Hamiltonians in quantum field theory and show that the master field satisfies a new type of commutation relations, the so-called entangled (or interacting) commutation relations. These relations extend the interacting Fock relations established earlier in non-relativistic QED and the free (or Boltzmann) commutation relations which have been found in the large $N$ limit of QCD. As an application of the stochastic limit method, we consider the photon splitting cascades in magnetic field and show that photons in cascades form entangled states (“triphons”) and obey a new type of statistics corresponding to the entangled commutation relations rather than the Bose statistics.