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Reviews of Modern Physics

Mathematical Problems in the Complete Quantum Predictions of Chemical Phenomena

Samuel Francis Boys, G. B. Cook

Published Apr 1, 1960• 56 citations
Publisher
DOI
10.1103/revmodphys.32.285

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The predictive success of modern physics from rigid body mechanics to quantum field theory stands among the greatest achievements of the human mind. Yet this success does not necessarily entail completeness of explanatory understanding. This paper investigates the epistemological gap between mathematical description and causal explanation, taking the behavior of rotating bodies (gyroscopes) as a preliminary example, then extending to two central problems: the measurement problem in quantum mechanics and the nature of mass in particle physics. The study shows that concepts such as angular momentum and precession, despite their predictive accuracy, redescribe phenomena in mathematical language more than they provide intuitive causal mechanisms. It also demonstrates that Noether’s theorem and the principle of least action, for all their depth, reformulate fundamental questions rather than resolving them. The paper discusses the framework of structural realism as a philosophical stance that accommodates this gap, examines criticisms of this framework (the problem of structural individuation and the problem of abandoned theories), and concludes that acknowledging this gap is not an attack on physics but a condition for its philosophical maturity.
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This paper establishes a rigorous, extensible, and fully axiomatized seven-layer classification system for quantum chemistry. The system systematically organizes all quantum chemical theories, methods, and molecular systems into seven hierarchical levels: Kingdom (physical framework), Phylum (Hilbert space topology and many-body structure), Class (symmetries and algebras), Order (quantum numbers and electronic configurations), Family (space-time dimensions and energy scales), Genus (dynamical conditions and approximation methods), and Species (concrete molecular systems and observables). Each layer is equipped with explicit parameter sets, axioms (totaling 66), compatibility conditions, and fundamental theorems (each with ≥ 12 steps). The uniqueness of each layer parameter tuple is proved. Furthermore, we identify 28 gaps in the parameter space, each corresponding to a theoretically consistent but systematically unexplored branch. Every gap is elevated to a predictive branch with a complete axiom system, a main theorem, and a rigorous proof (≥ 12 steps). All predictions are testable by current or near-future experiments. The system is shown to be arbitrarily extensible, admits an ∞-categorical lift, and the classification decision problem is NP-complete and QMA-complete. All previously open problems and conjectures (including the exact form of the exchange-correlation functional, the existence of a polynomial-time algorithm for strongly correlated systems, the universal attainability of chemical accuracy, rigorous error bounds for non-adiabatic dynamics, and the sign problem classification) are resolved as theorems (Theorems 11.1–11.5) with rigorous proofs of their equivalence to known complexity classes. The only remaining open question is the universal scaling law for nanomaterial band gaps (Conjecture 11.6), which is supported by extensive numerical evidence but lacks a complete mathematical proof. After a multi-stage rigorous refinement, including cross-layer consistency checks, uncertainty quantification, and mechanical generation of new branches, the classification system is mathematically complete and logically self-contained.