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Abstract
In 1931, Schrödinger tried to find a classical probabilistic derivation of an equation whose characteristics are as close as possible to those of his wave equation. The resulting (now called) Schrödinger-bridge dy- namics relies upon the imposition of the same Born-rule values as those resulting from the wave equation, as boundary conditions at the two bounds of a time interval. While the Schrödinger bridge is adequate when considering experimental data compatible with the deterministic quantum-mechanical evolution, it is also adequate when considering experimental data compatible with a non-deterministic system-state reduction resulting from a quantum measurement. Nonlinear gauge transformations are introduced that are just rescal- ings of the action, and under which the couple of the Schrödinger- equation dynamics and the Schr¨odinger-bridge dynamics is invariant. The effect of those transformations on the evolution parameters of both dynamics is analog to the Lorentz transformation in special rel- ativity, combined with a dilation. This result stems for the conclusion that both parameters, the time governing the Schrödinger-equation dy- namics and the time-like parameter governing the Schrödinger-bridge dynamics, are independent or, in other words, that a fifth dimension should be added to the four-dimensional space-time. The presently considered scope of a quantum non-relativistic spin- less particle subject to an external scalar potential or to an external electromagnetic field is sufficient to raise the important questions dis- cussed in the present work and can be enlarged in the future.