Time symmetry in wave-function collapse

The notion of a physical collapse of the wave function is embodied in dynamical collapse models. These involve a modification of the unitary evolution of the wave function so as to give a dynamical account of collapse. The resulting dynamics is at first sight time asymmetric for the simple reason that the wave function depends on those collapse events in the past but not those in the future. Here we show that dynamical wave-function collapse models admit a general description that has no built-in direction of time. Given some simple constraints, we show that there exist empirically equivalent pictures of collapsing wave functions in both time directions, each satisfying the same dynamical rules. A preferred direction is singled out only by the asymmetric initial and final time constraints on the state of the universe.

Figure 1

Beam-splitter experiment exposing the apparent time-reversal asymmetry of wave-function collapse. Particles are emitted from a source, $\mathsf{S}$. They pass through a 50-50 beam splitter, $\mathsf{B}$; particles subsequently traveling to the right are detected at $\mathsf{D}$ and particles reflected upwards reach the ceiling $\mathsf{C}$. The Born rule says that half of all particles emitted from $\mathsf{S}$ will be detected at $\mathsf{D}$. Considering the events of the experiment in the backward-in-time direction we have an interaction between particle and detector at $\mathsf{D}$ followed with certainty by an interaction between the particles and the source at $\mathsf{S}$. This is in contrast to the Born rule prediction applied backward in time which would claim that only half of the particles travel left to $\mathsf{S}$ while half travel down to the floor $\mathsf{F}$.