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The Dynamics of Discrete Fact: A Phase-Transition Theory of Wavefunction Collapse (V1)

Ahmed Hamid Mahmoud

Published Jan 27, 2026• 0 citations
Publisher

Abstract

(Note: Please refer to Version 3 (v3) of The Dynamics of Discrete Fact: A Phase-Transition Theory of Wavefunction Collapse for the updated and expanded presentation of the framework). _______________________ This is Version 2 of The Dynamics of Discrete Fact, presenting an expanded and clarified construction of wavefunction collapse as a genuine physical phenomenon: a nonequilibrium phase transition in a macroscopic measurement apparatus. As before, collapse is modeled as a critical instability in an emergent order parameter governed by time-dependent Ginzburg–Landau dynamics. Discrete outcomes arise through spontaneous symmetry breaking in the apparatus–environment system, while Born weights are preserved via probability-conserving basin flow in the associated Fokker–Planck description—without modifying the Schrödinger equation or introducing new constants. Relative to v1, v2 (i) makes the single-run (outcome-conditional) collapse dynamics explicit rather than implicit, (ii) strengthens the Born-rule preservation argument by tying basin structure more directly to apparatus symmetry, and (iii) tightens the experimental discussion by specifying the regimes in which critical signatures are and are not expected. The framework retains concrete signatures—critical slowing near threshold, hysteresis under parameter cycling, metastability, and outcome-correlated transient detector spikes—and positions collapse alongside emergent macroscopic instabilities such as superconductivity and ferromagnetism.

DOI
10.5281/zenodo.18385297

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This is the first single volume about the collapse theories of quantum mechanics, which is becoming a very active field of research in both physics and philosophy. In standard quantum mechanics, it is postulated that when the wave function of a quantum system is measured, it no longer follows the Schrödinger equation, but instantaneously and randomly collapses to one of the wave functions that correspond to definite measurement results. However, why and how a definite measurement result appears is unknown. A promising solution to this problem are collapse theories in which the collapse of the wave function is spontaneous and dynamical. Chapters written by distinguished physicists and philosophers of physics discuss the origin and implications of wave-function collapse, the controversies around collapse models and their ontologies, and new arguments for the reality of wave function collapse. This is an invaluable resource for students and researchers interested in the philosophy of physics and foundations of quantum mechanics.
Abstract The quantum measurement problem—the absence of any dynamical mechanism connecting continuous wavefunction evolution to discrete empirical outcomes—has persisted since the foundations of quantum theory. Decoherence explains interference suppression but cannot explain outcome selection: the diagonal density matrix remains an improper mixture until one outcome is actualized. We propose that collapse is a physical phase transition in the coupled system–apparatus field. The order parameter ψ, constructed as a coarse-grained collective coordinate of apparatus degrees of freedom, evolves under Ginzburg–Landau dynamicswith a symmetry-breaking potential. When the coherence pressure γ exceeds a critical threshold γc, the symmetric phase(superposition) becomes unstable and the field crystallizes into one of the discrete stable minima (eigenstates). We derive the critical coupling from microscopic system–apparatus Hamiltonians, obtaining γc ∼ (√Ng)−1 where N is the number of apparatus degrees of freedom and g is the coupling constant. This scaling explains why macroscopic apparatus collapse wavefunctions while microscopic interactions preserve coherence. The Born rule Pn = |cn|2 is preserved through this dynamical selection mechanism: attractor basin geometry under probability-conserving flow converts probabilistic weights into definite outcomes without altering the underlying probabilities. The theory yields four falsifiable predictions absent from standard quantum mechanics: critical slowing near γc, hysteresis in the collapse–recoherence cycle, metastable supercooled superpositions, and transient Jacobian spikes at the moment of collapse. We provide quantitative estimates for superconducting qubit readout, predicting collapsetimes of order 100 ns with critical slowing observable within current experimental precision. This framework provides a concrete existence proof, within a controlled effective-theory regime, that collapse dynamics can be constructed from standard quantum mechanics plus statistical mechanics, requiring no modification to the Schr¨odinger equation.
Phase Differential Theory (PDT) proposes that quantum mechanics emerges as an effective description of a deeper dynamical phase manifold. In this framework, four scalar phase fields define a locked background whose long‑wavelength envelope dynamics reproduce the Schrödinger equation. The Born rule arises as the natural conserved probability density compatible with locality and continuity of phase flow, and measurement is interpreted as phase synchronisation between subsystems rather than wavefunction collapse. This paper develops a controlled derivation of the Schrödinger equation as an envelope approximation, identifies the conserved probability current associated with phase flow, and shows that |ψ|² is the natural density selected by locality, positivity, and conservation. Measurement is modelled as a dynamical locking of relative phase between system and apparatus, yielding discrete outcomes without introducing non‑unitary dynamics at the fundamental level. All results are obtained within a low‑energy, long‑wavelength effective regime. The microscopic equations of motion for the phase manifold are intentionally deferred, as the effective envelope theory presented here is self‑contained and does not require explicit microscopic specification.
Description This work develops a foundational reformulation of quantum measurement by treating physical time as an emergent, dynamically produced quantity rather than as an external background parameter. Within the Discrete Expansion and Emergent Time (DEET) framework, physical duration is generated by the spacetime gradient of a scalar phase field, \Theta(x), with the local time flow defined by u_\mu = \nabla_\mu \Theta. Once time is produced rather than assumed, the weighting of mutually exclusive quantum outcomes requires a new structural interpretation. The paper introduces a time-production (allocation) density \rho(\psi), defined as a bookkeeping quantity for realized physical time rather than a probability postulate. By imposing only minimal and physically unavoidable requirements—global phase invariance, locality, separability under independent composition, and additivity under coarse-graining over exclusive alternatives—it is shown that this density is uniquely fixed (up to normalization) as \rho(\psi) \propto |\psi|^2. Measurement is modeled dynamically as a phase–gradient locking process, in which interaction with a macroscopic record sector enforces a shared realized time flow. Total time production is conserved during measurement, but redistributed across exclusive record sectors. Defining outcome weights as ratios of allocated time production yields the Born rule uniquely, without introducing stochastic collapse postulates, subjective probability, or additional interpretive assumptions. In this framework, state reduction corresponds to dynamical stabilization of a single record-compatible time flow rather than non-unitary projection, and apparent randomness arises from coarse-graining over inaccessible microscopic phase configurations. The work clarifies the structural origin of the Born rule, unifies time-dependent and timeless regimes of quantum theory, and identifies conservative falsifiability targets, including finite-time locking dynamics and correlated phase/timing residuals during measurement.