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Chaos Solitons & Fractals

‘Scale covariant’ representation of quantum mechanics. Energy in scale-relativity theory

Jean-Claude Pissondes

Published Jul 1, 1998• 16 citations
Publisher
DOI
10.1016/s0960-0779(97)00139-2

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By positing energy as intrinsic and independent—with momentum, time, and spatial properties emerging as functions of boost energy (either directly or indirectly)—E-Theory achieves a fully covariant, background-independent formulation for quantum mechanics. This paper demonstrates that by anchoring special relativity in the invariant energy–momentum relationship and reinterpreting energy as an independent variable rather than a dependent property, we transform the traditional four-vector—typically expressed via complex differential relationships—into three simple algebraic functions for momentum, time, and space. We can then transform the standard Time-Dependent Schrödinger Equation (TDSE) into a covariant, background-independent version simply by scaling the time derivative by a universal factor derived from the boost energy.This shift from the time domain (time-dependent) to the energy domain (energy-dependent) has profound consequences: a covariant TDSE emerges; both relativistic and non-relativistic dispersion are recovered from the standard non-relativistic Hamiltonian; inverse relativistic effects appear at ultra-low energies; and a single wave equation governs quantum evolution across all energy scales and particle spins. Most importantly, we free quantum mechanics from its reliance on a fixed temporal background and place it on equal footing with general relativity, thus laying the foundation for their ultimate unification.
We systematically analyze the conceptual structure of modern physics, i.e., the theory of relativity and quantum mechanics and quantum field theory, from the perspective of Leibniz’s three basic principles of identity, sufficient reason and continuity. We find that Leibniz’s way of thinking can still provide insight into fundamental problems of physics, such as the nature of physical constants, the relation between physical invariances and covariant coordinate representations, the role of the Planck scale and quantum mechanical indeterminism and the measurement problem. Leibniz himself could not possess all those insights, simply because physics was not yet sufficiently developed then, but he had already forged some conceptual tools with which some fundamental problems of modern physics can be addressed.