Collapse as a Dissipative Phase Transition: Liouvillian Gap Opening, Pointer Stability, and $\tau$-Flow Dynamics

Abstract

We present a rigorous operator-theoretic formulation of quantum collapse as a dissipative phase transition in the spectrum of a completely positive dynamical semigroup.

Within a finite-dimensional Hilbert space, we define collapse in terms of exponential contraction toward a pointer algebra induced by a Liouvillian spectral gap in the coherence sector. We prove that coherence decays at a rate determined by the minimal nonzero real part of the Liouvillian spectrum and establish a sharp stability inequality quantifying competition between dissipative gap opening and Hamiltonian coherence regeneration.

We further show that in macroscopic amplification regimes, the gap scales superlinearly in redundancy, producing thermodynamic sharpening analogous to conventional phase transitions. Finally, we demonstrate that this structure emerges naturally from a $ \tau $-recursive dynamical generator of the form $ \mathcal{A} = -iH - \Gamma $, where $ \Gamma $ is constructed from commutator tension $ [H,M]^\dagger[H,M] $.

Conclusion: Collapse thus appears not as a primitive stochastic postulate, nor as purely epistemic update, but as a dynamical transition in the spectral structure of the evolution generator.

Key Highlights & Physical Implications

• Novel Resolution to the Measurement Problem: Replaces axiomatic stochastic jumps with a deterministic, spectral phase transition.
• Commutator Tension Mechanism: Identifies $ [H,M]^\dagger[H,M] $ as the dynamical origin of quantum friction and decoherence.
• Macroscopic Scaling Laws: Proves superlinear gap growth in redundant apparatuses, explaining why large systems behave classically.
• Pointer Algebra Stability: Establishes a strict inequality ($ \Delta > \kappa $) that dictates when a system crosses from the coherent phase to the classical pointer phase.

Keywords & Index Terms

Quantum Measurement Problem, Decoherence Theory, Dissipative Phase Transitions, Open Quantum Systems, Liouvillian Spectral Gap, Pointer States, Lindblad Master Equation, Onto-Physical Entanglement, CPT-Coherence Theory, Commutator Tension, Quantum Trajectories, Thermodynamic Limit.