After completion of our recent work on dissipative quantum hydrodynamics coupled to general relativity we became aware of related results by Turski, who provides a mathematically consistent formulation of dissipation within quantum dynamics. This establishes that dissipation at the quantum level can be treated as a fundamental and well-posed modification rather than a phenomenological artifact. In this article we clarify how this mathematical foundation enables a closed physical framework when combined with a geometric interpretation of dissipation.We argue that dissipation in quantum mechanics enables the formation of localized matter structures, while dissipation into spacetime geometry produces volumetric expansion that dilutes dissipative effects and preserves global consistency.
After completion of our recent work on dissipative quantum hydrodynamics coupled to general relativity we became aware of related results by Turski, who provides a mathematically consistent formulation of dissipation within quantum dynamics. This establishes that dissipation at the quantum level can be treated as a fundamental and well-posed modification rather than a phenomenological artifact. In this article we clarify how this mathematical foundation enables a closed physical framework when combined with a geometric interpretation of dissipation.We argue that dissipation in quantum mechanics enables the formation of localized matter structures, while dissipation into spacetime geometry produces volumetric expansion that dilutes dissipative effects and preserves global consistency.
Abstract In general relativity, the dynamics of objects is governed by the curvature of spacetime, which is caused by the presence of matter and energy. In contrast, in quantum mechanics, the dynamics is governed by the wavefunction, which completely describes the behavior of the particles. There is an ongoing effort to explore analogs of space and spacetime curvature in the context of quantum mechanics. Such analogies may reveal a deeper structure of quantum reality and its possible relations with Einstein’s theory of gravity. In this note, by coupling the non-relativistic Schrödinger equation with the heat equation and using the hydrodynamical formulation of quantum mechanics, we find that the dynamics of the particle is fully characterized by the normalized curvature of the wavefunction’s amplitude. Such a curvature obtains an analogy to the Ricci curvature of curved space in a Riemannian manifold. The proposed geometric correspondence provides a new pathway to explore quantum dynamics through the lens of differential geometry, the language of general relativity.
AbstractThis work presents a unified geometric framework for quantum reality through four interconnected papers. It proposes that the coherent, non-separable correlations described by Hilbert space originate from an unobservable Energy Space, a fundamental substrate beyond spacetime that preserves perfect coherence and all conservation laws, thereby providing a geometric interpretation of quantum entanglement. Within this framework, the relation E = hf is interpreted as an encoding relation rather than a statement of physical energy actualization. During measurement, structural branching may occur, but Energy Space ensures that only one geometrically valid branch undergoes physical localization. Spacetime emerges from Energy Space as a passive geometric medium with intrinsic dynamic capabilities, giving rise to quantum objects and their interactions.Quantum objects are interpreted as dynamic geometric patterns of spacetime possessing an encoded geometric identity. Their physical properties, including mass, charge, spin, momentum, energy, wavelength, and phase, are fundamentally encoded in Energy Space and become physically actualized according to the interaction context and the degree of coherence or decoherence. The stability, transformation, and propagation of quantum objects are determined by their geometric compatibility with spacetime and with other quantum geometries.Electromagnetic, weak, and strong interactions are interpreted as direct interactions between compatible quantum-object geometries, whereas gravity is regarded as an indirect geometric interaction mediated through spacetime geometry. Geometric interactions may produce superposition, transformation into new quantum-object geometries, or the formation of larger composite geometric structures, providing a unified geometric interpretation of interactions, confinement, and composite systems.Within this geometric framework, a recursive geometric scaling parameter satisfying the invariant relation is proposed. Based on this hierarchy, phenomenological recursive scaling relations are developed for the lepton and quark families, suggesting a possible underlying geometric structure governing fermion masses, generational hierarchy, and relative stability.Finally, the Peak-Coupling Theory proposes that quantum localization and decoherence occur through discrete peak-coupling events between interacting quantum geometries at oscillatory extrema within a quantum wave packet. The wave envelope and phase continuously modulate the distribution of possible coupling events, while observable interaction probabilities may emerge from the squared magnitude of local coupling amplitudes, providing a possible geometric route toward the emergence of Born-rule probabilitiesTogether, these four papers present a unified conceptual framework in which Energy Space, spacetime, quantum objects, physical interactions, particle hierarchies, quantum measurement, and localization are interpreted as interconnected manifestations of an underlying geometric reality.These ideas are developed across four papers: Paper 1: Energy Space and Geometric Nature of Quantum Objects and Spacetime Paper 2: Quantum Geometry Paper 3: Lepton and Quark Generational Phenomenological Patterns; and Paper 4: Peak-Coupling Theory of Quantum Interaction
We present a geometric formulation of quantum mechanics based on the symplectic structure of the projective Hilbert space. Building upon the standard Kähler framework, we introduce an extension in which the symplectic structure is allowed to couple to a metric-affine background geometry, leading to a deformation of the Hamiltonian flow on the state space.
We show that, under suitable conditions, the deformed structure remains symplectic and defines a well-posed Hamiltonian system. The formulation reduces to standard Schrödinger dynamics in the limit where the geometric deformation vanishes.
