Born in an infinite universe: A cosmological interpretation of quantum mechanics

We study the quantum measurement problem in the context of an infinite, statistically uniform space, as could be generated by eternal inflation. It has recently been argued that when identical copies of a quantum measurement system exist, the standard projection operators and Born rule method for calculating probabilities must be supplemented by estimates of relative frequencies of observers. We argue that an infinite space actually renders the Born rule redundant, by physically realizing all outcomes of a quantum measurement in different regions, with relative frequencies given by the square of the wave-function amplitudes. Our formal argument hinges on properties of what we term the quantum confusion operator, which projects onto the Hilbert subspace where the Born rule fails, and we comment on its relation to the oft-discussed quantum frequency operator. This analysis unifies the classical and quantum levels of parallel universes that have been discussed in the literature, and has implications for several issues in quantum measurement theory. Replacing the standard hypothetical ensemble of measurements repeated ad infinitum by a concrete decohered spatial collection of experiments carried out in different distant regions of space provides a natural context for a statistical interpretation of quantum mechanics. It also shows how, even for a single measurement, probabilities may be interpreted as relative frequencies in unitary (Everettian) quantum mechanics. We also argue that after discarding a zero-norm part of the wave function, the remainder consists of a superposition of indistinguishable terms, so that arguably “collapse” of the wave function is irrelevant, and the “many worlds” of Everett’s interpretation are unified into one. Finally, the analysis suggests a “cosmological interpretation” of quantum theory in which the wave function describes the actual spatial collection of identical quantum systems, and quantum uncertainty is attributable to the observer’s inability to self-locate in this collection.

Figure 1

As a function of the spin up fraction $n/N$, the figure shows the scaled binomial distribution for $(n,p)=(500,1/3)$ (shaded curve) together with the spectra of the frequency operator (diagonal line) and the $ϵ=0.1$ confusion operator (shaded, equal to 1 for $|n/N-p|>ϵ$).

Figure 2

Generic conformal structure of eternal inflation and post-inflationary reheating surface in open, topological, or stochastic eternal inflation. A region in which inflation is eternal (i.e. contains timelike worldlines that can extend to infinite proper time while remaining in the inflation region) is bounded by an infinite surface of field value ${\varphi }_s$ at which eternality fails. Between this value and ${\varphi }_e$ lie some number of e-folds of slow-roll inflation. This slow-roll inflation ends on a spacelike surface that represents a natural equal-time surface for the subsequent evolution including reheating, etc. This surface is spatially infinite, with spatial infinity denoted by $i_0$. (The future infinities following the reheating surface are not depicted).