On the Constant e as a Bridge between Quantum and Relativistic Physics
Description
Within the two-projection operator framework, starting from two non-commuting orthogonal projections P and Q, we sketch how the natural constant e emerges as a common mathematical foundation for both quantum mechanics (unitary phase and dissipative decay) and relativity (the speed of light and Lorentz transformations). We show that:
· The exponential map e^{\phi J} of the complex structure J generates unitary rotations, of which the quantum phase e^{i\phi} is a special case;
· The double commutator term -\gamma[J,[J,\rho]] in the GKLS generator constructed from J produces exponential decay e^{-2\gamma t}, describing dissipation in open systems;
· From the radial expansion amplitude F(t,\theta) and the rank-flow attractor \theta = \pi/4, the propagation speed tends to a constant c, leading to the Minkowski metric and Lorentz transformations, with the speed of light c emerging as a fixed point of the attractor;
· The constant e serves as the base of all these exponential functions, providing a unified algebraic structure connecting quantum dynamics and relativistic kinematics. All derivations rely solely on intrinsic properties of the projection algebra, without external assumptions.
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Additional details
Additional titles
- Alternative title (English)
- A Sketch of Derivation within the Two-Projection Framework
Dates
- Issued
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2026-04-05This paper began with a simple, almost naive question: why does the natural constant e keep appearing in the phase factor of quantum mechanics, the exponential decay of open systems, and the Lorentz factor of relativity? Could it be more than a coincidence? Working within the two‑projection framework, I started to see a faint but consistent thread. From two non‑commuting projections, the complex structure J emerges naturally. Its exponential map gives the unitary phase; the same J constructs a GKLS generator that produces dissipation; and when we expand the zero‑dimensional origin into spacetime, a rank‑flow attractor fixes the speed of light to a constant c. This is not a finished theory of everything. It is a sketch of derivation — an attempt to build a small bridge between quantum and relativistic worlds using the simplest possible algebraic ingredients. The name of the bridge is e. I am sharing this sketch not to claim a final unification, but in the hope that someone might walk further along this bridge, perhaps noticing where it is shaky, perhaps seeing a wider landscape. If you do, please let me know. — Guanhua Yu, April 5, 2026