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Physics Letters A

Reduced role of the wavefunctions' curvature of quantum potentials in non-standard quantum systems

Tomer Shushi

Published Apr 18, 2023• 5 citations
Publisher
DOI
10.1016/j.physleta.2023.128850

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We argue that the quantum wavefunction already encodes a fundamentally relational structure that has not been fully recognized in standard interpretations of quantum mechanics. Writing the wavefunction in polar form, \psi(x,t)=|\psi(x,t)|e^{i\theta(x,t)}, We show that well-established phenomena such as the Berry phase, the Aharonov–Bohm effect, and quantum entanglement naturally admit a unified interpretation in terms of relational holonomy and non-factorizable relational structure. In particular, geometric phases correspond to cycle holonomies in relational space, while entangled states represent irreducible relational configurations rather than correlations between independently existing objects. Building on this reinterpretation, we introduce an operational relational curvature proxy constructed from phase structure and argue that phase gradients can act as geometric sources beyond their standard kinematic role in quantum currents. We formulate conservative, testable predictions in engineered quantum systems, emphasizing deviations from standard Gross–Pitaevskii dynamics that scale quadratically with phase curvature rather than linearly with phase gradients. This work does not propose a modification of quantum mechanics at atomic scales. Instead, it identifies a previously overlooked relational layer already present in the wavefunction formalism, providing a conceptual and mathematical bridge between quantum mechanics, emergent geometry, and relational approaches to quantum gravity.
Bohmian mechanics offers a deterministic alternative to conventional quantum theory through well-defined particle trajectories. While successful in nonrelativistic contexts, its extension to curved spacetime-and hence quantum gravity-remains unresolved. Here, we develop a covariant extension of Bohmian mechanics in curved spacetime that removes the need for metric quantization. From a Lagrangian formulation, we derive a generalized guidance equation in which Bohmian trajectories generate hidden curvature, replacing metric superposition with a statistical ensemble constrained by Heisenberg uncertainty, offering a novel perspective on quantum gravity. Consequently, in our approach, measuring the gravitational potential at a point unveils a pre-existing trajectory and its associated curvature-a departure from the observer-centric paradigm of standard quantum mechanics-providing an alternative in which gravitational effects emerge from deterministic quantum trajectories rather than wavefunction collapse. Numerical simulations in Robertson-Walker and cigar soliton spacetimes reveal that while quantum interference is curvature-sensitive, Zitterbewegung remains invariant, distinguishing fundamental quantum effects. Moreover, deviations from the Born rule in inhomogeneous spacetimes are observed and suggest gravity-induced quantum non-equilibrium. This new approach has far-reaching implications for the role of determinism and potential observational signatures of quantum non-equilibrium in cosmology.
We propose an extension of general relativity and quantum mechanics in which the field of attention Φ(x)Φ(x), defined as the quantum expectation value of an operator O^O^, serves both as a source of spacetime curvature and as the driver of wavefunction collapse. Time is treated as a counting parameter for acts of signification, not as a fundamental flow. Mass is interpreted as local informational entropy. The model reduces to standard physics in the limit of vanishing attention. It predicts massive scalar gravitational modes (Yukawa correction to Newtonian potential), a geometry-dependent decoherence rate in double-slit experiments with macromolecules, and a cosmological role for ΦΦ as either dark energy or dark matter, depending on the mass parameter mm. The theory is falsifiable and offers concrete numerical predictions testable with current or near-future experiments.