Shearing Off the Tree: Emerging Branch Structure and Born's Rule in an Equilibrated Multiverse

Abstract

Within the many worlds interpretation (MWI) it is believed that, as time passes on, the linearity of the Schrödinger equation together with decoherence generate an exponentially growing tree of branches where "everything happens", provided the branches are defined for a decohering basis. By studying an example, using exact numerical diagonalization of the Schrödinger equation to compute the decoherent histories functional, we find that this picture needs revision. Our example shows decoherence for histories defined at a few times, but a significant fraction (often the vast majority) of branches shows strong interference effects for histories of many times. In a sense made precise below, the histories independently sample an equilibrated quantum process, and, remarkably, we find that only histories that sample frequencies in accordance with Born's rule remain decoherent. Our results suggest that there is more structure in the many worlds tree than previously anticipated, influencing arguments of both proponents and opponents of the MWI.

Figures (4)

• Figure 1: (a) Tree structure of the Multiverse for a repeated binary "0 or 1" measurement: The standard account posits the reality of all branches (dashed and solid lines), but our results show that only a subset decoheres (solid lines). (b) Trajectory for the probability to obtain outcome 'one' conditioned on one branch (pink line in (a)). The numerical parameters of (b) are the same as in Fig. \ref{['fig 5050']} with $D=25000$.
• Figure 2: Coherence measure $\epsilon(n)$ (blue circles) and probabilities $p(n)$ (solid purple line) and $q(n)$ (black stars) as a function of $n$. Here and in all figures: ${\langle {n}\rangle} = Ld_1/D$ is the expected number of ones according to $p(n)$, we always rescale $p(n)$ and $q(n)$ by dividing by $p_\text{max} = \max_n p(n)$, the $x$ axis is rescaled to display $n$ on an interval of size two, and the gray area corresponds to one standard deviation of $p(n)$.
• Figure 3: Coherence measure $\epsilon(n)$ for $D=250$ (orange squares) and $D=25000$ (blue circles) and probabilities $p(n)$ and $q(n)$ (as in Fig. \ref{['fig 5050']}) for $D=25000$ as a function of $n$.
• Figure 4: Plot of $\epsilon(n)$, $p(n)$ and $q(n)$ for $L=250$ and $D=25000$ (legend as in Fig. \ref{['fig 5050']}). For (b) we rescaled $\tau$ to $10\tau$ to ensure equilibration between different trials.