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

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.