This document presents a structured exposition of the mathematical framework underlying quantum mechanics. The text revisits the emergence of Hilbert spaces as the natural setting for quantum theory and discusses the development of the subject through the ideas of Hilbert, Fréchet, and von Neumann. Particular attention is given to the sequence space $\ell^2$ , orthogonal decompositions, completeness, and the role of self-adjoint operators and spectral theory in the mathematical formulation of quantum mechanics. The aim of the document is to provide a clear and rigorous reference text for students approaching the mathematical structures of quantum theory. The presentation emphasizes conceptual clarity while maintaining the formal precision required in functional analysis and mathematical physics.This document presents a structured exposition of the mathematical framework underlying quantum mechanics. The text revisits the emergence of Hilbert spaces as the natural setting for quantum theory and discusses the development of the subject through the ideas of Hilbert, Fréchet, and von Neumann. Particular attention is given to the sequence space $\ell^2$, orthogonal decompositions, completeness, and the role of self-adjoint operators and spectral theory in the mathematical formulation of quantum mechanics. The aim of the document is to provide a clear and rigorous reference text for students approaching the mathematical structures of quantum theory. The presentation emphasizes conceptual clarity while maintaining the formal precision required in functional analysis and mathematical physics.

Mar 6, 20260 citations

Reconstruction Physics: A Mathematical Framework for Green Functions, Hilbert Spaces, Gauge Transport, and Emergent Geometry

Borros Arneth

Modern theoretical physics is formulated using several highly successful but conceptually distinct mathematical structures. Quantum mechanics is based on Hilbert spaces and linear operators, quantum field theory is formulated through Green functions and propagators, gauge theories rely on fiber bundles and holonomy, while general relativity describes gravitation through differential geometry. Although these mathematical frameworks are deeply interconnected, they are traditionally introduced as independent axiomatic constructions. In this work, we propose a unified mathematical reconstruction framework in which Green functions, Hilbert spaces, gauge transport, and effective geometry emerge from a single primitive transport architecture. The fundamental object is neither a wavefunction nor a spacetime manifold, but an operator-valued reconstruction architecture consisting of directed transport graphs, interaction vertices, closure relations, and recursive consistency maps. Green operators arise naturally as inverse transport operators, Hilbert spaces appear as completions of admissible architectural amplitudes, gauge fields emerge from holonomy of closed transport loops, and effective geometry is reconstructed as the continuum limit of transport consistency. A central Reconstruction Theorem is formulated, establishing that every admissible reconstruction architecture induces Green operators, Hilbert reconstruction spaces, holonomy observables, effective gauge connections, and continuum geometry. As an application, a discrete set of architectural quantum numbers (a,b,c,d) is introduced to characterize propagation, interaction, topology, and recursion. A phenomenological mass reconstruction formula is subsequently derived as an application rather than a fundamental assumption. The proposed framework does not replace established physical theories. Instead, it offers a common mathematical origin from which several of their fundamental structures arise naturally. The work therefore represents a structural reformulation of theoretical physics based on operator-valued transport architectures.Modern theoretical physics is formulated using several highly successful but conceptually distinct mathematical structures. Quantum mechanics is based on Hilbert spaces and linear operators, quantum field theory is formulated through Green functions and propagators, gauge theories rely on fiber bundles and holonomy, while general relativity describes gravitation through differential geometry. Although these mathematical frameworks are deeply interconnected, they are traditionally introduced as independent axiomatic constructions. In this work, we propose a unified mathematical reconstruction framework in which Green functions, Hilbert spaces, gauge transport, and effective geometry emerge from a single primitive transport architecture. The fundamental object is neither a wavefunction nor a spacetime manifold, but an operator-valued reconstruction architecture consisting of directed transport graphs, interaction vertices, closure relations, and recursive consistency maps. Green operators arise naturally as inverse transport operators, Hilbert spaces appear as completions of admissible architectural amplitudes, gauge fields emerge from holonomy of closed transport loops, and effective geometry is reconstructed as the continuum limit of transport consistency. A central Reconstruction Theorem is formulated, establishing that every admissible reconstruction architecture induces Green operators, Hilbert reconstruction spaces, holonomy observables, effective gauge connections, and continuum geometry. As an application, a discrete set of architectural quantum numbers (a,b,c,d) is introduced to characterize propagation, interaction, topology, and recursion. A phenomenological mass reconstruction formula is subsequently derived as an application rather than a fundamental assumption. The proposed framework does not replace established physical theories. Instead, it offers a common mathematical origin from which several of their fundamental structures arise naturally. The work therefore represents a structural reformulation of theoretical physics based on operator-valued transport architectures.

