An unexpected theoretical structure that could explain quantum-mechanics postulates like the Born rule and the wave-function reduction

TL;DR

The work proposes that the entire structure of quantum mechanics can be derived from a single postulate: the invariance of the Heisenberg uncertainty product under a group of nonlinear gauge transformations (NLGT). This leads, starting from classical Hamilton–Jacobi and continuity equations, to the quantum average-energy functional, the Schrödinger equation, and the Born rule; it further introduces an analytic continuation that reveals a second, beyond-quantum energy-like functional and, via Schrödinger-bridge dynamics, provides an explanatory mechanism for wave-function collapse in a space-like dimension. The unitary Schrödinger evolution and the Schrödinger-bridge process become intertwined under NLGT, linked by a hyperbolic rotation between time t and the auxiliary parameter τ, with collapse appearing as a boundary-value, non-causal transition in τ. The framework suggests a unified, geometric view of quantum dynamics and measurement and points to extensions to potentials, many-body systems, and quantum field theory, where non-quantum phenomena may reside in the additional space-like dimension.

Abstract

A unique postulate is shown to underly the whole quantum mechanics theory: the invariance of the Heisenberg uncertainty inequality under a group of special nonlinear gauge transformations (NLGT). With this postulate, the quantum mechanics of a free particle is derived from classical mechanics, including the statements of the postulates of quantum mechanics, except for the wave-function-collapse postulate. An explanatory mechanism for the latter postulate is derived by performing an analytical continuation of the NLGTs. This extension results in a Schrödinger-bridge process, intertwined under the NLGT with the standard unitary quantum evolution, and revealing non-quantum (or beyond-quantum) phenomena. Mechanisms of that latter kind, like the ones associated to the quantum measurement process, occur in a new space-like dimension and hence are non causal in nature, in opposition to a time evolution. The present exercice focusses on the free particle in order to highlight the features of the performed derivation in the simplest possible way. Work is in progress to extend the performed derivation beyond that simple case.