Quantum transitions in interacting fields

In recent papers we have discussed the meaning of dressed excited quantum states. We have shown that these states can be formulated in terms of distribution functions outside the Hilbert space. Our approach applies to “nonintegrable” systems in the sense of Poincaré. It involves the analytic continuation of the unitary operator U describing the transformation from bare to dressed stable states. This leads to a “star-unitary” operator Λ. Interacting fields are nonintegrable systems obtained as a result of resonances. It is therefore natural to expect that our previous results remain valid for interacting fields. We consider a simple example, which corresponds to an extension of the usual Friedrichs model. This involves a local field in interaction with a bilocal scalar field. This model has been previously studied by one of the co-authors (I.P.). It is a simplified version of the model describing A⃗B+C transition with quadratic interaction. The usual Bogoliubov transformation eliminates the field corresponding to A, while our method leads to strictly exponential decay of unstable field. The dressed field A corresponds to a singular distribution function outside the Liouville-Hilbert space. As in the Friedrichs model there exists two time scales—one for the preparation of a dressed state and depending on initial conditions called “Zeno period” and the other universal for the decay.