Wavefunctions localization, and the Wigner's Friend Paradox in a Framework of Discrete-Space Hypothesis

Abstract: We present a resolution of the Wigner's Friend paradox within a framework of quantum mechanics (QM) on the hybrid space RxQ_{p}, where Q_{p} denotes the field of p-adic numbers, regarded as a model of discrete microscopic space at the Planck-Bronstein scale. In this framework, wavefunction collapse is not an independent postulate but a dynamical consequence of the Schrödinger equation with non-local Hamiltonians: wavefunctions localize onto compact supports during measurement interactions, producing definite pointer readings without the intervention of observers or the exchange of information between subsystems. We model both Wigner and his Friend as classical apparatuses and show that each produces a definite reading through independent applications of the collapse mechanism, thereby eliminating the conflict between their descriptions of reality. The framework is consistent with the principal no-go theorems in finite- and infinite-dimensional Hilbert spaces associated with extended Wigner's Friend scenarios -- including those of Frauchiger-Renner, Brukner, Bong et al., and Guérin et al. -- since it requires no agents capable of recording or reasoning about outcomes, thereby vacating the observer-dependent assumptions that drive those theorems. We illustrate the collapse mechanism explicitly through a toy model of a particle in a box, comparing the standard description with the new one. The non-locality intrinsic to QM on L2(RxQ_{p}) permits realism at the cost of locality, and the Absoluteness of Observed Events holds in our framework without requiring observer independence.

• The paper presents a deterministic mechanism for wavefunction collapse emerging from non-local Hamiltonians in a discrete-space quantum model.
• It reformulates the Wigner’s Friend paradox by treating measurement devices as classical apparatuses that yield observer-independent outcomes.
• The approach reconciles quantum measurement with fundamental spatial discreteness at the Planck scale, offering new insights for quantum information protocols.

Quantum Measurement, Wavefunction Localization, and the Wigner's Friend Paradox in Discrete-Space Quantum Mechanics

Introduction

This work proposes a resolution of the Wigner's Friend paradox within the framework of quantum mechanics formulated on the hybrid space $\mathbb{R} \times \mathbb{Q}_p$, with $\mathbb{Q}_p$ the field of $p$-adic numbers. The model is motivated by the space discreteness hypothesis at the Planck-Bronstein scale, where physical space is conjectured to be fundamentally discrete and radically distinct from its macroscopic, manifold-like structure. Within this formalism, wavefunction collapse emerges as a deterministic consequence of Schrödinger dynamics governed by non-local Hamiltonians, rather than as an independent measurement axiom. Wigner and his Friend are treated as classical apparatuses, each producing definite outcomes via independent applications of the collapse mechanism. Notably, the framework vacates the observer-dependent postulates that underlie recent no-go theorems for Wigner's Friend scenarios and demonstrates that absoluteness of measurement events can be preserved under the structural non-locality induced by the $p$-adic model.

Discrete-Space Hypothesis and Mathematical Foundation

The discrete-space hypothesis, extending from the Bronstein inequality, posits a fundamental minimal length $l_B$ beyond which the smooth manifold structure of space breaks down. This framework models physical space as $\mathbb{R} \times \mathbb{Q}_p$, with $\mathbb{Q}_p$ functioning as a totally disconnected, ultrametric space at the microscale. The choice is further justified by Volovich's conjecture of $p$-adic spacetime at the Planck scale and by connections between $p$-adic quantum mechanics (QM) and quantum computing (e.g., random walks on graphs and the Jackiw-Rebbi model).

Quantum states reside in the Hilbert space $L^2(\mathbb{R} \times \mathbb{Q}_p)$, and system evolution is determined by Schrödinger equations with non-local Hamiltonians. The model recovers standard QM on $\mathbb{Q}_p$0 by restricting to the macroscopic coordinate and p-adic QM on $\mathbb{Q}_p$1 in the other limit. A critical innovation allows any discrete Schrödinger equation on $\mathbb{Q}_p$2 to be equivalently formulated in $\mathbb{Q}_p$3, the unit ball in $\mathbb{Q}_p$4, preserving the spectrum and dynamical structure.

