Discrete Hilbert Space, the Born Rule, and Quantum Gravity

TL;DR

This paper explores how quantum gravity implies a discrete Hilbert space with a minimum norm, which is crucial for deriving the Born Rule without collapse in quantum mechanics, linking spacetime discreteness to quantum foundations.

Contribution

It proposes that quantum gravity leads to a discrete Hilbert space with a minimum norm, providing a new perspective on the emergence of the Born Rule in no-collapse quantum mechanics.

Findings

Discrete models suggest a minimum norm in Hilbert space.

Quantum gravity implies a minimal length scale affects quantum states.

Discreteness of Hilbert space relates to decoherent histories of spacetime.

Abstract

Quantum gravitational effects suggest a minimal length, or spacetime interval, of order the Planck length. This in turn suggests that Hilbert space itself may be discrete rather than continuous. One implication is that quantum states with norm below some very small threshold do not exist. The exclusion of what Everett referred to as maverick branches is necessary for the emergence of the Born Rule in no collapse quantum mechanics. We discuss this in the context of quantum gravity, showing that discrete models (such as simplicial or lattice quantum gravity) indeed suggest a discrete Hilbert space with minimum norm. These considerations are related to the ultimate level of fine-graining found in decoherent histories (of spacetime geometry plus matter fields) produced by quantum gravity.