Mathematical formalism of many-worlds quantum mechanics

TL;DR

This paper develops a mathematical formalism for many-worlds quantum mechanics by defining worlds as orthonormal bases of a Hilbert space, integrating ideas from Dirac, Everett, and 't Hooft, and deriving the Born rule within this framework.

Contribution

It introduces a novel formalism where worlds are orthonormal bases, providing a new perspective on quantum states and their evolution, and connects the many-worlds interpretation with a rigorous mathematical structure.

Findings

Defines worlds as orthonormal bases of Hilbert space

Shows evolution governed by Schrödinger's equation for worlds

Derives the Born rule using the Copenhagen interpretation within this framework

Abstract

We combine the ideas of Dirac's orthonormal representation, Everett's relative state, and 't Hooft's ontological basis to define the notion of a world for quantum mechanics. Mathematically, for a quantum system $\mathcal{Q}$ with an associated Hilbert space $\mathbb{H},$ a world of $\mathcal{Q}$ is defined to be an orthonormal basis of $\mathbb{H}.$ The evolution of the system is governed by Schr\"{o}dinger's equation for the worlds of it. An observable in a certain world is a self-adjoint operator diagonal under the corresponding basis. Moreover, a state is defined in an associated world but can be uniquely extended to the whole system as proved recently by Marcus, Spielman, and Srivastava. Although the states described by unit vectors in $\mathbb{H}$ may be determined in different worlds, there are the so-called topology-compact states which must be determined by the totality of a world. We can apply the Copenhagen interpretation to a world for regarding a quantum state as an external observation, and obtain the Born rule of random outcomes. Therefore, we present a mathematical formalism of quantum mechanics based on the notion of a world instead of a quantum state.