Dynamical Collapse Mechanism

A central claim is that wavefunction collapse is a derived, deterministic process resulting from the non-local dynamics in the $\mathbb{Q}_p$5-adic sector, and not an extra-postulated measurement rule. Explicitly, when a quantum system $\mathbb{Q}_p$6 interacting with an apparatus $\mathbb{Q}_p$7 (both modeled within $\mathbb{Q}_p$8) undergoes a measurement, its joint wavefunction localizes compactly in its configuration space. This localization is mediated by continuous, surjective maps $\mathbb{Q}_p$9 (e.g., the Monna map), which "project" $p$0-adic spatial structures into the macroscopic domain.

The model is distinct from GRW-type collapse models in several aspects:

• Collapse is deterministic, in contrast to the stochastic mechanism of GRW.
• It arises naturally from the geometry of space, with no need for new physical constants or nonlinear/semi-stochastic modifications to the Schrödinger equation.
• The "physical" observable is exclusively the probability measure defined by the wavefunction; the wavefunction itself need not be regarded ontologically.

Application to the Wigner's Friend Paradox

The Wigner's Friend thought experiment highlights the incompatibility in standard QM between the description by an internal observer (who perceives definite outcomes) and an external observer (who describes the system as a superposition). The present formulation reframes both Friend and Wigner as classical apparatuses, each evolving according to the same collapse dynamics without invoking observer-dependent state assignment.

• In Scenario I ($p$1), Friend's pointer yields a definite outcome via wavefunction localization at $p$2; subsequently, Wigner’s measurement localizes the combined state at $p$3, with no causal or informational connection between the two events.
• In Scenario II ($p$4), the same structure holds, with the order of measurements reversed: the collapse is always attributed to non-local spatial structure, not observer action.

In both cases, definite outcomes are produced, and at no point is it required to treat either Friend or Wigner as an agent capable of information processing, memory, or logical inference. This dynamic collapses the standard paradox, as mutually exclusive descriptions are never simultaneously maintained.

Consistency with Quantum No-Go Theorems

The framework is consistent with key no-go theorems (Frauchiger–Renner, Brukner, Bong et al., Guérin et al.) that elaborate the incompatibility between certain combinations of the universality of quantum theory, observer-independent facts, agent rationality, and persistent observer records. In the $p$5 model:

• Measurement outcomes are absolute and observer-independent due to the collapse mechanism.
• No rational agents are postulated; only classical measurement devices exist.
• The universal applicability of the Schrödinger equation is sustained, though locality is unambiguously surrendered due to the intrinsic non-locality of p-adic operators.

Recent relativistic and infinite-dimensional extensions of the Wigner's Friend paradox, which introduce further complications via Lorentz invariance and agent-dependence, are also sidestepped, as the present framework is manifestly non-relativistic in its $p$6-adic sector and contains no observer-agents.

Implications and Future Directions

The principal theoretical implication is the feasibility of a fully deterministic, observer-independent account of quantum measurement, under the umbrella of a discrete, fundamentally non-local spatial model at the Planck scale. By situating collapse as a structural feature of non-local Hamiltonians rather than a separate axiom, the approach preserves both unitarity and definiteness of outcomes without reliance on stochastic modifications or agent-centric interpretations.

Practical consequences are, at present, indirect: reconciliation of certain cryptographic protocols, quantum information paradoxes, and device-independent models may benefit from a formalism where measurement is strictly a matter of spatial structure and dynamical law.

Open questions persist regarding:

• The explicit selection of the totally disconnected space $p$7 and the macroscopic-projection map $p$8, which lack unique empirical or theoretical justification.
• The extension of the collapse mechanism to more general topologies or alternative forms of microscopic non-locality.
• Validation or falsification via experimental or phenomenological consequences at the interface of Planck-scale physics and quantum information.

A rigorous classification of quantum scenarios that can be consistently modeled in $p$9, and the connection to higher-dimensional operator algebras (or p-adic field theories), remains a promising direction.

Conclusion

Quantum mechanics on $p$0, as developed in this framework, delivers a coherent, deterministic account of wavefunction localization that addresses classical paradoxes of measurement, notably the Wigner's Friend scenario. By operationalizing collapse as a property of space-time structure, the formulation sidesteps the binary of observer-relativity versus objective fact, preserving the absoluteness of observed events at the cost of spatial locality. This approach situates the measurement problem within the terrain of quantum field theory on non-trivial topologies, suggesting novel routes for reconciling foundational issues and setting a new agenda for theoretical and experimental investigation of quantum non-locality and the structure of physical space.