Quantum transitions and dressed unstable states - art. no. 052106

We consider the problem of the meaning of quantum unstable states including their dressing. According to both Dirac and Heitler this problem has not b een solved in the usual formulation of quantum mechanics, A precise definit ion of excited states is still needed to describe quantum transitions. We u se our formulation given in terms of density matrices outside the Hilbert s pace. We obtain a dressed unstable state for the Friedrichs model, which is the simplest model that incorporates both bare and dressed quantum states, The excited unstable state is derived from the stable states through analy tic continuation. It is given by an irreducible density matrix with broken time symmetry. It can be expressed by a superposition of Gamow density oper ators. The main difference from previous studies is that excited states are not factorizable into wave functions. The dressed unstable state satisfies all the criteria that we can expect: it has a real average energy and a no nvanishing trace. The average energy differs from Green's function energy b y a small effect starting with fourth order in the coupling constant. Our s tate decays following a Markovian equation, There are no deviations from ex ponential decay neither for short nor for long times, as is the case for th e bare state. The dressed state satisfies an uncertainty relation between e nergy and lifetime. We can also define dressed photon states and describe h ow the energy of the excited state is transmitted to the photons. There is another very important aspect: deviations from exponential decay would be i n contradiction with indiscernibility as one could define, e.g., old mesons and young mesons according to their lifetime. This problem is solved by sh owing that quantum transitions are the result of two processes: a dressing process, discussed in a previous publication, and a decay process, which is much slower for electrodynamic systems. During the dressing process the un stable state is prepared. Then the dressed state decays in a purely exponen tial way, In the Hilbert space the two processes are not separated. Therefo re it is not astonishing that we obtain for the unstable dressed state an i rreducible density matrix outside the Liouville-Hilbert-space. This is a li mit of Hilbert space states that are arbitrarily close to the decaying stat e. There are experiments that could verify our proposal. A typical one woul d be the study of the line shape, which is due to the superposition of the short-time process and die long-time process. The long-time process tah-en separately leads to a much sharper line shape, and avoids the divergence of the fluctuation predicted by the Lorentz line shape.