A Geometric Substructure for Quantum Dynamics

TL;DR

The paper proposes a geometric substructure for quantum dynamics in which, between measurements, a state point moves with constant velocity in an underlying complex space, linking $|\,\psi\rangle$ to a trajectory governed by $v^i=d z^i/dt$ and $z^i(t)=z_0^i+v^i(t-t_0)$. Upon measurement, the direction changes to an orthogonal velocity $w_K^{\,i}=V\,\beta_K^{\,i}$ with Born probabilities $P_K=\sum_i|\overline{\beta_K^{\,i}}\alpha^i|^2$, producing a mixture that collapses to a definite trajectory when observed. The framework is extended to a complex Riemannian (potentially Kähler) manifold for $S$, with $g_{ij}=\partial^2 B/\partial \bar{z}^i \partial z^j$ and geodesic motion guiding state evolution, suggesting curvature could influence dynamics. A further coupling to spacetime via a non-factorizable metric $g_{ij\mu\nu}$ implies entropy-driven changes in the environment could interact with gravity, offering a route toward a unified view of quantum dynamics and gravitation that warrants future investigation.

Abstract

The description of a closed quantum system is extended with the identification of an underlying substructure enabling an expanded formulation of dynamics in the Heisenberg picture. Between measurements a ``state point" moves in an underlying multi-dimensional complex projective space with constant velocity determined by the quantum state vector. Following a measurement the point changes direction and moves with new constant velocity along one of several possible new orthogonal paths with probabilities determined by the Born Interpretation of the state vector. From this previously hidden substructure a new picture of quantum dynamics and quantum measurements emerges without violating existing no-gotheorems regarding hidden variables. A natural generalisation to a Riemannian substructure is proposed, determined by the entropy of the background environment. This leads to a suggestedinteraction between the substructure of quantum dynamics and the background gravitational field.