On the Constant e as a Bridge between Quantum and Relativistic Physics

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 eφ J of the complex structure J generates unitary rotations, of which the quantum phase eiφ is a special case; · The double commutator term -γ[J,[J,ρ]] in the GKLS generator constructed from J produces exponential decay e-2γ t, describing dissipation in open systems; · From the radial expansion amplitude F(t,θ) and the rank-flow attractor θ = π/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.