Abstract

We put forward a quantum model of cosmology that is exactly periodic but avoids the Boltzmann Brain problem. If the universe is described by a quantum state evolving unitarily in a finite-dimensional Hilbert space, its evolution will be recurrent: given enough time, the state will return arbitrarily close to its initial state. There is a worry that such a scenario cannot be phenomenologically acceptable, because the state will spend most of its time in a high-entropy equilibrium macrostate, with rare fluctuations downward in entropy, and the vast majority of observers will be minimal fluctuations away from equilibrium, or “Boltzmann Brains.” Here we show that this is not necessarily true. If the differences in energy eigenvalues are commensurable, the evolution is not simple recurrent, but exactly periodic. Moreover, if the state starts at minimum thermodynamic entropy, its evolution can feature a distinguished entropy excursion that is much more pronounced than one would expect from the conventional expression P(∆S)∝exp(−∆S). This excursion could represent our Big Bang, with relatively few Boltzmann fluctuations occurring in the subsequent equilibrium phase before a Big Crunch occurs and the cycle begins again. We speculate on the spacetime interpretation of this kind of quantum universe.