Time symmetry and quantum dephasing

We first stress that the time symmetry in quantum mechanics manifests itself in the analytical properties of the Fourier transform of the evolution operator in the complex E-plane (E being the variable conjugate to time), in such a way that all singularities are distributed only on the real axis in the first Riemannian sheet, and new poles appear, in the N infinite limit (N standing for the number of degrees of freedom of detector or instrument concerned), on the second Riemannian sheet in a symmetric way with respect to the real axis. We then examine the symmetry-breaking phenomenon, such as decay or dissipation, by setting up the initial value problem: The temporal evolution of the transition probability is divided into three parts, the first being Gaussian for very short times, the second exponential for intermediate times and the third of the power type for very long times. We know that the Gaussian decay is directly connected to the so-called quantum Zeno effect, the exponential decay corresponds to a sort of dephasing process, because the time rate of the total transition probability becomes a sum of time rates of partial probabilities, and both the Gaussian-like and power-like decay will disappear, leaving only the exponential one, in the van Hove limit. The dominance of the exponential decay is equivalent to the appearance of a master equation, which tells us that we have no phase-correlation but decoherence or dephasing. All temporal behaviors of quantum-mechanical transition probability and related physics are reflected in the analytical property of the Fourier transform of the evolution operator.