Derivation of Hilbert space structure

Paul Busch, PEKKA J. LAHTI, Juha-Pekka Pellonpää +1 more

Fiona Mcgeough

From Structural Persistence to Quantum Probability: An SN Derivation of Hilbert-Space Measurement

David Sigtermans

We show that the mathematical structure of Hilbert space—typically postulated in quantum theory—emerges as a consequence of two fundamental principles derived within the TEQ framework: (1) entropy as a generative structural principle; (2) a minimal principle selecting stable distinctions. These principles, and the resulting entropy-weighted action functional, are derived from first principles in prior work. Here, we analyze the entropy-weighted path integral and the geometry of entropy curvature, and prove that the space of entropy-stabilized modes satisfies the axioms of a Hilbert space: linearity, inner product structure, norm completeness, and probabilistic interpretation. In this view, the Born rule itself emerges from entropy-weighted path selection. Importantly, this result shows that the TEQ framework fully recovers the standard Hilbert space formalism—not by postulate, but as a structural consequence of its core principles—thus preserving compatibility with quantum theory while providing it with a deeper generative foundation. Our results strengthen the explanatory power of the TEQ framework and offer a structural derivation of core quantum features from a common thermodynamic-geometric basis.

Carlo Alabiso, Ittay Weiss

Giacomo Mauro D’Ariano

In the present paper I show how it is possible to derive the Hilbert space formulation of Quantum Mechanics from a comprehensive definition of physical experiment and assuming experimental accessibility and simplicity as specified by five simple Postulates. This accomplishes the program presented in form of conjectures in the previous paper. Pivotal roles are played by the local observability principle, which reconciles the holism of nonlocality with the reductionism of local observation, and by the postulated existence of informationally complete observables and of a symmetric faithful state. This last notion allows one to introduce an operational definition for the real version of the “adjoint”—i. e. the transposition—from which one can derive a real Hilbert‐space structure via either the Mackey‐Kakutani or the Gelfand‐Naimark‐Segal constructions. Here I analyze in detail only the Gelfand‐Naimark‐Segal construction, which leads to a real Hilbert space structure analogous to that of (classes of generally unbounded) selfadjoint operators in Quantum Mechanics. For finite dimensions, general dimensionality theorems that can be derived from a local observability principle, allow us to represent the elements of the real Hilbert space as operators over an underlying complex Hilbert space (see, however, a still open problem at the end of the paper). The route for the present operational axiomatization was suggested by novel ideas originated from Quantum Tomography.

M. Dalla Chiara, Roberto Giuntini, Richard J. Greechie