A Time-Symmetric Variational Reformulation of Nonrelativistic Quantum Mechanics

TL;DR

This paper introduces a unified, time-symmetric variational framework for nonrelativistic quantum mechanics, deriving Schrödinger dynamics and Born-rule statistics without separate postulates or collapse, emphasizing boundary conditions and hydrodynamic variables.

Contribution

It presents a novel variational reformulation that derives quantum dynamics from boundary conditions, eliminating the dualism of evolution and collapse, and offers a new perspective on quantum uncertainty.

Findings

Schrödinger equation emerges from Fisher-information regularization.

Quantum statistics arise without collapse through boundary-based variational principles.

Time-symmetric formulation recovers key quantum features without additional postulates.

Abstract

Standard quantum mechanics relies on two distinct dynamical principles: unitary evolution and collapse. A mathematically self-contained variational framework is presented that replaces this dualism with a single principle, in which nonrelativistic Schr\"odinger dynamics are not postulated but emerge as an admissible optimality condition of a primal-dual boundary-value problem. By expressing the state in terms of hydrodynamic variables $(\rho,\mathbf{j})$ subject to a continuity constraint, it is shown that Fisher-information regularization yields the linear Schr\"odinger equation within the admissible single-valued variational class. Rather than evolving an initial state forward in time, the dynamics arise from minimizing a global action that connects the initial and final boundary constraints, with the selected solution corresponding to a specific hydrodynamic flow within an ensemble of admissible histories. A von Neumann pointer model illustrates how Born-rule statistics for recorded outcomes arise without introducing a separate collapse law. Within this formulation, quantum uncertainty is interpreted as effective randomness over boundary-compatible histories rather than as a fundamental stochastic postulate. The resulting framework provides a nonrelativistic proof of concept for how a single time-symmetric variational reformulation can recover key features of quantum theory.