Arrow of time in quantum mechanics

Why does the time flow only in one particular direction? This is a very big and fundamental question of physics, called the problem of "the arrow of time." There are a few types of the arrow of time. The cosmological arrow of time refers to the direction of time in our universe as we see it. The psychological arrow of time refers to the direction of time that we perceive. The arrow of time that I am going to discuss in the present talk is the dynamical arrow of time, which refers to the following problem. The fundamental equations of motion, namely the Newton equation for classical mechanics and the Schroedinger equation for quantum mechanics, are symmetric if we flip the arrow of time. Nonetheless, we see the excited states always decay and the entropy always increases. How can this inconsistency arise? There have been many answers to it but most of them attribute the reason to the limitation of our ability of measuring physical systems. My collaborator Gonzalo Ordonez and I have reservations because it almost sounds like the dynamical arrow of time could be flipped or eliminated if we have more abilities! We answer the question of the dynamical arrow of time in the realm of quantum mechanics in the following way. We succeeded in deriving a time-reversal symmetric decomposition of the time evolution operator which contains both decaying states and growing states symmetrically. If we watch the time evolution starting from an initial condition, we pick all the decaying states, whereas if we watch the time evolution ending in a terminal condition, we pick all the growing states. This shows that the arrow of time is generated by the difference in the way we watch the time evolution.