Quantum geometry from coordinate transformations relating quantum observers

Thomas Thiemann

It is mandatory to know how to operationally define and translate a reference frame into mathematics, in order that a physical interpretation of theory calculations in terms of observational data is possible. The situation is particularly challenging for gauge systems such as General Relativity where spacetime coordinates are subject to spacetime diffeomorphisms considered as gauge transformations turning coordinates into non-observables. This motivates the idea of operationally defined (material) reference frames which specify coordinates in terms of matter or geometry reference fields leading to the concept of relational observables, relational reference frames and gauge reduction. Upon quantisation, all fields become operator valued distributions. Now new conceptual and technical questions arise such as: Should one reduce before or after quantisation and how are the reference fields quantised respectively in either route? Is a reference frame itself subject to quantisation and how are different quantum reference frames related? How does the gauge reduction fit into this, i.e. how can it be that a certain reference field is considered a non-observable in one reference frame and an observable in another which upon quantisation even displays fluctuations? How precisely are gauge dependent fields interpreted in terms of the relational observables in a given reference frame? What is the relative dynamics, e.g. how exactly are physical Hamiltonians of two relational reference frames related? The present conceptual work addresses these and related questions in a non-perturbative field theory context of sufficient generality to cover General Relativity coupled to standard matter. A central role is played by the concept of the relational reference frame transformation (RRFT) for which a general formula is derived and its properties are explored.

James Thomas Wheeler

Veeravalli S. Varadarajan

Boris Leaf

Geometrical properties of space are described by relations between quantum observables, functions of generalized coordinate and momentum operators of a physical system. Space comprises the set of eigenvalues of the coordinate observables, which may be a discrete set of points instead of the continuum of classical geometry. Differentiation on coordinates and momenta is defined by commutation operations. Well-known formulas from classical geometry for gradient and divergence operators, connections, covariant differentiation, tangent vectors, geodesics, and the Riemann curvature tensor are derived as relations between quantum operators. The quantum formulation retains the symmetry between coordinate and momentum representations required by the principle of complementarity.

Veeravalli S. Varadarajan

Mortaza Zabihi

The transition from the Organization-Based Framework to the Constraint-Based Framework requires a fundamental reinterpretation of geometric description. In Phase 1, physical systems were reconstructed through the Organizational Coordinate System, where the execution vector χ represented the continuous realization of recursive organization. This geometry provided a complete generative description of how stable physical structures emerge through recursive execution. However, the generative geometry of recursive execution is not identical to the geometry directly accessible through observation. The reason is not that the Organizational Coordinate System becomes incomplete or incorrect. Rather, recursive execution contains multiple complementary aspects that are not simultaneously preserved within a single observable projection. This paper introduces the concept of Complementary Projection Spaces as the geometric foundation of the Constraint-Based Framework. The central argument is that the execution vector χ contains two fundamental components: the execution component associated with continuous recursive progression, and the organizational component associated with the constraints preserved during that progression. Projection does not eliminate either component. Instead, it transforms the observer’s geometric perspective by making different aspects of recursive execution explicitly observable. The organizational component of χ gives rise to the Quantum Projection Space, where projected constraints and localized organizational realizations become the primary observable structures. Within this geometry, continuous execution remains implicit, and time appears only as an ordering relation between projected realizations. The execution component of χ gives rise to the Relative Projection Space, where recursive progression and relational continuity become explicitly represented. Within this geometry, time is no longer merely an ordering parameter but becomes a fundamental component of observable relational structure. The existence of these complementary projection spaces provides a geometric explanation for the apparent conceptual difference between quantum mechanics and relativity. They do not represent separate physical realities or independent generative mechanisms. Instead, they represent different observable geometries obtained from the same underlying recursive execution. The introduction of Complementary Projection Spaces therefore establishes the geometric foundation of Phase 2. The Organizational Coordinate System remains the hidden generative geometry from which physical systems emerge. The Quantum Projection Space and Relative Projection Space become complementary observable geometries through which constraint-based descriptions of quantum and relativistic systems can be constructed. This article is part of a broader theoretical program investigating the emergence of spacetime, geometry, and physical structure from the ψ–Θ–χ continuum. Earlier works in this program explore different layers and stages of the framework, ranging from foundational pre-structural regimes to geometric realization: Generative Paper 1: Infinite Generative Field as the Origin of Physical Reality Generative Paper 2: Vacuum as the Minimal Condition of Definability Generative Paper 3: Primitive Recursive Functions as the Architecture of Recursive Organization Generative Paper 4: From Recursive Organization to the ψ–Θ–χ Architecture Generative Paper 5: Organizational Space and Organizational Fields Generative Paper 6: Elementary Fiber Dynamics — The Building Block of Photon Formation Generative Paper 7: Persistent Cyclic Organization Toward Photon Formation Generative Paper 8: Photon Dynamics Through Cyclic Reconstruction Generative Paper 9: From Photon to Electromagnetism Generative Review 1: From a Single Generative Field to Photon and Electromagnetism Generative Paper 10: Organizational Constraints: The Foundation of the Constraint-Based Framework Generative Paper 11 — Constraint Architecture in Generative Framework Generative Paper 12 — Quantum and Relative Projection Spaces: From Organizational Geometry to Quantum & Relative Geometry Generative Paper 13 — Vector Point & Projected History: Architecture of Quantum Projection Geometry Generative Paper 14 — Presentation-Level Domain Constraint: From Symmetry through Hilbert Space & State Space to Quantum State Generative Paper 15 — Presentation-Level Mapping Constraint: From Operational Rules to Measurement