Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

On Measurement and Invariant Structure in Quantum Mechanics

Fredrik Jensen

Published Feb 12, 2026• 0 citations
Publisher

Abstract

This paper examines measurement and representational limits within a geometric formulation of quantum mechanics based on connection geometry and holonomy. Measurement is interpreted as projection onto invariant quantities associated with comparison of states under parallel transport. This projection preserves invariant phase relations while discarding non-invariant information. As a result, measurement outcomes appear discrete and probabilistic at the representational level. Correlations between subsystems arise from shared geometric structure and exhibit nonlocal consistency without introducing additional dynamical mechanisms. The analysis clarifies how measurement and classical behavior can be understood within a geometric account of quantum kinematics.

DOI
10.5281/zenodo.18623394

Related papers

On Amplitudes, Interference, and Probability in Quantum Mechanics

Fredrik Jensen

This paper examines the geometric structure underlying amplitudes, interference, and probability in quantum mechanics. Building on the interpretation of quantum phase as holonomy, amplitudes are analyzed in terms of parallel transport in complex line bundles. Interference is interpreted as arising from the composition of alternative transport paths, and probability emerges from the invariant comparison of amplitudes under phase transformations. Under general assumptions of linearity and phase invariance, the squared-norm structure associated with probability is shown to arise naturally from consistency conditions on amplitude composition. The analysis focuses on kinematic structure and does not assume specific dynamical models.

•Feb 12, 2026•0 citations

Action, Holonomy, and the Geometry of Quantum Phase

Fredrik Jensen

Quantum mechanics is formulated in terms of complex phase and action-dependent amplitudes. In this work, quantum phase is identified with the holonomy of a geometric connection, and Planck's constant appears as the conversion scale between geometric transport and physical action. This provides a geometric formulation of quantum kinematics in which phase accumulation, noncommutativity, and quantization arise naturally from connection geometry and topology. This paper is Part I of a series which develops a geometric framework for quantum mechanics based on holonomy and transport structure. Subsequent papers develop operator representations and dynamical structure. This Zenodo record establishes priority and archival timestamp. Public release via arXiv is in progress.

•Feb 11, 2026•0 citations

On the Geometric Structure of State Space and Operators in Quantum Mechanics

Fredrik Jensen

This paper examines the geometric structure underlying quantum state space and operator representations. Building on the interpretation of quantum phase in terms of holonomy and parallel transport, Hilbert space is analyzed as a representation of transport structure in complex line and vector bundles. Quantum states are interpreted as equivalence classes under phase transformations, and operators arise as generators of infinitesimal transformations associated with the connection. The construction provides a geometric perspective on state space, inner products, and operator structure within standard quantum mechanics.

•Feb 12, 2026•0 citations

Composite Systems, Holonomy, and Decoherence

Fredrik Jensen

This paper analyzes composite quantum systems from the perspective of bundle connections and holonomy. A geometric framework for composing subsystems is developed, and the behavior of phase information under restriction to subsystems is examined. It is shown that when access is limited to a subsystem, interference terms associated with relative phase can be suppressed in the reduced description, leading to additive weights for distinguishable components. This provides a geometric account of decoherence and the emergence of effectively classical behavior in composite systems. The analysis is formulated in terms of connection geometry and holonomy structure.

•Feb 12, 2026•0 citations

Classical and quantum mechanics across representations: an operational reading of the Wigner Weyl correspondence

Samuel Schlegel, Borivoje Dakić, Flavio Del Santo

The physical content of a theory is not intrinsically tied to any single mathematical formalism. Both classical and quantum mechanics admit equivalent representations, notably in phase space and in Hilbert space, related by the Wigner-Weyl correspondence. While this correspondence has long been studied in mathematical physics, its foundational and operational implications are often left implicit. Here we give a systematic account of what changes, and what does not, when classical and quantum theories are expressed in each other's native language. This representational viewpoint separates artifacts (such as the appearance of non-positivity or negativity under certain maps) from robust structural distinctions that persist across representations, in particular noncommutativity and its $\hbar$-dependent $\star$-deformation of the classical algebra. We develop the comparison at the level of states, kinematics, and dynamics, and extend it to measurement by formulating both outcome statistics and state-update rules within the same framework.

quant-pharXiv:2607.13135•Physical Review A•Jul 16, 2026•0 citations

Quantum Mechanics as Constraint Geometry: A Structural Interpretation of Superposition, Measurement, and Nonlocality

Denis Bailey

Quantum mechanics is mathematically complete but conceptually burdened by a discrete‑state ontology that the formalism itself does not require. This paper reframes superposition, measurement, and nonlocality as structural features of unresolved and resolved constraint geometry in continuous fields. In this interpretation, a quantum state is not a catalogue of simultaneous possibilities but the orientation of an unresolved constraint region; measurement is the stabilization of that region under new constraints; and entanglement reflects a single unresolved region spanning multiple subsystems. The standard Hilbert‑space formalism remains unchanged—linearity, projection, the Born rule, and tensor structure all arise naturally from the geometry of unresolved continuity. This framework dissolves the paradoxes associated with collapse, wavefunction ontology, and nonlocal influence, yielding a unified, minimal, and physically intuitive account of quantum behavior that preserves the full empirical power of quantum mechanics while eliminating its metaphysical burdens.

