The Geometric Part of Decoherence: Quasi-Orthogonality in High-Dimensional Hilbert Spaces

TL;DR

This paper identifies a geometric mechanism in high-dimensional Hilbert spaces that explains the effectiveness of decoherence through quasi-orthogonality of environmental states, complementing dynamical explanations.

Contribution

It isolates a geometric factor—quasi-orthogonality in high dimensions—that enhances understanding of decoherence without altering quantum interpretive frameworks.

Findings

Almost all state vectors in high-dimensional spaces are nearly orthogonal.

This geometry provides a vast capacity for environmental records, facilitating decoherence.

The mechanism explains why macroscopic superpositions fail to exhibit interference.

Abstract

We isolate a geometric mechanism that complements the dynamical suppression of macroscopic interference: In a high-dimensional Hilbert space, almost all state vectors are nearly orthogonal, accommodating an exponentially large reservoir of mutually quasi-orthogonal environmental records. This geometry explains why macroscopic alternatives fail to exhibit visible interference once such records are populated. The argument is conditional and finite-dimensional, and it leaves the interpretive core of quantum mechanics untouched: geometry alone does not select a pointer basis, does not guarantee that a given Hamiltonian drives the system into typical regions of the accessible subspace, and does not turn an improper mixture into a proper one. It merely supplies the vast Hilbert-space capacity that makes decoherence so overwhelmingly effective for all practical purposes.