A Stability-Based Interpretation of Quantum State Evolution

A Recursive Consistency Framework for Quantum Mechanics

**By: **Rahmon Randall

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Abstract

Quantum mechanics provides extraordinarily accurate predictions but remains conceptually incomplete in its interpretation of state selection, wavefunction collapse, and decoherence.

This article introduces a stability-based interpretation in which quantum state evolution is governed by recursive internal consistency under environmental constraints.

Rather than treating collapse as fundamentally observer-driven or intrinsically random, this framework models quantum evolution as a process in which only internally self-consistent state configurations persist under recursive interaction with external degrees of freedom.

Measurement, decoherence, and entanglement are reinterpreted as dynamical processes of constraint-driven state resolution.

This interpretation preserves all standard quantum mechanical formalisms while offering a unifying perspective on state selection, coherence decay, and correlated systems.

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1. The Recursive Consistency Principle

We define a recursive state-selection operator:

S(ψ) = projection onto internally consistent components of ψ

Quantum state evolution is proposed to follow:

ψₜ₊₁ = S( ψₜ ∘ C(ψₜ) )

Where:

• ψₜ = quantum state at time t
• • C(ψ) = environmental and interaction-induced constraints
• • ∘ = nonlinear interaction composition
• • S(·) = consistency projection operator

This defines a recursive process in which incompatible or dynamically unstable components of the state are suppressed.

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2. Reinterpreting the Wavefunction

In standard quantum mechanics: |ψ(x,t)|²

is interpreted as a probability density.

In this framework: |ψ|²

is interpreted as a relative dynamical stability density, representing the persistence strength of a given state component under recursive interactions.

Probability emerges from differential stability under repeated environmental coupling.

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3. Superposition as Multi-Stable State Structure

A general superposition:

ψ = Σᵢ cᵢ |i⟩

is interpreted as a set of multiple dynamically permissible configurations coexisting prior to constraint resolution.

Each basis state |i⟩ represents a dynamically viable configuration until interaction-induced constraints suppress incompatible components.

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4. Wavefunction Collapse as Stability Resolution

Measurement is modeled as the introduction of strong external constraints:

ψ → S(ψ)

Collapse corresponds to projection onto the subset of components that remain dynamically self-consistent under measurement coupling.

This preserves standard projection postulates while offering a dynamical rationale for state selection.

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5. Decoherence as Stability Degradation

Decoherence arises as:

ψ → ψₘᵢₓₑd

due to increasing coupling with environmental degrees of freedom.

In this interpretation, decoherence reflects the progressive elimination of internally coherent superposition components as external constraints accumulate.

Decoherence time scales correspond to rates of consistency loss under environmental interaction.

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6. Entanglement as Joint Stability Constraint

For entangled states:

|ψ⟩ = Σᵢⱼ cᵢⱼ |i⟩ₐ |j⟩ᵦ

The composite system is treated as a single dynamical stability structure.

Correlations arise from joint consistency requirements on the composite state, rather than signal exchange.

Measurement on subsystem A alters the constraint structure of the composite system, resulting in correlated resolution in subsystem B.

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7. Vacuum Fluctuations and State Viability

Vacuum fluctuations are interpreted as transient state components that fail to maintain dynamical consistency over time.

Observable particles correspond to configurations that achieve sustained dynamical stability.

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8. Relationship to Standard Quantum Formalism

This framework:

• Preserves Schrödinger dynamics
• • Preserves Born rule predictions
• • Preserves standard operator formalism
• • Adds a dynamical interpretation for projection and decoherence

It does not modify quantum mechanics — it supplements interpretation.

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9. Unified Stability Evolution Equation

The proposed interpretive evolution rule:

ψₜ₊₁ = S( ψₜ ∘ C(ψₜ) )

Where S projects onto dynamically self-consistent state components under interaction constraints.

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10. Implications for Quantum Engineering

This framework suggests new perspectives for:

• Coherence optimization via constraint control
• • Noise-resistant state design
• • Stability-based interpretation of decoherence times
• • Constraint-aware quantum system simulation
• • Dynamical modeling of collapse without observer dependence

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11. Summary (Plain Language)

• Quantum states evolve under recursive interaction
• • Measurement introduces constraints that select consistent components
• • Collapse is a dynamical projection process
• • Decoherence reflects consistency loss
• • Entanglement reflects joint constraint structure
• • Probability reflects relative dynamical persistence

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Formal Name:

Randall’s Recursive Consistency Interpretation (RRCI)