•Mar 18, 2026•0 citations
This paper examines measurement and representational limits within a geometric formulation of quantum mechanics based on connection geometry and holonomy. Measurement is interpreted as projection onto invariant quantities associated with comparison of states under parallel transport. This projection preserves invariant phase relations while discarding non-invariant information. As a result, measurement outcomes appear discrete and probabilistic at the representational level. Correlations between subsystems arise from shared geometric structure and exhibit nonlocal consistency without introducing additional dynamical mechanisms. The analysis clarifies how measurement and classical behavior can be understood within a geometric account of quantum kinematics.
This paper examines the geometric structure underlying amplitudes, interference, and probability in quantum mechanics. Building on the interpretation of quantum phase as holonomy, amplitudes are analyzed in terms of parallel transport in complex line bundles. Interference is interpreted as arising from the composition of alternative transport paths, and probability emerges from the invariant comparison of amplitudes under phase transformations. Under general assumptions of linearity and phase invariance, the squared-norm structure associated with probability is shown to arise naturally from consistency conditions on amplitude composition. The analysis focuses on kinematic structure and does not assume specific dynamical models.
Quantum mechanics is formulated in terms of complex phase and action-dependent amplitudes. In this work, quantum phase is identified with the holonomy of a geometric connection, and Planck's constant appears as the conversion scale between geometric transport and physical action. This provides a geometric formulation of quantum kinematics in which phase accumulation, noncommutativity, and quantization arise naturally from connection geometry and topology. This paper is Part I of a series which develops a geometric framework for quantum mechanics based on holonomy and transport structure. Subsequent papers develop operator representations and dynamical structure. This Zenodo record establishes priority and archival timestamp. Public release via arXiv is in progress.
This paper examines the geometric structure underlying quantum state space and operator representations. Building on the interpretation of quantum phase in terms of holonomy and parallel transport, Hilbert space is analyzed as a representation of transport structure in complex line and vector bundles. Quantum states are interpreted as equivalence classes under phase transformations, and operators arise as generators of infinitesimal transformations associated with the connection. The construction provides a geometric perspective on state space, inner products, and operator structure within standard quantum mechanics.
This paper analyzes composite quantum systems from the perspective of bundle connections and holonomy. A geometric framework for composing subsystems is developed, and the behavior of phase information under restriction to subsystems is examined. It is shown that when access is limited to a subsystem, interference terms associated with relative phase can be suppressed in the reduced description, leading to additive weights for distinguishable components. This provides a geometric account of decoherence and the emergence of effectively classical behavior in composite systems. The analysis is formulated in terms of connection geometry and holonomy structure.
The physical content of a theory is not intrinsically tied to any single mathematical formalism. Both classical and quantum mechanics admit equivalent representations, notably in phase space and in Hilbert space, related by the Wigner-Weyl correspondence. While this correspondence has long been studied in mathematical physics, its foundational and operational implications are often left implicit. Here we give a systematic account of what changes, and what does not, when classical and quantum theories are expressed in each other's native language. This representational viewpoint separates artifacts (such as the appearance of non-positivity or negativity under certain maps) from robust structural distinctions that persist across representations, in particular noncommutativity and its ℏ\hbarℏ-dependent ⋆\star⋆-deformation of the classical algebra. We develop the comparison at the level of states, kinematics, and dynamics, and extend it to measurement by formulating both outcome statistics and state-update rules within the same framework.
Quantum mechanics is mathematically complete but conceptually burdened by a discrete‑state ontology that the formalism itself does not require. This paper reframes superposition, measurement, and nonlocality as structural features of unresolved and resolved constraint geometry in continuous fields. In this interpretation, a quantum state is not a catalogue of simultaneous possibilities but the orientation of an unresolved constraint region; measurement is the stabilization of that region under new constraints; and entanglement reflects a single unresolved region spanning multiple subsystems. The standard Hilbert‑space formalism remains unchanged—linearity, projection, the Born rule, and tensor structure all arise naturally from the geometry of unresolved continuity. This framework dissolves the paradoxes associated with collapse, wavefunction ontology, and nonlocal influence, yielding a unified, minimal, and physically intuitive account of quantum behavior that preserves the full empirical power of quantum mechanics while eliminating its metaphysical burdens.