The Geometry of Quantum Flow

James Thomas Wheeler

Stuart R. Hameroff

Abstract In neuroscience, the flow of time is a conscious experience produced by the brain. But in physics, time is either a process, or a dimension in four-dimensional spacetime geometry. Could all three explanations be correct? The Penrose–Hameroff ‘Orch OR’ theory suggests consciousness is a sequence of discrete, irreversible quantum state reductions occurring at an objective threshold (‘objective reduction’, ‘OR’) a process in fundamental spacetime geometry. These ‘self-collapses’ of the quantum wavefunction are ‘orchestrated’ in microtubules inside brain neurons. Each Orch OR event selects microtubule states which purposefully regulate neuronal functions, and provide sequences of ‘NOW’ moments of conscious experience. Connected to fundamental spacetime geometry by Penrose ‘OR’, consciousness is quantum state reduction, a set of irreversible steps which ‘ratchet forward’ in the fine scale geometry of the universe, creating a flow of time.

Julian Schwinger

Jincheng Zhang

Quantum mechanics traditionally focuses on dynamical variables such as energy, phase, and entanglement to characterize and manipulate quantum states. Recent advances in quantum information theory have revealed that the geometry of quantum state space, endowed with the Fubini–Study metric, carries rich structural information that can be exploited for control and resource theories. In this paper we propose a new dynamical framework, the Quantum State Curvature Flow (QSCF), which treats the curvature of a quantum state as an evolving quantity. We define a curvature scalar Kψ associated with a normalized state vector |ψ⟩ in projective Hilbert space, derived from the Riemannian curvature of the Fubini–Study metric. We then postulate a partial differential equation for the time evolution of Kψ, analogous to curvature flow equations in differential geometry, and examine its properties. The QSCF formalism uncovers novel invariants, links curvature dynamics to entanglement evolution, and suggests new strategies for continuous quantum control. While our work is purely theoretical, it opens a pathway toward a deeper understanding of quantum state geometry and its role in quantum technologies.

Luigi Accardi, Anilesh Mohari, Centro V. Volterra

Ariel Caticha

The mathematical formalism of quantum mechanics is derived or “reconstructed” from more basic considerations of the probability theory and information geometry. The starting point is the recognition that probabilities are central to QM; the formalism of QM is derived as a particular kind of flow on a finite dimensional statistical manifold—a simplex. The cotangent bundle associated to the simplex has a natural symplectic structure and it inherits its own natural metric structure from the information geometry of the underlying simplex. We seek flows that preserve (in the sense of vanishing Lie derivatives) both the symplectic structure (a Hamilton flow) and the metric structure (a Killing flow). The result is a formalism in which the Fubini–Study metric, the linearity of the Schrödinger equation, the emergence of complex numbers, Hilbert spaces and the Born rule are derived rather than postulated.