Everettian Mechanics with Hyperfinitely Many Worlds

Abstract

The present paper shows how one might model Everettian quantum mechanics using hyperfinitely many worlds. A hyperfinite model allows one to consider idealized measurements of observables with continuous-valued spectra where different outcomes are associated with possibly infinitesimal probabilities. One can also prove hyperfinite formulations of Everett’s limiting relative-frequency and randomness properties, theorems he considered central to his formulation of quantum mechanics. Finally, this model provides an intuitive framework in which to consider no-collapse formulations of quantum mechanics more generally.

Conquering Mount Everett: Branch Counting versus the Born Rule

“…This, I take it, is the received view: DeWitt (1971) explicitly asserts that there are finitely many branches, and Wallace (2012) assumes as much in his decision-theoretic justification of the Born Rule. I will also make this assumption; but notice that Barrett and Goldbring (2022) have recently developed a model of Everettian Quantum Mechanics with hyperfinitely many branches, for observables with continuous spectra, that allows us to talk about ratios in cardinalities. I suspect such a model will be compatible with branch-counting, wherein certain self-locating propositions will receive infinitesimal probability.…”

Hypothetical Frequencies as Approximations

“…Hájek's "parting offer" of a hyperfinite version of frequentism is an exercise that fully aligns with this attitude. 14 Indeed, it might find purchase on the nonstandard approach to quantum mechanics recently illustrated by Barrett and Goldbring (2021). 15…”