The Theory of Homotopical Quantum Mechanics and the Measurement Problem

We present Homotopical Quantum Mechanics (HQM), a new formulation of quantum theory that resolves the quantum measurement problem using Homotopy Type Theory (HoTT) and \(\infty\)-category theory. In HQM, the pre-measurement state of a system is a homotopy class of paths in an \(\infty\)-topos, with all representatives physically equivalent and capable of interference. Measurement is modeled as a homotopy pullback that, when an information-bearing interaction occurs, deterministically contracts the entire homotopy class to a single representative path via a functor rendering the type contractible. This contraction is driven by entanglement between system and observer, not by a stochastic collapse postulate. We introduce a complete dynamical model of contraction, triggered when apparatus–system entanglement entropy exceeds a critical value \(S_{\mathrm{crit}}\) or when pointer states become operationally distinguishable beyond a detector tolerance \(\epsilon\). Using a Hamiltonian pointer model, we show that contraction maps under a functor \(\mathcal{F}:\mathcal{HQT}\to\mathbf{Hilb}\) to the Lüders update, guaranteeing agreement with the Born rule via the Busch–Gleason theorem and preserving no-signaling. Thus, this formulation eliminates the ad hoc stochastic collapse postulate, unifies system and observer in a single topological and logical structure, and provides a physically grounded, operationally testable criterion for outcome definiteness. HQM thus offers both a mathematically rigorous foundation for quantum measurement and new experimental signatures—such as finite contraction delay—that distinguish it from standard quantum mechanics.