Jul 1, 20260 citations

The Mathematical Structure of Quantum Mechanics

Philip L. Bowers

Cambridge University Press eBooksAug 27, 20200 citations

03 - Emergence of Quantum Mechanics and Born Measure from a Coherence-Driven TwoField Discrete Framework

Enrico Ferrero

This series presents a structured and logically sequential formulation of a discrete informational framework governed by global coherence constraints. Although it originates from the same research program developed in the earlier exploratory preprint series (P-series), the works presented here constitute an autonomous and self-contained series, organized in a progressively axiomatic form and focused on the explicit derivation of the framework’s effective structures. The goal of the construction is to investigate whether structures commonly associated with quantum mechanics, thermodynamics, and geometry may emerge from a discrete relational substrate without being introduced as fundamental postulates. Each article addresses a specific level of the theoretical derivation and forms part of a progressive logical structure based on results established in the preceding works of the series. This article develops the emergence of Hilbert-space structure and Schrödinger-type effective dynamics from the coherent two-field sector established in Paper 02. The work demonstrates that the global coherence constraints imposed on the discrete informational substrate naturally generate an effective complex representation endowed with linear superposition properties and unitary-preserving evolution. The analysis shows that the two-component coherent sector admits a mathematically stable reformulation equivalent to a Hilbert-space structure, and that the effective evolution of the coherent amplitudes leads to a Schrödinger-type dynamical equation without introducing quantum mechanics as an initial assumption. The article further derives the probabilistic interpretation associated with the effective amplitudes and clarifies how quantum-like statistical behavior emerges from the internal coherence conditions of the underlying discrete system. No quantum postulates are assumed at the foundational level. The purpose of the article is to demonstrate that the formal structures associated with quantum mechanics can emerge as effective consequences of coherence-preserving informational dynamics. The article does not assume quantum mechanics as a foundational theory and does not postulate the Hilbert-space formalism a priori. Instead, the work demonstrates that, within the internal hypotheses and coherence constraints of the proposed discrete informational framework, structures mathematically equivalent to Hilbert-space representations and Schrödinger-type evolution can arise as effective emergent descriptions. The purpose of the derivation is therefore not to reproduce standard quantum mechanics by assumption, but to show that quantum-like formal structures may consistently emerge from deeper coherence-preserving informational dynamics.This series presents a structured and logically sequential formulation of a discrete informational framework governed by global coherence constraints. Although it originates from the same research program developed in the earlier exploratory preprint series (P-series), the works presented here constitute an autonomous and self-contained series, organized in a progressively axiomatic form and focused on the explicit derivation of the framework’s effective structures. The goal of the construction is to investigate whether structures commonly associated with quantum mechanics, thermodynamics, and geometry may emerge from a discrete relational substrate without being introduced as fundamental postulates. Each article addresses a specific level of the theoretical derivation and forms part of a progressive logical structure based on results established in the preceding works of the series. This article develops the emergence of Hilbert-space structure and Schrödinger-type effective dynamics from the coherent two-field sector established in Paper 02. The work demonstrates that the global coherence constraints imposed on the discrete informational substrate naturally generate an effective complex representation endowed with linear superposition properties and unitary-preserving evolution. The analysis shows that the two-component coherent sector admits a mathematically stable reformulation equivalent to a Hilbert-space structure, and that the effective evolution of the coherent amplitudes leads to a Schrödinger-type dynamical equation without introducing quantum mechanics as an initial assumption. The article further derives the probabilistic interpretation associated with the effective amplitudes and clarifies how quantum-like statistical behavior emerges from the internal coherence conditions of the underlying discrete system. No quantum postulates are assumed at the foundational level. The purpose of the article is to demonstrate that the formal structures associated with quantum mechanics can emerge as effective consequences of coherence-preserving informational dynamics. The article does not assume quantum mechanics as a foundational theory and does not postulate the Hilbert-space formalism a priori. Instead, the work demonstrates that, within the internal hypotheses and coherence constraints of the proposed discrete informational framework, structures mathematically equivalent to Hilbert-space representations and Schrödinger-type evolution can arise as effective emergent descriptions. The purpose of the derivation is therefore not to reproduce standard quantum mechanics by assumption, but to show that quantum-like formal structures may consistently emerge from deeper coherence-preserving informational dynamics.

May 13, 20260 citations

The Mathematical Structure of Elementary Quantum Mechanics

J. M. Jauch

Springer eBooksJan 1, 19731 citation

\ell^2

, orthogonal decompositions, completeness, and the role of self-adjoint operators and spectral theory in the mathematical formulation of quantum mechanics. The aim of the document is to provide a clear and rigorous reference text for students approaching the mathematical structures of quantum theory. The presentation emphasizes conceptual clarity while maintaining the formal precision required in functional analysis and mathematical physics.