About the Concept of Entropy for Unstable Quantum States: on the Role of Resonances in Quantum Statistical Mechanics

The standard formalism of quantum statistical mechanics assumes real energy eigenstates belonging to the system under study. This is indeed true when one is working in the Hamiltonian (e.g, Hermitian) representation, in terms of Hermitian operators and wave functions with real energies. Alternatively one may think of representations where the Hamiltonians are non-Hermitian but which could have real eigenvalues under the action of certain symmetries, like PT, although this class of operators do not belong to extensions of the Hilbert space. In other words, PT symmetric non-Hermitian Hamiltonians and those belonging to extensions of the Hilbert space are not the same. Working with Gel’fand triplets allow us to deal with states of complex energy. The existence of unstable states, that is to say states of complex energy like Gamow states, has been demonstrated for a variety of systems, then it is a reasonable question to investigate the consequences upon the conventional formulation of statistical observables in basis where bound states and Gamow’s states belong to the spectrum. Then, if one formulates transition amplitudes, instead of probabilities, in basis with real and complex energy states, one gets mean values which are also complex. Therefore, the conventional interpretation of the entropy, as a purely real component of the free energy, is not longer valid. In order to get a purely real free energy, the imaginary part of the product TS should be equal to the imaginary part of E, being the temperature T real. In this work we shall extend on this notion and advance a physical interpretation of complex values of the entropy by means of some examples consisting of the quantum mechanical treatment of systems which exhibit Gamow’s states in their spectra. The interest in such studies has been raised recently by theoretical analysis of the role of Gamow states on the thermodynamics of black holes, particularly in dealing with the case of Gamow states gravitationally bound to a Reissner-Nordström black hole. Here, we shall analyse the case of a mini-black hole, to illustrate and interpret the role of the real and imaginary parts of thermodynamical potentials like the entropy. The formalism is applicable to a physical system where its energy spectrum, resulting from the treatment of a central potential under confining boundary conditions, exhibits bound states and resonances.