Which Space Is Real? A Review of Mamun's "Algebraic Quantum Mechanics and Hilbert Space Realism: The Ontological and Epistemological Nexus (Vol-I)

This review examines Dr. Syed Muntasir Mamun’s paper “Algebraic Quantum Mechanics and Hilbert Space Realism: The Ontological and Epistemological Nexus (Vol-I)” (2026). The paper asks what the mathematics of quantum theory represents. It explains the Gelfand–Naimark–Segal (GNS) construction, by which a Hilbert space representation is generated from an abstract C*-algebra of observables and a state; discusses the interpretive problem posed by unitarily inequivalent representations in systems with infinitely many degrees of freedom, drawing on Ruetsche’s Coalescence Approach; and compares positions that treat different mathematical structures as real, from Algebraic Imperialism to Wavefunction Realism, Spacetime State Realism and Hilbert Space Realism (“Mad-Dog Everettianism”). It closes with phenomenological critiques and directions for future research. This review identifies three strengths: a clear exposition of the GNS construction and its philosophical significance; a well-organised map of competing ontologies; and a useful agenda for future research. It then raises four points for further discussion: a few technical statements that could be made more precise; the balance between the positions discussed; the choice of sources; and presentation. It closes with broader implications for interdisciplinary work in the foundations of physics.