Information Theoretic Characterization of Physical Theories with Projective State Space

Hans Halvorson

Thanh Minh Nguyen

We propose a constraint-based formalism in which physical systems are characterized by operators acting on a state space. In this framework, constraint operators encode admissibility, projection, and structural restriction in a unified manner. We analyze the algebraic properties of these operators and show that, under natural conditions, they form a projection-based structure. This perspective provides a common structural language that accommodates diverse physical theories. In particular, projection operators in quantum mechanics may be interpreted as instances of constraint operators, and measurement processes may be viewed as constraint-driven transformations. In statistical and information-theoretic contexts, entropy can be interpreted as a coarse-grained measure of the size of constraint-compatible state spaces. The proposed formalism does not aim to replace existing theories, but to provide a minimal and theory-independent framework for describing how physical structure may arise from constraints. This work provides a structural bridge between constraint-based formulations and projection structures in quantum mechanics.

Gabriele Carcassi, Tobias Thrien, Christine A. Aidala

Reverse Physics is a methodology that breaks physical theories into separate mathematical and physical conditions to establish their logical relationships. To showcase the power of the methodology, we present several results for quantum mechanics and their related insights. The standard Hilbert-space formulation conflicts with basic physical requirements, while a minimal topological modification can solve these problems. The ensemble space, rather than the pure-state space, distinguishes classical from quantum systems. The Born rule is an additional assumption linking orthogonality, mutual exclusivity and information entropy. Under explicit background conditions, unitary evolution is equivalent to deterministic and reversible evolution. Nonselective projective measurements can be characterized as Lindblad equilibration processes, while unitary evolution can be characterized as a limit of infinitesimal projective processes. Classical mechanics is recovered as the high-entropy limit of quantum mechanics, and every quantum state, pure or mixed, is a dynamic, spectral and thermodynamic equilibrium. These results are self-contained, use the standard vector-space representation and can thus be used as common tools and constraints for teaching, interpretations, reconstructions and future theories.

A.I. Trinity Lab

This work introduces Unified Relational Projection Dynamics (URPD), a minimal information-theoretic framework in which physical states are represented as relational frames in the complex projective state space CPN−1\mathbb{C}P^{N-1}CPN−1.The theory defines evolution as entropy-maximizing transitions between frames under constraints of informational displacement, providing a unified geometric origin for quantum probabilities, temporal flow, and causality.URPD naturally ensures information preservation through non-vanishing frame overlaps induced by the Fubini–Study metric, offering a non-holographic explanation of the Page curve and black-hole information retention.This framework establishes a compact, relational, and geometry-based foundation for quantum mechanics and spacetime structure.

Markus P. Müller, Lluís Masanes

Rob Clifton, Jeffrey Bub, Hans Halvorson