[Paper Review] Quantum mechanics as the dynamical geometry of trajectories

Philipp Roser|arXiv (Cornell University)|Jul 31, 2015

Quantum Mechanics and Applications30 references3 citations

TL;DR

This paper proposes a trajectory-only formulation of non-relativistic quantum mechanics by treating configuration space as a dynamical Weyl geometry, where quantum trajectories emerge from an action principle combining classical matter dynamics and curvature. It resolves Wallstrom's objection—concerning quantized angular momentum in wavefunction-free theories—by generalizing the Lagrangian to allow non-contractible loops, recovering de Broglie-Bohm trajectories in equilibrium without a physical wavefunction.

ABSTRACT

We illustrate how non-relativistic quantum mechanics may be recovered from a dynamical Weyl geometry on configuration space and an `ensemble' of trajectories (or `worlds'). The theory, which is free of a physical wavefunction, is presented starting from a classical `many-systems' action to which a curvature term is added. In this manner the equations of equilibrium de~Broglie-Bohm theory are recovered. However, naïvely the set of solution precludes solutions with non-zero angular momentum (a version of a problem raised by Wallstrom). This is remedied by a slight extension of the action, leaving the equations of motion unchanged.

Motivation & Objective

• To develop a quantum theory without a physical wavefunction, relying solely on trajectories in configuration space.
• To resolve Wallstrom's objection—why angular momentum is quantized in trajectory-only models—by introducing a geometric structure with non-contractible loops.
• To unify dynamical Weyl geometry with the de Broglie-Bohm ensemble approach, recovering equilibrium quantum mechanics phenomenologically.
• To establish a physical ontology based on a manifold of trajectories rather than abstract configuration space, improving philosophical coherence.

Proposed method

• Formulate a classical 'many-systems' action over an ensemble of trajectories in configuration space.
• Introduce a curvature term in the action analogous to the Einstein-Hilbert term, but in configuration space, leading to a dynamical Weyl geometry.
• Derive equations of motion from the total action, showing they match de Broglie-Bohm trajectories under equilibrium conditions (ρ = |ψ|²).
• Generalize the total time derivative in the Lagrangian to allow for phase-like functions W defined on a manifold of trajectories.
• Introduce a manifold of trajectories instead of a single configuration space to allow for non-contractible loops, enabling quantized loop integrals.
• Use the Weyl connection and conformal coupling to ensure the geometric structure supports quantized observables like angular momentum.

Experimental results

Research questions

• RQ1 How can quantum mechanics be derived without a physical wavefunction, relying only on trajectories in configuration space?
• RQ2 Why do trajectory-only models fail to reproduce quantized angular momentum, and how can this be resolved?
• RQ3 Can a dynamical geometry of configuration space—specifically Weyl geometry—recover the de Broglie-Bohm trajectories in equilibrium?
• RQ4 What is the physical role of the manifold of trajectories, and how does it differ from configuration space in terms of topology and observables?
• RQ5 Can the geometric approach to quantum mechanics be extended to quantum field theory, particularly in the presence of nodes in the field functional?

Key findings

• The theory recovers the de Broglie-Bohm trajectories in equilibrium, matching all predictions of standard quantum mechanics.
• The inclusion of a curvature term in the action leads to a dynamical Weyl geometry on configuration space, with singularities at wavefunction nodes.
• The introduction of a manifold of trajectories allows for non-contractible loops, enabling non-zero, quantized values of ∮∂iW·dli = 2πℏ·m for m∈ℤ.
• Wallstrom's objection is resolved by generalizing the Lagrangian to include a Weyl-invariant term involving a phase function W, which becomes multi-valued on non-contractible loops.
• The theory is formally equivalent to Santamato’s action but derived from a different starting point, without relying on expectation value minimization.
• The geometric approach suggests a path toward a unified foundation for quantum field theory and potentially quantum gravity, especially via superspace trajectories.