Explicit analytical examples are constructed to illustrate the effect of the deformation. In particular, curvature-dependent deformations lead to a rescaling of Hamiltonian flows, while torsion-induced contributions produce direction-dependent corrections. In addition, geometric phases acquire corrections determined by the deformed symplectic structure.
These results provide a mathematically consistent framework for exploring geometric modifications of quantum evolution induced by background curvature and affine structure.
Abstract This work presents a unified geometric framework for quantum reality through four interconnected papers. It proposes that the coherent, non-separable correlations described by Hilbert space originate from an unobservable Energy Space, a fundamental substrate beyond spacetime that preserves perfect coherence and all conservation laws, thereby providing a geometric interpretation of quantum entanglement. Within this framework, the relation E = hf is interpreted as an encoding relation rather than a statement of physical energy actualization. During measurement, structural branching may occur, but Energy Space ensures that only one geometrically valid branch undergoes physical localization. Spacetime emerges from Energy Space as a passive geometric medium with intrinsic dynamic capabilities, giving rise to quantum objects and their interactions. Quantum objects are interpreted as dynamic geometric patterns of spacetime possessing an encoded geometric identity. Their physical properties, including mass, charge, spin, momentum, energy, wavelength, and phase, are fundamentally encoded in Energy Space and become physically actualized according to the interaction context and the degree of coherence or decoherence. The stability, transformation, and propagation of quantum objects are determined by their geometric compatibility with spacetime and with other quantum geometries. Electromagnetic, weak, and strong interactions are interpreted as direct interactions between compatible quantum-object geometries, whereas gravity is regarded as an indirect geometric interaction mediated through spacetime geometry. Geometric interactions may produce superposition, transformation into new quantum-object geometries, or the formation of larger composite geometric structures, providing a unified geometric interpretation of interactions, confinement, and composite systems. Within this geometric framework, a recursive geometric scaling parameter satisfying the invariant relation is proposed. Based on this hierarchy, phenomenological recursive scaling relations are developed for the lepton and quark families, suggesting a possible underlying geometric structure governing fermion masses, generational hierarchy, and relative stability. Finally, the Peak-Coupling Theory proposes that quantum localization and decoherence occur through discrete peak-coupling events between interacting quantum geometries at oscillatory extrema within a quantum wave packet. The wave envelope and phase continuously modulate the distribution of possible coupling events, while observable interaction probabilities may emerge from the squared magnitude of local coupling amplitudes, providing a possible geometric route toward the emergence of Born-rule probabilities Together, these four papers present a unified conceptual framework in which Energy Space, spacetime, quantum objects, physical interactions, particle hierarchies, quantum measurement, and localization are interpreted as interconnected manifestations of an underlying geometric reality. These ideas are developed across four papers: Paper 1: Energy Space and Geometric Nature of Quantum Objects and SpacetimePaper 2: Quantum GeometryPaper 3: Lepton and Quark Generational Phenomenological Patterns; and Paper 4: Peak-Coupling Theory of Quantum Interaction
This work investigates the emergence of state-dependent dissipative structures in open quantum systems and develops a geometric framework for constructing physically admissible non-linear extensions of Markovian quantum dynamics. Building upon the coarse-grained completely positive generators developed in previous work, we consider the effects of finite and dynamically polarizable environments, leading naturally to state-dependent Kossakowski structures and non-linear dissipative flows. To constrain such non-linear dynamics and prevent the appearance of unphysical states, we formulate the evolution as a gradient flow on the manifold of quantum states endowed with the Bures metric. A locality-preserving Extension Consistency Condition is introduced to ensure the admissibility of non-linear lifts to composite systems and to eliminate non-local ancillary back-action. Within this framework, we establish a Boundary Repulsion Theorem showing that the rank-deficient boundary of the positive state manifold is located at an infinite geometric distance along the flow trajectories. As a consequence, positivity of the density operator is preserved for all finite evolution times. A worked example based on a three-level V-system demonstrates the regularizing effect of the Bures geometry in the presence of state-dependent dissipation. The results suggest that information-geometric structures may provide a natural route toward physically consistent non-linear quantum master equations while preserving the operational integrity of quantum state evolution.
In recent years trajectory based methodologies have become increasingly popular for evaluating the time evolution of quantum systems. A revival of the de Broglie Bohm interpretation of quantum mechanics has spawned several such techniques for examining quantum dynamics from a hydrodynamic perspective. Using techniques similar to those found in computational fluid dynamics one can construct the wavefunction of a quantum system at any time from the trajectories of a discrete ensemble of hydrodynamic fluid elements (Bohm particles) which evolve according to non-classical equations of motion. Until very recently these schemes have been limited to closed systems. In this paper we present our methodology for including the effects of a thermal environment into the hydrodynamic formulation of quantum dynamics. We derive hydrodynamic equations of motion from the Caldeira-Leggett master equation for the reduced density matrix and give a brief overview of our computational scheme which incorporates an adaptive Lagrangian mesh. Our applications focus upon the dissipative dynamics of open unbounded quantum systems. Using both the Wigner phase space representation and the linear entropy we probe the breakdown of the Markov approximation of the bath dynamics at low temperatures. We suggest a criteria for rationalizing the validity of the Markov approximation in open unbound systems and discuss decoherence, energy relaxation, and quantum/classical correspondence in the context of the Bohmian paths.