Higher-order categorical coherence breakdown: a geometric framework for nonlinear quantum mechanics and its applications to strongly correlated electron systems

Abstract

We introduce a higher quantum mechanics whose fundamental structure arises from the breakdown of categorical coherence beyond the first order. In our formulation, standard quantum mechanics itself emerges from first-order categorical coherence breakdown, corresponding to the familiar non-commutativity of observables and described geometrically by the Uhlmann gauge connection on the purification bundle. By promoting this to a higher categorical and higher gauge framework, we show that breakdown at higher coherence levels corresponds to the emergence of higher Uhlmann curvatures-geometric obstruction classes whose state-dependent structure induces intrinsic nonlinearities in the quantum equations of motion. We provide a concrete categorical model based on a 2-category of contexts generated by projective-valued measures (PVMs) with coarse-grainings, construct the Uhlmann bundle-gerbe over the manifold of full-rank density operators, and compute its Deligne class. A rigorous transgression functor from the path 2-groupoid of contexts to the holonomy 2-group of the gerbe yields curvature-weighted Magnus/Chen expansions, from which we derive explicit nonlinear correction functionals \(\mathcal {N}_{j}[\rho ]\) for æ =2,3. These nonlinear terms are the direct quantum-mechanical analog of interaction terms in gauge field theory, but arise here from multi-way measurement incompatibilities rather than external interactions. We argue that this higher-order geometric structure provides a natural theoretical framework for regimes where standard linear quantum mechanics is insufficient-particularly in quantum chemistry, multi-electron strongly correlated systems, and nonadiabatic dynamics at conical intersections. Applications are discussed for catalytic processes, chaotic electron dynamics, and materials with strong electron correlation, where our theory predicts experimentally testable deviations from linear quantum predictions.