Spontaneous Decoherence from Logarithmic Spectral Phase Deformations

Abstract

We examine a mechanism of spontaneous decoherence in which the generator of quantum dynamics is deformed to a logarithmically modified self-adjoint operator \begin{equation*} F_β(H) = H + βH \log \frac{H}{E_*} \end{equation*} for a positive self-adjoint Hamiltonian $H$ and a fixed reference scale $E_* > 0$. Dynamical phases acquire energy-dependent factors $\exp[-itβE \log(E/E_*)]$, whose rapid variation across the spectrum suppresses interference between distinct energies through a non-stationary-phase mechanism. Stationary-phase analysis shows that oscillatory contributions to amplitudes decay at least as $\mathcal{O}(1/|β|)$ when $|β|$ is large. Since $F_β(H)$ is self-adjoint for every real $β$, the evolution operator $U_β(t) = \exp[-itF_β(H)]$ is unitary. The kinematical structure of quantum mechanics -- Hilbert-space inner products, projection operators, the Born rule -- remains unchanged. Decoherence arises as suppression of interference terms in coarse-grained observables and decoherence functionals, not as norm loss or stochastic collapse. Physical motivation for logarithmic spectral deformations comes from clock imperfections, renormalization-group and effective-action corrections introducing $\log E$ terms, and semiclassical gravity analyses with complex actions generating spectral factors involving $\log(E/E_{\text{P}})$. The mechanism is illustrated with two-level systems, quartic oscillators, FRW minisuperspace models, and Schwarzschild-interior-type Hamiltonians. Current superconducting-qubit coherence times constrain $|β| \lesssim 10^{-5}$; trapped ions, NV centers, and cold atoms could strengthen this to $|β| \lesssim 10^{-8}$.